How to use _diff method in PyHamcrest

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test_exponential.py

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1import sys2from sympy.external import import_module3matchpy = import_module("matchpy")4if not matchpy:5 #bin/​test will not execute any tests now6 disabled = True7if sys.version_info[:2] < (3, 6):8 disabled = True9from sympy.integrals.rubi.rubi import rubi_integrate10from sympy.functions import log, sqrt, exp, cos, sin, tan, sec, csc, cot, sinh, sech, atan, asin, acos, atanh, asinh, acosh11from sympy import acsch as arccsch, acsc as arccsc12from sympy.integrals.rubi.utility_function import (EllipticE, EllipticF, Int, ArcCsch, ArcCsc, Gamma,13 hypergeom, rubi_test, AppellF1, EllipticPi, Log, Sqrt, ArcTan, ArcTanh, ArcSin, ArcSinh, ArcCosh, ArcTanh, ArcCos, Hypergeometric2F1)14from sympy import pi15from sympy import S, hyper, I, simplify, exp_polar, symbols, Ei,erf, erfi,gamma,uppergamma, polylog, Integral, exp16from sympy.testing.pytest import SKIP17a, b, c, d, e, f, m, n, x, u , k, p, r, s, t= symbols('a b c d e f m n x u k p r s t')18A, B, C, D, a, b, c, d, e, f, g, h, i, y, z, m, n, p, q, u, v, w, E, F, G, H = symbols('A B C D a b c d e f g h i y z m n p q u v w E F G H')19def test_1():20 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d + e*x)**m, x), x, F**(c*(a - b*d/​e))*(-b*c*(d + e*x)*log(F)/​e)**(-m)*(d + e*x)**m*Gamma(m + S(1), -b*c*(d + e*x)*log(F)/​e)/​(b*c*log(F)), expand=True, _diff=True, _numerical=True)21 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d + e*x)**S(4), x), x, F**(c*(a + b*x))*(d + e*x)**S(4)/​(b*c*log(F)) - S(4)*F**(c*(a + b*x))*e*(d + e*x)**S(3)/​(b**S(2)*c**S(2)*log(F)**S(2)) + S(12)*F**(c*(a + b*x))*e**S(2)*(d + e*x)**S(2)/​(b**S(3)*c**S(3)*log(F)**S(3)) - S(24)*F**(c*(a + b*x))*e**S(3)*(d + e*x)/​(b**S(4)*c**S(4)*log(F)**S(4)) + S(24)*F**(c*(a + b*x))*e**S(4)/​(b**S(5)*c**S(5)*log(F)**S(5)), expand=True, _diff=True, _numerical=True)22 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d + e*x)**S(3), x), x, F**(c*(a + b*x))*(d + e*x)**S(3)/​(b*c*log(F)) - S(3)*F**(c*(a + b*x))*e*(d + e*x)**S(2)/​(b**S(2)*c**S(2)*log(F)**S(2)) + S(6)*F**(c*(a + b*x))*e**S(2)*(d + e*x)/​(b**S(3)*c**S(3)*log(F)**S(3)) - S(6)*F**(c*(a + b*x))*e**S(3)/​(b**S(4)*c**S(4)*log(F)**S(4)), expand=True, _diff=True, _numerical=True)23 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d + e*x)**S(2), x), x, F**(c*(a + b*x))*(d + e*x)**S(2)/​(b*c*log(F)) - S(2)*F**(c*(a + b*x))*e*(d + e*x)/​(b**S(2)*c**S(2)*log(F)**S(2)) + S(2)*F**(c*(a + b*x))*e**S(2)/​(b**S(3)*c**S(3)*log(F)**S(3)), expand=True, _diff=True, _numerical=True)24 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d + e*x), x), x, F**(c*(a + b*x))*(d + e*x)/​(b*c*log(F)) - F**(c*(a + b*x))*e/​(b**S(2)*c**S(2)*log(F)**S(2)), expand=True, _diff=True, _numerical=True)25 assert rubi_test(rubi_integrate(F**(c*(a + b*x)), x), x, F**(c*(a + b*x))/​(b*c*log(F)), expand=True, _diff=True, _numerical=True)26 assert rubi_test(rubi_integrate(F**(c*(a + b*x))/​(d + e*x), x), x, F**(c*(a - b*d/​e))*Ei(b*c*(d + e*x)*log(F)/​e)/​e, expand=True, _diff=True, _numerical=True)27 assert rubi_test(rubi_integrate(F**(c*(a + b*x))/​(d + e*x)**S(2), x), x, -F**(c*(a + b*x))/​(e*(d + e*x)) + F**(c*(a - b*d/​e))*b*c*log(F)*Ei(b*c*(d + e*x)*log(F)/​e)/​e**S(2), expand=True, _diff=True, _numerical=True)28 assert rubi_test(rubi_integrate(F**(c*(a + b*x))/​(d + e*x)**S(3), x), x, -F**(c*(a + b*x))*b*c*log(F)/​(S(2)*e**S(2)*(d + e*x)) - F**(c*(a + b*x))/​(S(2)*e*(d + e*x)**S(2)) + F**(c*(a - b*d/​e))*b**S(2)*c**S(2)*log(F)**S(2)*Ei(b*c*(d + e*x)*log(F)/​e)/​(S(2)*e**S(3)), expand=True, _diff=True, _numerical=True)29 assert rubi_test(rubi_integrate(F**(c*(a + b*x))/​(d + e*x)**S(4), x), x, -F**(c*(a + b*x))*b**S(2)*c**S(2)*log(F)**S(2)/​(S(6)*e**S(3)*(d + e*x)) - F**(c*(a + b*x))*b*c*log(F)/​(S(6)*e**S(2)*(d + e*x)**S(2)) - F**(c*(a + b*x))/​(S(3)*e*(d + e*x)**S(3)) + F**(c*(a - b*d/​e))*b**S(3)*c**S(3)*log(F)**S(3)*Ei(b*c*(d + e*x)*log(F)/​e)/​(S(6)*e**S(4)), expand=True, _diff=True, _numerical=True)30 assert rubi_test(rubi_integrate(F**(c*(a + b*x))/​(d + e*x)**S(5), x), x, -F**(c*(a + b*x))*b**S(3)*c**S(3)*log(F)**S(3)/​(S(24)*e**S(4)*(d + e*x)) - F**(c*(a + b*x))*b**S(2)*c**S(2)*log(F)**S(2)/​(S(24)*e**S(3)*(d + e*x)**S(2)) - F**(c*(a + b*x))*b*c*log(F)/​(S(12)*e**S(2)*(d + e*x)**S(3)) - F**(c*(a + b*x))/​(S(4)*e*(d + e*x)**S(4)) + F**(c*(a - b*d/​e))*b**S(4)*c**S(4)*log(F)**S(4)*Ei(b*c*(d + e*x)*log(F)/​e)/​(S(24)*e**S(5)), expand=True, _diff=True, _numerical=True)31 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d**S(4) + S(4)*d**S(3)*e*x + S(6)*d**S(2)*e**S(2)*x**S(2) + S(4)*d*e**S(3)*x**S(3) + e**S(4)*x**S(4)), x), x, F**(c*(a + b*x))*(d + e*x)**S(4)/​(b*c*log(F)) - S(4)*F**(c*(a + b*x))*e*(d + e*x)**S(3)/​(b**S(2)*c**S(2)*log(F)**S(2)) + S(12)*F**(c*(a + b*x))*e**S(2)*(d + e*x)**S(2)/​(b**S(3)*c**S(3)*log(F)**S(3)) - S(24)*F**(c*(a + b*x))*e**S(3)*(d + e*x)/​(b**S(4)*c**S(4)*log(F)**S(4)) + S(24)*F**(c*(a + b*x))*e**S(4)/​(b**S(5)*c**S(5)*log(F)**S(5)), expand=True, _diff=True, _numerical=True)32 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d**S(3) + S(3)*d**S(2)*e*x + S(3)*d*e**S(2)*x**S(2) + e**S(3)*x**S(3)), x), x, F**(c*(a + b*x))*(d + e*x)**S(3)/​(b*c*log(F)) - S(3)*F**(c*(a + b*x))*e*(d + e*x)**S(2)/​(b**S(2)*c**S(2)*log(F)**S(2)) + S(6)*F**(c*(a + b*x))*e**S(2)*(d + e*x)/​(b**S(3)*c**S(3)*log(F)**S(3)) - S(6)*F**(c*(a + b*x))*e**S(3)/​(b**S(4)*c**S(4)*log(F)**S(4)), expand=True, _diff=True, _numerical=True)33 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d**S(2) + S(2)*d*e*x + e**S(2)*x**S(2)), x), x, F**(c*(a + b*x))*(d + e*x)**S(2)/​(b*c*log(F)) - S(2)*F**(c*(a + b*x))*e*(d + e*x)/​(b**S(2)*c**S(2)*log(F)**S(2)) + S(2)*F**(c*(a + b*x))*e**S(2)/​(b**S(3)*c**S(3)*log(F)**S(3)), expand=True, _diff=True, _numerical=True)34 assert rubi_test(rubi_integrate(F**(c*(a + b*x))/​(d**S(2) + S(2)*d*e*x + e**S(2)*x**S(2)), x), x, -F**(c*(a + b*x))/​(e*(d + e*x)) + F**(c*(a - b*d/​e))*b*c*log(F)*Ei(b*c*(d + e*x)*log(F)/​e)/​e**S(2), expand=True, _diff=True, _numerical=True)35 assert rubi_test(rubi_integrate(F**(c*(a + b*x))/​(d**S(3) + S(3)*d**S(2)*e*x + S(3)*d*e**S(2)*x**S(2) + e**S(3)*x**S(3)), x), x, -F**(c*(a + b*x))*b*c*log(F)/​(S(2)*e**S(2)*(d + e*x)) - F**(c*(a + b*x))/​(S(2)*e*(d + e*x)**S(2)) + F**(c*(a - b*d/​e))*b**S(2)*c**S(2)*log(F)**S(2)*Ei(b*c*(d + e*x)*log(F)/​e)/​(S(2)*e**S(3)), expand=True, _diff=True, _numerical=True)36 assert rubi_test(rubi_integrate(F**(c*(a + b*x))/​(d**S(4) + S(4)*d**S(3)*e*x + S(6)*d**S(2)*e**S(2)*x**S(2) + S(4)*d*e**S(3)*x**S(3) + e**S(4)*x**S(4)), x), x, -F**(c*(a + b*x))*b**S(2)*c**S(2)*log(F)**S(2)/​(S(6)*e**S(3)*(d + e*x)) - F**(c*(a + b*x))*b*c*log(F)/​(S(6)*e**S(2)*(d + e*x)**S(2)) - F**(c*(a + b*x))/​(S(3)*e*(d + e*x)**S(3)) + F**(c*(a - b*d/​e))*b**S(3)*c**S(3)*log(F)**S(3)*Ei(b*c*(d + e*x)*log(F)/​e)/​(S(6)*e**S(4)), expand=True, _diff=True, _numerical=True)37 assert rubi_test(rubi_integrate(F**(c*(a + b*x))/​(d**S(5) + S(5)*d**S(4)*e*x + S(10)*d**S(3)*e**S(2)*x**S(2) + S(10)*d**S(2)*e**S(3)*x**S(3) + S(5)*d*e**S(4)*x**S(4) + e**S(5)*x**S(5)), x), x, -F**(c*(a + b*x))*b**S(3)*c**S(3)*log(F)**S(3)/​(S(24)*e**S(4)*(d + e*x)) - F**(c*(a + b*x))*b**S(2)*c**S(2)*log(F)**S(2)/​(S(24)*e**S(3)*(d + e*x)**S(2)) - F**(c*(a + b*x))*b*c*log(F)/​(S(12)*e**S(2)*(d + e*x)**S(3)) - F**(c*(a + b*x))/​(S(4)*e*(d + e*x)**S(4)) + F**(c*(a - b*d/​e))*b**S(4)*c**S(4)*log(F)**S(4)*Ei(b*c*(d + e*x)*log(F)/​e)/​(S(24)*e**S(5)), expand=True, _diff=True, _numerical=True)38 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*((d + e*x)**n)**m, x), x, F**(c*(a - b*d/​e))*(-b*c*(d + e*x)*log(F)/​e)**(-m*n)*((d + e*x)**n)**m*Gamma(m*n + S(1), -b*c*(d + e*x)*log(F)/​e)/​(b*c*log(F)), expand=True, _diff=True, _numerical=True)39 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d**S(4) + S(4)*d**S(3)*e*x + S(6)*d**S(2)*e**S(2)*x**S(2) + S(4)*d*e**S(3)*x**S(3) + e**S(4)*x**S(4))**m, x), x, F**(c*(a - b*d/​e))*(-b*c*(d + e*x)*log(F)/​e)**(-S(4)*m)*((d + e*x)**S(4))**m*Gamma(S(4)*m + S(1), -b*c*(d + e*x)*log(F)/​e)/​(b*c*log(F)), expand=True, _diff=True, _numerical=True)40 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d**S(3) + S(3)*d**S(2)*e*x + S(3)*d*e**S(2)*x**S(2) + e**S(3)*x**S(3))**m, x), x, F**(c*(a - b*d/​e))*(-b*c*(d + e*x)*log(F)/​e)**(-S(3)*m)*((d + e*x)**S(3))**m*Gamma(S(3)*m + S(1), -b*c*(d + e*x)*log(F)/​e)/​(b*c*log(F)), expand=True, _diff=True, _numerical=True)41 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d**S(2) + S(2)*d*e*x + e**S(2)*x**S(2))**m, x), x, F**(c*(a - b*d/​e))*(-b*c*(d + e*x)*log(F)/​e)**(-S(2)*m)*((d + e*x)**S(2))**m*Gamma(S(2)*m + S(1), -b*c*(d + e*x)*log(F)/​e)/​(b*c*log(F)), expand=True, _diff=True, _numerical=True)42 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d + e*x)**m, x), x, F**(c*(a - b*d/​e))*(-b*c*(d + e*x)*log(F)/​e)**(-m)*(d + e*x)**m*Gamma(m + S(1), -b*c*(d + e*x)*log(F)/​e)/​(b*c*log(F)), expand=True, _diff=True, _numerical=True)43 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d + e*x)**(-m), x), x, F**(c*(a - b*d/​e))*(-b*c*(d + e*x)*log(F)/​e)**m*(d + e*x)**(-m)*Gamma(-m + S(1), -b*c*(d + e*x)*log(F)/​e)/​(b*c*log(F)), expand=True, _diff=True, _numerical=True)44 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d**S(2) + S(2)*d*e*x + e**S(2)*x**S(2))**(-m), x), x, F**(c*(a - b*d/​e))*(-b*c*(d + e*x)*log(F)/​e)**(S(2)*m)*((d + e*x)**S(2))**(-m)*Gamma(-S(2)*m + S(1), -b*c*(d + e*x)*log(F)/​e)/​(b*c*log(F)), expand=True, _diff=True, _numerical=True)45 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d**S(3) + S(3)*d**S(2)*e*x + S(3)*d*e**S(2)*x**S(2) + e**S(3)*x**S(3))**(-m), x), x, F**(c*(a - b*d/​e))*(-b*c*(d + e*x)*log(F)/​e)**(S(3)*m)*((d + e*x)**S(3))**(-m)*Gamma(-S(3)*m + S(1), -b*c*(d + e*x)*log(F)/​e)/​(b*c*log(F)), expand=True, _diff=True, _numerical=True)46 assert rubi_test(rubi_integrate(F**(S(5)*x + S(2)), x), x, F**(S(5)*x + S(2))/​(S(5)*log(F)), expand=True, _diff=True, _numerical=True)47 assert rubi_test(rubi_integrate(F**(a + b*x), x), x, F**(a + b*x)/​(b*log(F)), expand=True, _diff=True, _numerical=True)48 assert rubi_test(rubi_integrate(S(10)**(S(5)*x + S(2)), x), x, S(2)**(S(5)*x + S(2))*S(5)**(S(5)*x + S(1))/​log(S(10)), expand=True, _diff=True, _numerical=True)49 assert rubi_test(rubi_integrate(F**(a + b*x)*x**(S(7)/​2), x), x, S(105)*sqrt(pi)*F**a*erfi(sqrt(b)*sqrt(x)*sqrt(log(F)))/​(S(16)*b**(S(9)/​2)*log(F)**(S(9)/​2)) + F**(a + b*x)*x**(S(7)/​2)/​(b*log(F)) - S(7)*F**(a + b*x)*x**(S(5)/​2)/​(S(2)*b**S(2)*log(F)**S(2)) + S(35)*F**(a + b*x)*x**(S(3)/​2)/​(S(4)*b**S(3)*log(F)**S(3)) - S(105)*F**(a + b*x)*sqrt(x)/​(S(8)*b**S(4)*log(F)**S(4)), expand=True, _diff=True, _numerical=True)50 assert rubi_test(rubi_integrate(F**(a + b*x)*x**(S(5)/​2), x), x, -S(15)*sqrt(pi)*F**a*erfi(sqrt(b)*sqrt(x)*sqrt(log(F)))/​(S(8)*b**(S(7)/​2)*log(F)**(S(7)/​2)) + F**(a + b*x)*x**(S(5)/​2)/​(b*log(F)) - S(5)*F**(a + b*x)*x**(S(3)/​2)/​(S(2)*b**S(2)*log(F)**S(2)) + S(15)*F**(a + b*x)*sqrt(x)/​(S(4)*b**S(3)*log(F)**S(3)), expand=True, _diff=True, _numerical=True)51 assert rubi_test(rubi_integrate(F**(a + b*x)*x**(S(3)/​2), x), x, S(3)*sqrt(pi)*F**a*erfi(sqrt(b)*sqrt(x)*sqrt(log(F)))/​(S(4)*b**(S(5)/​2)*log(F)**(S(5)/​2)) + F**(a + b*x)*x**(S(3)/​2)/​(b*log(F)) - S(3)*F**(a + b*x)*sqrt(x)/​(S(2)*b**S(2)*log(F)**S(2)), expand=True, _diff=True, _numerical=True)52 assert rubi_test(rubi_integrate(F**(a + b*x)*sqrt(x), x), x, -sqrt(pi)*F**a*erfi(sqrt(b)*sqrt(x)*sqrt(log(F)))/​(S(2)*b**(S(3)/​2)*log(F)**(S(3)/​2)) + F**(a + b*x)*sqrt(x)/​(b*log(F)), expand=True, _diff=True, _numerical=True)53 assert rubi_test(rubi_integrate(F**(a + b*x)/​sqrt(x), x), x, sqrt(pi)*F**a*erfi(sqrt(b)*sqrt(x)*sqrt(log(F)))/​(sqrt(b)*sqrt(log(F))), expand=True, _diff=True, _numerical=True)54 assert rubi_test(rubi_integrate(F**(a + b*x)/​x**(S(3)/​2), x), x, S(2)*sqrt(pi)*F**a*sqrt(b)*sqrt(log(F))*erfi(sqrt(b)*sqrt(x)*sqrt(log(F))) - S(2)*F**(a + b*x)/​sqrt(x), expand=True, _diff=True, _numerical=True)55 assert rubi_test(rubi_integrate(F**(a + b*x)/​x**(S(5)/​2), x), x, S(4)*sqrt(pi)*F**a*b**(S(3)/​2)*log(F)**(S(3)/​2)*erfi(sqrt(b)*sqrt(x)*sqrt(log(F)))/​S(3) - S(4)*F**(a + b*x)*b*log(F)/​(S(3)*sqrt(x)) - S(2)*F**(a + b*x)/​(S(3)*x**(S(3)/​2)), expand=True, _diff=True, _numerical=True)56 assert rubi_test(rubi_integrate(F**(a + b*x)/​x**(S(7)/​2), x), x, S(8)*sqrt(pi)*F**a*b**(S(5)/​2)*log(F)**(S(5)/​2)*erfi(sqrt(b)*sqrt(x)*sqrt(log(F)))/​S(15) - S(8)*F**(a + b*x)*b**S(2)*log(F)**S(2)/​(S(15)*sqrt(x)) - S(4)*F**(a + b*x)*b*log(F)/​(S(15)*x**(S(3)/​2)) - S(2)*F**(a + b*x)/​(S(5)*x**(S(5)/​2)), expand=True, _diff=True, _numerical=True)57 assert rubi_test(rubi_integrate(F**(a + b*x)/​x**(S(9)/​2), x), x, S(16)*sqrt(pi)*F**a*b**(S(7)/​2)*log(F)**(S(7)/​2)*erfi(sqrt(b)*sqrt(x)*sqrt(log(F)))/​S(105) - S(16)*F**(a + b*x)*b**S(3)*log(F)**S(3)/​(S(105)*sqrt(x)) - S(8)*F**(a + b*x)*b**S(2)*log(F)**S(2)/​(S(105)*x**(S(3)/​2)) - S(4)*F**(a + b*x)*b*log(F)/​(S(35)*x**(S(5)/​2)) - S(2)*F**(a + b*x)/​(S(7)*x**(S(7)/​2)), expand=True, _diff=True, _numerical=True)58 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d + e*x)**(S(7)/​2), x), x, F**(c*(a + b*x))*(d + e*x)**(S(7)/​2)/​(b*c*log(F)) - S(7)*F**(c*(a + b*x))*e*(d + e*x)**(S(5)/​2)/​(S(2)*b**S(2)*c**S(2)*log(F)**S(2)) + S(35)*F**(c*(a + b*x))*e**S(2)*(d + e*x)**(S(3)/​2)/​(S(4)*b**S(3)*c**S(3)*log(F)**S(3)) - S(105)*F**(c*(a + b*x))*e**S(3)*sqrt(d + e*x)/​(S(8)*b**S(4)*c**S(4)*log(F)**S(4)) + S(105)*sqrt(pi)*F**(c*(a - b*d/​e))*e**(S(7)/​2)*erfi(sqrt(b)*sqrt(c)*sqrt(d + e*x)*sqrt(log(F))/​sqrt(e))/​(S(16)*b**(S(9)/​2)*c**(S(9)/​2)*log(F)**(S(9)/​2)), expand=True, _diff=True, _numerical=True)59 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d + e*x)**(S(5)/​2), x), x, F**(c*(a + b*x))*(d + e*x)**(S(5)/​2)/​(b*c*log(F)) - S(5)*F**(c*(a + b*x))*e*(d + e*x)**(S(3)/​2)/​(S(2)*b**S(2)*c**S(2)*log(F)**S(2)) + S(15)*F**(c*(a + b*x))*e**S(2)*sqrt(d + e*x)/​(S(4)*b**S(3)*c**S(3)*log(F)**S(3)) - S(15)*sqrt(pi)*F**(c*(a - b*d/​e))*e**(S(5)/​2)*erfi(sqrt(b)*sqrt(c)*sqrt(d + e*x)*sqrt(log(F))/​sqrt(e))/​(S(8)*b**(S(7)/​2)*c**(S(7)/​2)*log(F)**(S(7)/​2)), expand=True, _diff=True, _numerical=True)60 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d + e*x)**(S(3)/​2), x), x, F**(c*(a + b*x))*(d + e*x)**(S(3)/​2)/​(b*c*log(F)) - S(3)*F**(c*(a + b*x))*e*sqrt(d + e*x)/​(S(2)*b**S(2)*c**S(2)*log(F)**S(2)) + S(3)*sqrt(pi)*F**(c*(a - b*d/​e))*e**(S(3)/​2)*erfi(sqrt(b)*sqrt(c)*sqrt(d + e*x)*sqrt(log(F))/​sqrt(e))/​(S(4)*b**(S(5)/​2)*c**(S(5)/​2)*log(F)**(S(5)/​2)), expand=True, _diff=True, _numerical=True)61 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*sqrt(d + e*x), x), x, F**(c*(a + b*x))*sqrt(d + e*x)/​(b*c*log(F)) - sqrt(pi)*F**(c*(a - b*d/​e))*sqrt(e)*erfi(sqrt(b)*sqrt(c)*sqrt(d + e*x)*sqrt(log(F))/​sqrt(e))/​(S(2)*b**(S(3)/​2)*c**(S(3)/​2)*log(F)**(S(3)/​2)), expand=True, _diff=True, _numerical=True)62 assert rubi_test(rubi_integrate(F**(c*(a + b*x))/​sqrt(d + e*x), x), x, sqrt(pi)*F**(c*(a - b*d/​e))*erfi(sqrt(b)*sqrt(c)*sqrt(d + e*x)*sqrt(log(F))/​sqrt(e))/​(sqrt(b)*sqrt(c)*sqrt(e)*sqrt(log(F))), expand=True, _diff=True, _numerical=True)63 assert rubi_test(rubi_integrate(F**(c*(a + b*x))/​(d + e*x)**(S(3)/​2), x), x, -S(2)*F**(c*(a + b*x))/​(e*sqrt(d + e*x)) + S(2)*sqrt(pi)*F**(c*(a - b*d/​e))*sqrt(b)*sqrt(c)*sqrt(log(F))*erfi(sqrt(b)*sqrt(c)*sqrt(d + e*x)*sqrt(log(F))/​sqrt(e))/​e**(S(3)/​2), expand=True, _diff=True, _numerical=True)64 assert rubi_test(rubi_integrate(F**(c*(a + b*x))/​(d + e*x)**(S(5)/​2), x), x, -S(4)*F**(c*(a + b*x))*b*c*log(F)/​(S(3)*e**S(2)*sqrt(d + e*x)) - S(2)*F**(c*(a + b*x))/​(S(3)*e*(d + e*x)**(S(3)/​2)) + S(4)*sqrt(pi)*F**(c*(a - b*d/​e))*b**(S(3)/​2)*c**(S(3)/​2)*log(F)**(S(3)/​2)*erfi(sqrt(b)*sqrt(c)*sqrt(d + e*x)*sqrt(log(F))/​sqrt(e))/​(S(3)*e**(S(5)/​2)), expand=True, _diff=True, _numerical=True)65 assert rubi_test(rubi_integrate(F**(c*(a + b*x))/​(d + e*x)**(S(7)/​2), x), x, -S(8)*F**(c*(a + b*x))*b**S(2)*c**S(2)*log(F)**S(2)/​(S(15)*e**S(3)*sqrt(d + e*x)) - S(4)*F**(c*(a + b*x))*b*c*log(F)/​(S(15)*e**S(2)*(d + e*x)**(S(3)/​2)) - S(2)*F**(c*(a + b*x))/​(S(5)*e*(d + e*x)**(S(5)/​2)) + S(8)*sqrt(pi)*F**(c*(a - b*d/​e))*b**(S(5)/​2)*c**(S(5)/​2)*log(F)**(S(5)/​2)*erfi(sqrt(b)*sqrt(c)*sqrt(d + e*x)*sqrt(log(F))/​sqrt(e))/​(S(15)*e**(S(7)/​2)), expand=True, _diff=True, _numerical=True)66 assert rubi_test(rubi_integrate(F**(c*(a + b*x))/​(d + e*x)**(S(9)/​2), x), x, -S(16)*F**(c*(a + b*x))*b**S(3)*c**S(3)*log(F)**S(3)/​(S(105)*e**S(4)*sqrt(d + e*x)) - S(8)*F**(c*(a + b*x))*b**S(2)*c**S(2)*log(F)**S(2)/​(S(105)*e**S(3)*(d + e*x)**(S(3)/​2)) - S(4)*F**(c*(a + b*x))*b*c*log(F)/​(S(35)*e**S(2)*(d + e*x)**(S(5)/​2)) - S(2)*F**(c*(a + b*x))/​(S(7)*e*(d + e*x)**(S(7)/​2)) + S(16)*sqrt(pi)*F**(c*(a - b*d/​e))*b**(S(7)/​2)*c**(S(7)/​2)*log(F)**(S(7)/​2)*erfi(sqrt(b)*sqrt(c)*sqrt(d + e*x)*sqrt(log(F))/​sqrt(e))/​(S(105)*e**(S(9)/​2)), expand=True, _diff=True, _numerical=True)67 assert rubi_test(rubi_integrate(x**(S(13)/​2)*exp(-b*x), x), x, -x**(S(13)/​2)*exp(-b*x)/​b - S(13)*x**(S(11)/​2)*exp(-b*x)/​(S(2)*b**S(2)) - S(143)*x**(S(9)/​2)*exp(-b*x)/​(S(4)*b**S(3)) - S(1287)*x**(S(7)/​2)*exp(-b*x)/​(S(8)*b**S(4)) - S(9009)*x**(S(5)/​2)*exp(-b*x)/​(S(16)*b**S(5)) - S(45045)*x**(S(3)/​2)*exp(-b*x)/​(S(32)*b**S(6)) - S(135135)*sqrt(x)*exp(-b*x)/​(S(64)*b**S(7)) + S(135135)*sqrt(pi)*erf(sqrt(b)*sqrt(x))/​(S(128)*b**(S(15)/​2)), expand=True, _diff=True, _numerical=True)68 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d + e*x)**(S(4)/​3), x), x, -F**(c*(a - b*d/​e))*e*(d + e*x)**(S(1)/​3)*Gamma(S(7)/​3, -b*c*(d + e*x)*log(F)/​e)/​(b**S(2)*c**S(2)*(-b*c*(d + e*x)*log(F)/​e)**(S(1)/​3)*log(F)**S(2)), expand=True, _diff=True, _numerical=True)69 assert rubi_test(rubi_integrate((d + e*x)**(S(4)/​3)*(F**(c*(a + b*x)))**n, x), x, -F**(-c*n*(a + b*x) + c*n*(a - b*d/​e))*e*(d + e*x)**(S(1)/​3)*(F**(c*(a + b*x)))**n*Gamma(S(7)/​3, -b*c*n*(d + e*x)*log(F)/​e)/​(b**S(2)*c**S(2)*n**S(2)*(-b*c*n*(d + e*x)*log(F)/​e)**(S(1)/​3)*log(F)**S(2)), expand=True, _diff=True, _numerical=True)70 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d + e*x), x), x, F**(c*(a + b*x))*(d + e*x)/​(b*c*log(F)) - F**(c*(a + b*x))*e/​(b**S(2)*c**S(2)*log(F)**S(2)), expand=True, _diff=True, _numerical=True)71 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d + e*x + f*x**S(2)), x), x, F**(c*(a + b*x))*d/​(b*c*log(F)) + F**(c*(a + b*x))*e*x/​(b*c*log(F)) + F**(c*(a + b*x))*f*x**S(2)/​(b*c*log(F)) - F**(c*(a + b*x))*e/​(b**S(2)*c**S(2)*log(F)**S(2)) - S(2)*F**(c*(a + b*x))*f*x/​(b**S(2)*c**S(2)*log(F)**S(2)) + S(2)*F**(c*(a + b*x))*f/​(b**S(3)*c**S(3)*log(F)**S(3)), expand=True, _diff=True, _numerical=True)72 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d + e*x + f*x**S(2) + g*x**S(3)), x), x, F**(c*(a + b*x))*d/​(b*c*log(F)) + F**(c*(a + b*x))*e*x/​(b*c*log(F)) + F**(c*(a + b*x))*f*x**S(2)/​(b*c*log(F)) + F**(c*(a + b*x))*g*x**S(3)/​(b*c*log(F)) - F**(c*(a + b*x))*e/​(b**S(2)*c**S(2)*log(F)**S(2)) - S(2)*F**(c*(a + b*x))*f*x/​(b**S(2)*c**S(2)*log(F)**S(2)) - S(3)*F**(c*(a + b*x))*g*x**S(2)/​(b**S(2)*c**S(2)*log(F)**S(2)) + S(2)*F**(c*(a + b*x))*f/​(b**S(3)*c**S(3)*log(F)**S(3)) + S(6)*F**(c*(a + b*x))*g*x/​(b**S(3)*c**S(3)*log(F)**S(3)) - S(6)*F**(c*(a + b*x))*g/​(b**S(4)*c**S(4)*log(F)**S(4)), expand=True, _diff=True, _numerical=True)73 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(d + e*x + f*x**S(2) + g*x**S(3) + h*x**S(4)), x), x, F**(c*(a + b*x))*d/​(b*c*log(F)) + F**(c*(a + b*x))*e*x/​(b*c*log(F)) + F**(c*(a + b*x))*f*x**S(2)/​(b*c*log(F)) + F**(c*(a + b*x))*g*x**S(3)/​(b*c*log(F)) + F**(c*(a + b*x))*h*x**S(4)/​(b*c*log(F)) - F**(c*(a + b*x))*e/​(b**S(2)*c**S(2)*log(F)**S(2)) - S(2)*F**(c*(a + b*x))*f*x/​(b**S(2)*c**S(2)*log(F)**S(2)) - S(3)*F**(c*(a + b*x))*g*x**S(2)/​(b**S(2)*c**S(2)*log(F)**S(2)) - S(4)*F**(c*(a + b*x))*h*x**S(3)/​(b**S(2)*c**S(2)*log(F)**S(2)) + S(2)*F**(c*(a + b*x))*f/​(b**S(3)*c**S(3)*log(F)**S(3)) + S(6)*F**(c*(a + b*x))*g*x/​(b**S(3)*c**S(3)*log(F)**S(3)) + S(12)*F**(c*(a + b*x))*h*x**S(2)/​(b**S(3)*c**S(3)*log(F)**S(3)) - S(6)*F**(c*(a + b*x))*g/​(b**S(4)*c**S(4)*log(F)**S(4)) - S(24)*F**(c*(a + b*x))*h*x/​(b**S(4)*c**S(4)*log(F)**S(4)) + S(24)*F**(c*(a + b*x))*h/​(b**S(5)*c**S(5)*log(F)**S(5)), expand=True, _diff=True, _numerical=True)74 assert rubi_test(rubi_integrate(x**m*(a + b*x)**S(3)*exp(-a - b*x), x), x, -a**S(3)*x**m*(b*x)**(-m)*Gamma(m + S(1), b*x)*exp(-a)/​b - S(3)*a**S(2)*x**m*(b*x)**(-m)*Gamma(m + S(2), b*x)*exp(-a)/​b - S(3)*a*x**m*(b*x)**(-m)*Gamma(m + S(3), b*x)*exp(-a)/​b - x**m*(b*x)**(-m)*Gamma(m + S(4), b*x)*exp(-a)/​b, expand=True, _diff=True, _numerical=True)75 assert rubi_test(rubi_integrate(x**S(3)*(a + b*x)**S(3)*exp(-a - b*x), x), x, -a**S(3)*x**S(3)*exp(-a - b*x)/​b - S(3)*a**S(3)*x**S(2)*exp(-a - b*x)/​b**S(2) - S(6)*a**S(3)*x*exp(-a - b*x)/​b**S(3) - S(6)*a**S(3)*exp(-a - b*x)/​b**S(4) - S(3)*a**S(2)*x**S(4)*exp(-a - b*x) - S(12)*a**S(2)*x**S(3)*exp(-a - b*x)/​b - S(36)*a**S(2)*x**S(2)*exp(-a - b*x)/​b**S(2) - S(72)*a**S(2)*x*exp(-a - b*x)/​b**S(3) - S(72)*a**S(2)*exp(-a - b*x)/​b**S(4) - S(3)*a*b*x**S(5)*exp(-a - b*x) - S(15)*a*x**S(4)*exp(-a - b*x) - S(60)*a*x**S(3)*exp(-a - b*x)/​b - S(180)*a*x**S(2)*exp(-a - b*x)/​b**S(2) - S(360)*a*x*exp(-a - b*x)/​b**S(3) - S(360)*a*exp(-a - b*x)/​b**S(4) - b**S(2)*x**S(6)*exp(-a - b*x) - S(6)*b*x**S(5)*exp(-a - b*x) - S(30)*x**S(4)*exp(-a - b*x) - S(120)*x**S(3)*exp(-a - b*x)/​b - S(360)*x**S(2)*exp(-a - b*x)/​b**S(2) - S(720)*x*exp(-a - b*x)/​b**S(3) - S(720)*exp(-a - b*x)/​b**S(4), expand=True, _diff=True, _numerical=True)76 assert rubi_test(rubi_integrate(x**S(2)*(a + b*x)**S(3)*exp(-a - b*x), x), x, -a**S(3)*x**S(2)*exp(-a - b*x)/​b - S(2)*a**S(3)*x*exp(-a - b*x)/​b**S(2) - S(2)*a**S(3)*exp(-a - b*x)/​b**S(3) - S(3)*a**S(2)*x**S(3)*exp(-a - b*x) - S(9)*a**S(2)*x**S(2)*exp(-a - b*x)/​b - S(18)*a**S(2)*x*exp(-a - b*x)/​b**S(2) - S(18)*a**S(2)*exp(-a - b*x)/​b**S(3) - S(3)*a*b*x**S(4)*exp(-a - b*x) - S(12)*a*x**S(3)*exp(-a - b*x) - S(36)*a*x**S(2)*exp(-a - b*x)/​b - S(72)*a*x*exp(-a - b*x)/​b**S(2) - S(72)*a*exp(-a - b*x)/​b**S(3) - b**S(2)*x**S(5)*exp(-a - b*x) - S(5)*b*x**S(4)*exp(-a - b*x) - S(20)*x**S(3)*exp(-a - b*x) - S(60)*x**S(2)*exp(-a - b*x)/​b - S(120)*x*exp(-a - b*x)/​b**S(2) - S(120)*exp(-a - b*x)/​b**S(3), expand=True, _diff=True, _numerical=True)77 assert rubi_test(rubi_integrate(x*(a + b*x)**S(3)*exp(-a - b*x), x), x, a*(a + b*x)**S(3)*exp(-a - b*x)/​b**S(2) + S(3)*a*(a + b*x)**S(2)*exp(-a - b*x)/​b**S(2) + S(6)*a*(a + b*x)*exp(-a - b*x)/​b**S(2) + S(6)*a*exp(-a - b*x)/​b**S(2) - (a + b*x)**S(4)*exp(-a - b*x)/​b**S(2) - S(4)*(a + b*x)**S(3)*exp(-a - b*x)/​b**S(2) - S(12)*(a + b*x)**S(2)*exp(-a - b*x)/​b**S(2) - S(24)*(a + b*x)*exp(-a - b*x)/​b**S(2) - S(24)*exp(-a - b*x)/​b**S(2), expand=True, _diff=True, _numerical=True)78 assert rubi_test(rubi_integrate((a + b*x)**S(3)*exp(-a - b*x), x), x, -(a + b*x)**S(3)*exp(-a - b*x)/​b - S(3)*(a + b*x)**S(2)*exp(-a - b*x)/​b - S(6)*(a + b*x)*exp(-a - b*x)/​b - S(6)*exp(-a - b*x)/​b, expand=True, _diff=True, _numerical=True)79 assert rubi_test(rubi_integrate((a + b*x)**S(3)*exp(-a - b*x)/​x, x), x, a**S(3)*exp(-a)*Ei(-b*x) - S(3)*a**S(2)*exp(-a - b*x) - S(3)*a*b*x*exp(-a - b*x) - S(3)*a*exp(-a - b*x) - b**S(2)*x**S(2)*exp(-a - b*x) - S(2)*b*x*exp(-a - b*x) - S(2)*exp(-a - b*x), expand=True, _diff=True, _numerical=True)80 assert rubi_test(rubi_integrate((a + b*x)**S(3)*exp(-a - b*x)/​x**S(2), x), x, -a**S(3)*b*exp(-a)*Ei(-b*x) - a**S(3)*exp(-a - b*x)/​x + S(3)*a**S(2)*b*exp(-a)*Ei(-b*x) - S(3)*a*b*exp(-a - b*x) - b**S(2)*x*exp(-a - b*x) - b*exp(-a - b*x), expand=True, _diff=True, _numerical=True)81 assert rubi_test(rubi_integrate((a + b*x)**S(3)*exp(-a - b*x)/​x**S(3), x), x, a**S(3)*b**S(2)*exp(-a)*Ei(-b*x)/​S(2) + a**S(3)*b*exp(-a - b*x)/​(S(2)*x) - a**S(3)*exp(-a - b*x)/​(S(2)*x**S(2)) - S(3)*a**S(2)*b**S(2)*exp(-a)*Ei(-b*x) - S(3)*a**S(2)*b*exp(-a - b*x)/​x + S(3)*a*b**S(2)*exp(-a)*Ei(-b*x) - b**S(2)*exp(-a - b*x), expand=True, _diff=True, _numerical=True)82 assert rubi_test(rubi_integrate((a + b*x)**S(3)*exp(-a - b*x)/​x**S(4), x), x, -a**S(3)*b**S(3)*exp(-a)*Ei(-b*x)/​S(6) - a**S(3)*b**S(2)*exp(-a - b*x)/​(S(6)*x) + a**S(3)*b*exp(-a - b*x)/​(S(6)*x**S(2)) - a**S(3)*exp(-a - b*x)/​(S(3)*x**S(3)) + S(3)*a**S(2)*b**S(3)*exp(-a)*Ei(-b*x)/​S(2) + S(3)*a**S(2)*b**S(2)*exp(-a - b*x)/​(S(2)*x) - S(3)*a**S(2)*b*exp(-a - b*x)/​(S(2)*x**S(2)) - S(3)*a*b**S(3)*exp(-a)*Ei(-b*x) - S(3)*a*b**S(2)*exp(-a - b*x)/​x + b**S(3)*exp(-a)*Ei(-b*x), expand=True, _diff=True, _numerical=True)83 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x))*x**m*(e + f*x)**S(2), x), x, F**(a + b*c)*e**S(2)*x**m*(-b*d*x*log(F))**(-m)*Gamma(m + S(1), -b*d*x*log(F))/​(b*d*log(F)) - S(2)*F**(a + b*c)*e*f*x**m*(-b*d*x*log(F))**(-m)*Gamma(m + S(2), -b*d*x*log(F))/​(b**S(2)*d**S(2)*log(F)**S(2)) + F**(a + b*c)*f**S(2)*x**m*(-b*d*x*log(F))**(-m)*Gamma(m + S(3), -b*d*x*log(F))/​(b**S(3)*d**S(3)*log(F)**S(3)), expand=True, _diff=True, _numerical=True)84 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x))*x**S(3)*(e + f*x)**S(2), x), x, F**(a + b*c + b*d*x)*e**S(2)*x**S(3)/​(b*d*log(F)) + S(2)*F**(a + b*c + b*d*x)*e*f*x**S(4)/​(b*d*log(F)) + F**(a + b*c + b*d*x)*f**S(2)*x**S(5)/​(b*d*log(F)) - S(3)*F**(a + b*c + b*d*x)*e**S(2)*x**S(2)/​(b**S(2)*d**S(2)*log(F)**S(2)) - S(8)*F**(a + b*c + b*d*x)*e*f*x**S(3)/​(b**S(2)*d**S(2)*log(F)**S(2)) - S(5)*F**(a + b*c + b*d*x)*f**S(2)*x**S(4)/​(b**S(2)*d**S(2)*log(F)**S(2)) + S(6)*F**(a + b*c + b*d*x)*e**S(2)*x/​(b**S(3)*d**S(3)*log(F)**S(3)) + S(24)*F**(a + b*c + b*d*x)*e*f*x**S(2)/​(b**S(3)*d**S(3)*log(F)**S(3)) + S(20)*F**(a + b*c + b*d*x)*f**S(2)*x**S(3)/​(b**S(3)*d**S(3)*log(F)**S(3)) - S(6)*F**(a + b*c + b*d*x)*e**S(2)/​(b**S(4)*d**S(4)*log(F)**S(4)) - S(48)*F**(a + b*c + b*d*x)*e*f*x/​(b**S(4)*d**S(4)*log(F)**S(4)) - S(60)*F**(a + b*c + b*d*x)*f**S(2)*x**S(2)/​(b**S(4)*d**S(4)*log(F)**S(4)) + S(48)*F**(a + b*c + b*d*x)*e*f/​(b**S(5)*d**S(5)*log(F)**S(5)) + S(120)*F**(a + b*c + b*d*x)*f**S(2)*x/​(b**S(5)*d**S(5)*log(F)**S(5)) - S(120)*F**(a + b*c + b*d*x)*f**S(2)/​(b**S(6)*d**S(6)*log(F)**S(6)), expand=True, _diff=True, _numerical=True)85 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x))*x**S(2)*(e + f*x)**S(2), x), x, F**(a + b*c + b*d*x)*e**S(2)*x**S(2)/​(b*d*log(F)) + S(2)*F**(a + b*c + b*d*x)*e*f*x**S(3)/​(b*d*log(F)) + F**(a + b*c + b*d*x)*f**S(2)*x**S(4)/​(b*d*log(F)) - S(2)*F**(a + b*c + b*d*x)*e**S(2)*x/​(b**S(2)*d**S(2)*log(F)**S(2)) - S(6)*F**(a + b*c + b*d*x)*e*f*x**S(2)/​(b**S(2)*d**S(2)*log(F)**S(2)) - S(4)*F**(a + b*c + b*d*x)*f**S(2)*x**S(3)/​(b**S(2)*d**S(2)*log(F)**S(2)) + S(2)*F**(a + b*c + b*d*x)*e**S(2)/​(b**S(3)*d**S(3)*log(F)**S(3)) + S(12)*F**(a + b*c + b*d*x)*e*f*x/​(b**S(3)*d**S(3)*log(F)**S(3)) + S(12)*F**(a + b*c + b*d*x)*f**S(2)*x**S(2)/​(b**S(3)*d**S(3)*log(F)**S(3)) - S(12)*F**(a + b*c + b*d*x)*e*f/​(b**S(4)*d**S(4)*log(F)**S(4)) - S(24)*F**(a + b*c + b*d*x)*f**S(2)*x/​(b**S(4)*d**S(4)*log(F)**S(4)) + S(24)*F**(a + b*c + b*d*x)*f**S(2)/​(b**S(5)*d**S(5)*log(F)**S(5)), expand=True, _diff=True, _numerical=True)86 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x))*x*(e + f*x)**S(2), x), x, F**(a + b*c + b*d*x)*e**S(2)*x/​(b*d*log(F)) + S(2)*F**(a + b*c + b*d*x)*e*f*x**S(2)/​(b*d*log(F)) + F**(a + b*c + b*d*x)*f**S(2)*x**S(3)/​(b*d*log(F)) - F**(a + b*c + b*d*x)*e**S(2)/​(b**S(2)*d**S(2)*log(F)**S(2)) - S(4)*F**(a + b*c + b*d*x)*e*f*x/​(b**S(2)*d**S(2)*log(F)**S(2)) - S(3)*F**(a + b*c + b*d*x)*f**S(2)*x**S(2)/​(b**S(2)*d**S(2)*log(F)**S(2)) + S(4)*F**(a + b*c + b*d*x)*e*f/​(b**S(3)*d**S(3)*log(F)**S(3)) + S(6)*F**(a + b*c + b*d*x)*f**S(2)*x/​(b**S(3)*d**S(3)*log(F)**S(3)) - S(6)*F**(a + b*c + b*d*x)*f**S(2)/​(b**S(4)*d**S(4)*log(F)**S(4)), expand=True, _diff=True, _numerical=True)87 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x))*(e + f*x)**S(2), x), x, F**(a + b*c + b*d*x)*(e + f*x)**S(2)/​(b*d*log(F)) - S(2)*F**(a + b*c + b*d*x)*f*(e + f*x)/​(b**S(2)*d**S(2)*log(F)**S(2)) + S(2)*F**(a + b*c + b*d*x)*f**S(2)/​(b**S(3)*d**S(3)*log(F)**S(3)), expand=True, _diff=True, _numerical=True)88 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x))*(e + f*x)**S(2)/​x, x), x, F**(a + b*c)*e**S(2)*Ei(b*d*x*log(F)) + S(2)*F**(a + b*c + b*d*x)*e*f/​(b*d*log(F)) + F**(a + b*c + b*d*x)*f**S(2)*x/​(b*d*log(F)) - F**(a + b*c + b*d*x)*f**S(2)/​(b**S(2)*d**S(2)*log(F)**S(2)), expand=True, _diff=True, _numerical=True)89 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x))*(e + f*x)**S(2)/​x**S(2), x), x, F**(a + b*c)*b*d*e**S(2)*log(F)*Ei(b*d*x*log(F)) + S(2)*F**(a + b*c)*e*f*Ei(b*d*x*log(F)) - F**(a + b*c + b*d*x)*e**S(2)/​x + F**(a + b*c + b*d*x)*f**S(2)/​(b*d*log(F)), expand=True, _diff=True, _numerical=True)90 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x))*(e + f*x)**S(2)/​x**S(3), x), x, F**(a + b*c)*b**S(2)*d**S(2)*e**S(2)*log(F)**S(2)*Ei(b*d*x*log(F))/​S(2) + S(2)*F**(a + b*c)*b*d*e*f*log(F)*Ei(b*d*x*log(F)) + F**(a + b*c)*f**S(2)*Ei(b*d*x*log(F)) - F**(a + b*c + b*d*x)*b*d*e**S(2)*log(F)/​(S(2)*x) - F**(a + b*c + b*d*x)*e**S(2)/​(S(2)*x**S(2)) - S(2)*F**(a + b*c + b*d*x)*e*f/​x, expand=True, _diff=True, _numerical=True)91 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x))*(e + f*x)**S(2)/​x**S(4), x), x, F**(a + b*c)*b**S(3)*d**S(3)*e**S(2)*log(F)**S(3)*Ei(b*d*x*log(F))/​S(6) + F**(a + b*c)*b**S(2)*d**S(2)*e*f*log(F)**S(2)*Ei(b*d*x*log(F)) + F**(a + b*c)*b*d*f**S(2)*log(F)*Ei(b*d*x*log(F)) - F**(a + b*c + b*d*x)*b**S(2)*d**S(2)*e**S(2)*log(F)**S(2)/​(S(6)*x) - F**(a + b*c + b*d*x)*b*d*e**S(2)*log(F)/​(S(6)*x**S(2)) - F**(a + b*c + b*d*x)*b*d*e*f*log(F)/​x - F**(a + b*c + b*d*x)*e**S(2)/​(S(3)*x**S(3)) - F**(a + b*c + b*d*x)*e*f/​x**S(2) - F**(a + b*c + b*d*x)*f**S(2)/​x, expand=True, _diff=True, _numerical=True)92 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x))*(e + f*x)**S(2)/​x**S(5), x), x, F**(a + b*c)*b**S(4)*d**S(4)*e**S(2)*log(F)**S(4)*Ei(b*d*x*log(F))/​S(24) + F**(a + b*c)*b**S(3)*d**S(3)*e*f*log(F)**S(3)*Ei(b*d*x*log(F))/​S(3) + F**(a + b*c)*b**S(2)*d**S(2)*f**S(2)*log(F)**S(2)*Ei(b*d*x*log(F))/​S(2) - F**(a + b*c + b*d*x)*b**S(3)*d**S(3)*e**S(2)*log(F)**S(3)/​(S(24)*x) - F**(a + b*c + b*d*x)*b**S(2)*d**S(2)*e**S(2)*log(F)**S(2)/​(S(24)*x**S(2)) - F**(a + b*c + b*d*x)*b**S(2)*d**S(2)*e*f*log(F)**S(2)/​(S(3)*x) - F**(a + b*c + b*d*x)*b*d*e**S(2)*log(F)/​(S(12)*x**S(3)) - F**(a + b*c + b*d*x)*b*d*e*f*log(F)/​(S(3)*x**S(2)) - F**(a + b*c + b*d*x)*b*d*f**S(2)*log(F)/​(S(2)*x) - F**(a + b*c + b*d*x)*e**S(2)/​(S(4)*x**S(4)) - S(2)*F**(a + b*c + b*d*x)*e*f/​(S(3)*x**S(3)) - F**(a + b*c + b*d*x)*f**S(2)/​(S(2)*x**S(2)), expand=True, _diff=True, _numerical=True)93 assert rubi_test(rubi_integrate((a + b*x)**S(4)*(c + d*x)**S(3)*exp(-a - b*x), x), x, -d**S(3)*(a + b*x)**S(7)*exp(-a - b*x)/​b**S(4) - S(7)*d**S(3)*(a + b*x)**S(6)*exp(-a - b*x)/​b**S(4) - S(42)*d**S(3)*(a + b*x)**S(5)*exp(-a - b*x)/​b**S(4) - S(210)*d**S(3)*(a + b*x)**S(4)*exp(-a - b*x)/​b**S(4) - S(840)*d**S(3)*(a + b*x)**S(3)*exp(-a - b*x)/​b**S(4) - S(2520)*d**S(3)*(a + b*x)**S(2)*exp(-a - b*x)/​b**S(4) - S(5040)*d**S(3)*(a + b*x)*exp(-a - b*x)/​b**S(4) - S(5040)*d**S(3)*exp(-a - b*x)/​b**S(4) - S(3)*d**S(2)*(a + b*x)**S(6)*(-a*d + b*c)*exp(-a - b*x)/​b**S(4) - S(18)*d**S(2)*(a + b*x)**S(5)*(-a*d + b*c)*exp(-a - b*x)/​b**S(4) - S(90)*d**S(2)*(a + b*x)**S(4)*(-a*d + b*c)*exp(-a - b*x)/​b**S(4) - S(360)*d**S(2)*(a + b*x)**S(3)*(-a*d + b*c)*exp(-a - b*x)/​b**S(4) - S(1080)*d**S(2)*(a + b*x)**S(2)*(-a*d + b*c)*exp(-a - b*x)/​b**S(4) - S(2160)*d**S(2)*(a + b*x)*(-a*d + b*c)*exp(-a - b*x)/​b**S(4) - S(2160)*d**S(2)*(-a*d + b*c)*exp(-a - b*x)/​b**S(4) - S(3)*d*(a + b*x)**S(5)*(-a*d + b*c)**S(2)*exp(-a - b*x)/​b**S(4) - S(15)*d*(a + b*x)**S(4)*(-a*d + b*c)**S(2)*exp(-a - b*x)/​b**S(4) - S(60)*d*(a + b*x)**S(3)*(-a*d + b*c)**S(2)*exp(-a - b*x)/​b**S(4) - S(180)*d*(a + b*x)**S(2)*(-a*d + b*c)**S(2)*exp(-a - b*x)/​b**S(4) - S(360)*d*(a + b*x)*(-a*d + b*c)**S(2)*exp(-a - b*x)/​b**S(4) - S(360)*d*(-a*d + b*c)**S(2)*exp(-a - b*x)/​b**S(4) - (a + b*x)**S(4)*(-a*d + b*c)**S(3)*exp(-a - b*x)/​b**S(4) - S(4)*(a + b*x)**S(3)*(-a*d + b*c)**S(3)*exp(-a - b*x)/​b**S(4) - S(12)*(a + b*x)**S(2)*(-a*d + b*c)**S(3)*exp(-a - b*x)/​b**S(4) - S(24)*(a + b*x)*(-a*d + b*c)**S(3)*exp(-a - b*x)/​b**S(4) - S(24)*(-a*d + b*c)**S(3)*exp(-a - b*x)/​b**S(4), expand=True, _diff=True, _numerical=True)94 assert rubi_test(rubi_integrate((a + b*x)**S(4)*(c + d*x)**S(2)*exp(-a - b*x), x), x, -d**S(2)*(a + b*x)**S(6)*exp(-a - b*x)/​b**S(3) - S(6)*d**S(2)*(a + b*x)**S(5)*exp(-a - b*x)/​b**S(3) - S(30)*d**S(2)*(a + b*x)**S(4)*exp(-a - b*x)/​b**S(3) - S(120)*d**S(2)*(a + b*x)**S(3)*exp(-a - b*x)/​b**S(3) - S(360)*d**S(2)*(a + b*x)**S(2)*exp(-a - b*x)/​b**S(3) - S(720)*d**S(2)*(a + b*x)*exp(-a - b*x)/​b**S(3) - S(720)*d**S(2)*exp(-a - b*x)/​b**S(3) - S(2)*d*(a + b*x)**S(5)*(-a*d + b*c)*exp(-a - b*x)/​b**S(3) - S(10)*d*(a + b*x)**S(4)*(-a*d + b*c)*exp(-a - b*x)/​b**S(3) - S(40)*d*(a + b*x)**S(3)*(-a*d + b*c)*exp(-a - b*x)/​b**S(3) - S(120)*d*(a + b*x)**S(2)*(-a*d + b*c)*exp(-a - b*x)/​b**S(3) - S(240)*d*(a + b*x)*(-a*d + b*c)*exp(-a - b*x)/​b**S(3) - S(240)*d*(-a*d + b*c)*exp(-a - b*x)/​b**S(3) - (a + b*x)**S(4)*(-a*d + b*c)**S(2)*exp(-a - b*x)/​b**S(3) - S(4)*(a + b*x)**S(3)*(-a*d + b*c)**S(2)*exp(-a - b*x)/​b**S(3) - S(12)*(a + b*x)**S(2)*(-a*d + b*c)**S(2)*exp(-a - b*x)/​b**S(3) - S(24)*(a + b*x)*(-a*d + b*c)**S(2)*exp(-a - b*x)/​b**S(3) - S(24)*(-a*d + b*c)**S(2)*exp(-a - b*x)/​b**S(3), expand=True, _diff=True, _numerical=True)95 assert rubi_test(rubi_integrate((a + b*x)**S(4)*(c + d*x)*exp(-a - b*x), x), x, -d*(a + b*x)**S(5)*exp(-a - b*x)/​b**S(2) - S(5)*d*(a + b*x)**S(4)*exp(-a - b*x)/​b**S(2) - S(20)*d*(a + b*x)**S(3)*exp(-a - b*x)/​b**S(2) - S(60)*d*(a + b*x)**S(2)*exp(-a - b*x)/​b**S(2) - S(120)*d*(a + b*x)*exp(-a - b*x)/​b**S(2) - S(120)*d*exp(-a - b*x)/​b**S(2) - (a + b*x)**S(4)*(-a*d + b*c)*exp(-a - b*x)/​b**S(2) - (a + b*x)**S(3)*(-S(4)*a*d + S(4)*b*c)*exp(-a - b*x)/​b**S(2) - (a + b*x)**S(2)*(-S(12)*a*d + S(12)*b*c)*exp(-a - b*x)/​b**S(2) - (a + b*x)*(-S(24)*a*d + S(24)*b*c)*exp(-a - b*x)/​b**S(2) - (-S(24)*a*d + S(24)*b*c)*exp(-a - b*x)/​b**S(2), expand=True, _diff=True, _numerical=True)96 assert rubi_test(rubi_integrate((a + b*x)**S(4)*exp(-a - b*x), x), x, -(a + b*x)**S(4)*exp(-a - b*x)/​b - S(4)*(a + b*x)**S(3)*exp(-a - b*x)/​b - S(12)*(a + b*x)**S(2)*exp(-a - b*x)/​b - S(24)*(a + b*x)*exp(-a - b*x)/​b - S(24)*exp(-a - b*x)/​b, expand=True, _diff=True, _numerical=True)97 assert rubi_test(rubi_integrate((a + b*x)**S(4)*exp(-a - b*x)/​(c + d*x), x), x, -(a + b*x)**S(3)*exp(-a - b*x)/​d - S(3)*(a + b*x)**S(2)*exp(-a - b*x)/​d - S(6)*(a + b*x)*exp(-a - b*x)/​d - S(6)*exp(-a - b*x)/​d + (a + b*x)**S(2)*(-a*d + b*c)*exp(-a - b*x)/​d**S(2) + (a + b*x)*(-S(2)*a*d + S(2)*b*c)*exp(-a - b*x)/​d**S(2) + (-S(2)*a*d + S(2)*b*c)*exp(-a - b*x)/​d**S(2) - (a + b*x)*(-a*d + b*c)**S(2)*exp(-a - b*x)/​d**S(3) - (-a*d + b*c)**S(2)*exp(-a - b*x)/​d**S(3) + (-a*d + b*c)**S(3)*exp(-a - b*x)/​d**S(4) + (-a*d + b*c)**S(4)*exp(-a + b*c/​d)*Ei(-b*(c + d*x)/​d)/​d**S(5), expand=True, _diff=True, _numerical=True)98 assert rubi_test(rubi_integrate((a + b*x)**S(4)*exp(-a - b*x)/​(c + d*x)**S(2), x), x, -b**S(3)*(c + d*x)**S(2)*exp(-a - b*x)/​d**S(4) - S(2)*b**S(2)*(c + d*x)*exp(-a - b*x)/​d**S(3) + S(4)*b**S(2)*(c + d*x)*(-a*d + b*c)*exp(-a - b*x)/​d**S(4) - S(2)*b*exp(-a - b*x)/​d**S(2) + S(4)*b*(-a*d + b*c)*exp(-a - b*x)/​d**S(3) - S(6)*b*(-a*d + b*c)**S(2)*exp(-a - b*x)/​d**S(4) - S(4)*b*(-a*d + b*c)**S(3)*exp(-a + b*c/​d)*Ei(-b*(c + d*x)/​d)/​d**S(5) - b*(-a*d + b*c)**S(4)*exp(-a + b*c/​d)*Ei(-b*(c + d*x)/​d)/​d**S(6) - (-a*d + b*c)**S(4)*exp(-a - b*x)/​(d**S(5)*(c + d*x)), expand=True, _diff=True, _numerical=True)99 assert rubi_test(rubi_integrate((a + b*x)**S(4)*exp(-a - b*x)/​(c + d*x)**S(3), x), x, -b**S(3)*x*exp(-a - b*x)/​d**S(3) - b**S(2)*exp(-a - b*x)/​d**S(3) + b**S(2)*(-S(4)*a*d + S(3)*b*c)*exp(-a - b*x)/​d**S(4) + S(6)*b**S(2)*(-a*d + b*c)**S(2)*exp(-a + b*c/​d)*Ei(-b*(c + d*x)/​d)/​d**S(5) + S(4)*b**S(2)*(-a*d + b*c)**S(3)*exp(-a + b*c/​d)*Ei(-b*(c + d*x)/​d)/​d**S(6) + b**S(2)*(-a*d + b*c)**S(4)*exp(-a + b*c/​d)*Ei(-b*(c + d*x)/​d)/​(S(2)*d**S(7)) + S(4)*b*(-a*d + b*c)**S(3)*exp(-a - b*x)/​(d**S(5)*(c + d*x)) + b*(-a*d + b*c)**S(4)*exp(-a - b*x)/​(S(2)*d**S(6)*(c + d*x)) - (-a*d + b*c)**S(4)*exp(-a - b*x)/​(S(2)*d**S(5)*(c + d*x)**S(2)), expand=True, _diff=True, _numerical=True)100 assert rubi_test(rubi_integrate((a + b*x)**S(4)*exp(-a - b*x)/​(c + d*x)**S(4), x), x, -b**S(3)*exp(-a - b*x)/​d**S(4) - S(4)*b**S(3)*(-a*d + b*c)*exp(-a + b*c/​d)*Ei(-b*(c + d*x)/​d)/​d**S(5) - S(6)*b**S(3)*(-a*d + b*c)**S(2)*exp(-a + b*c/​d)*Ei(-b*(c + d*x)/​d)/​d**S(6) - S(2)*b**S(3)*(-a*d + b*c)**S(3)*exp(-a + b*c/​d)*Ei(-b*(c + d*x)/​d)/​d**S(7) - b**S(3)*(-a*d + b*c)**S(4)*exp(-a + b*c/​d)*Ei(-b*(c + d*x)/​d)/​(S(6)*d**S(8)) - S(6)*b**S(2)*(-a*d + b*c)**S(2)*exp(-a - b*x)/​(d**S(5)*(c + d*x)) - S(2)*b**S(2)*(-a*d + b*c)**S(3)*exp(-a - b*x)/​(d**S(6)*(c + d*x)) - b**S(2)*(-a*d + b*c)**S(4)*exp(-a - b*x)/​(S(6)*d**S(7)*(c + d*x)) + S(2)*b*(-a*d + b*c)**S(3)*exp(-a - b*x)/​(d**S(5)*(c + d*x)**S(2)) + b*(-a*d + b*c)**S(4)*exp(-a - b*x)/​(S(6)*d**S(6)*(c + d*x)**S(2)) - (-a*d + b*c)**S(4)*exp(-a - b*x)/​(S(3)*d**S(5)*(c + d*x)**S(3)), expand=True, _diff=True, _numerical=True)101 assert rubi_test(rubi_integrate((a + b*x)**S(4)*exp(-a - b*x)/​(c + d*x)**S(5), x), x, b**S(4)*exp(-a + b*c/​d)*Ei(-b*(c + d*x)/​d)/​d**S(5) + S(4)*b**S(4)*(-a*d + b*c)*exp(-a + b*c/​d)*Ei(-b*(c + d*x)/​d)/​d**S(6) + S(3)*b**S(4)*(-a*d + b*c)**S(2)*exp(-a + b*c/​d)*Ei(-b*(c + d*x)/​d)/​d**S(7) + S(2)*b**S(4)*(-a*d + b*c)**S(3)*exp(-a + b*c/​d)*Ei(-b*(c + d*x)/​d)/​(S(3)*d**S(8)) + b**S(4)*(-a*d + b*c)**S(4)*exp(-a + b*c/​d)*Ei(-b*(c + d*x)/​d)/​(S(24)*d**S(9)) + S(4)*b**S(3)*(-a*d + b*c)*exp(-a - b*x)/​(d**S(5)*(c + d*x)) + S(3)*b**S(3)*(-a*d + b*c)**S(2)*exp(-a - b*x)/​(d**S(6)*(c + d*x)) + S(2)*b**S(3)*(-a*d + b*c)**S(3)*exp(-a - b*x)/​(S(3)*d**S(7)*(c + d*x)) + b**S(3)*(-a*d + b*c)**S(4)*exp(-a - b*x)/​(S(24)*d**S(8)*(c + d*x)) - S(3)*b**S(2)*(-a*d + b*c)**S(2)*exp(-a - b*x)/​(d**S(5)*(c + d*x)**S(2)) - S(2)*b**S(2)*(-a*d + b*c)**S(3)*exp(-a - b*x)/​(S(3)*d**S(6)*(c + d*x)**S(2)) - b**S(2)*(-a*d + b*c)**S(4)*exp(-a - b*x)/​(S(24)*d**S(7)*(c + d*x)**S(2)) + S(4)*b*(-a*d + b*c)**S(3)*exp(-a - b*x)/​(S(3)*d**S(5)*(c + d*x)**S(3)) + b*(-a*d + b*c)**S(4)*exp(-a - b*x)/​(S(12)*d**S(6)*(c + d*x)**S(3)) - (-a*d + b*c)**S(4)*exp(-a - b*x)/​(S(4)*d**S(5)*(c + d*x)**S(4)), expand=True, _diff=True, _numerical=True)102 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*x**m*(e*n + e*(b*c*x*log(F) + m + S(1))*log(d*x) + e)*log(d*x)**n, x), x, F**(c*(a + b*x))*e*x**(m + S(1))*log(d*x)**(n + S(1)), expand=True, _diff=True, _numerical=True)103 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*x**S(2)*(e*n + e*(b*c*x*log(F) + S(3))*log(d*x) + e)*log(d*x)**n, x), x, F**(c*(a + b*x))*e*x**S(3)*log(d*x)**(n + S(1)), expand=True, _diff=True, _numerical=True)104 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*x*(e*n + e*(b*c*x*log(F) + S(2))*log(d*x) + e)*log(d*x)**n, x), x, F**(c*(a + b*x))*e*x**S(2)*log(d*x)**(n + S(1)), expand=True, _diff=True, _numerical=True)105 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(e*n + e*(b*c*x*log(F) + S(1))*log(d*x) + e)*log(d*x)**n, x), x, F**(c*(a + b*x))*e*x*log(d*x)**(n + S(1)), expand=True, _diff=True, _numerical=True)106 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(b*c*e*x*log(F)*log(d*x) + e*n + e)*log(d*x)**n/​x, x), x, F**(c*(a + b*x))*e*log(d*x)**(n + S(1)), expand=True, _diff=True, _numerical=True)107 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(e*n + e*(b*c*x*log(F) + S(-1))*log(d*x) + e)*log(d*x)**n/​x**S(2), x), x, F**(c*(a + b*x))*e*log(d*x)**(n + S(1))/​x, expand=True, _diff=True, _numerical=True)108 assert rubi_test(rubi_integrate(F**(c*(a + b*x))*(e*n + e*(b*c*x*log(F) + S(-2))*log(d*x) + e)*log(d*x)**n/​x**S(3), x), x, F**(c*(a + b*x))*e*log(d*x)**(n + S(1))/​x**S(2), expand=True, _diff=True, _numerical=True)109 assert rubi_test(rubi_integrate(x**S(4)*sqrt(exp(a + b*x)), x), x, S(2)*x**S(4)*sqrt(exp(a + b*x))/​b - S(16)*x**S(3)*sqrt(exp(a + b*x))/​b**S(2) + S(96)*x**S(2)*sqrt(exp(a + b*x))/​b**S(3) - S(384)*x*sqrt(exp(a + b*x))/​b**S(4) + S(768)*sqrt(exp(a + b*x))/​b**S(5), expand=True, _diff=True, _numerical=True)110 assert rubi_test(rubi_integrate(x**S(3)*sqrt(exp(a + b*x)), x), x, S(2)*x**S(3)*sqrt(exp(a + b*x))/​b - S(12)*x**S(2)*sqrt(exp(a + b*x))/​b**S(2) + S(48)*x*sqrt(exp(a + b*x))/​b**S(3) - S(96)*sqrt(exp(a + b*x))/​b**S(4), expand=True, _diff=True, _numerical=True)111 assert rubi_test(rubi_integrate(x**S(2)*sqrt(exp(a + b*x)), x), x, S(2)*x**S(2)*sqrt(exp(a + b*x))/​b - S(8)*x*sqrt(exp(a + b*x))/​b**S(2) + S(16)*sqrt(exp(a + b*x))/​b**S(3), expand=True, _diff=True, _numerical=True)112 assert rubi_test(rubi_integrate(x*sqrt(exp(a + b*x)), x), x, S(2)*x*sqrt(exp(a + b*x))/​b - S(4)*sqrt(exp(a + b*x))/​b**S(2), expand=True, _diff=True, _numerical=True)113 assert rubi_test(rubi_integrate(sqrt(exp(a + b*x)), x), x, S(2)*sqrt(exp(a + b*x))/​b, expand=True, _diff=True, _numerical=True)114 assert rubi_test(rubi_integrate(sqrt(exp(a + b*x))/​x, x), x, exp(-b*x/​S(2))*sqrt(exp(a + b*x))*Ei(b*x/​S(2)), expand=True, _diff=True, _numerical=True)115 assert rubi_test(rubi_integrate(sqrt(exp(a + b*x))/​x**S(2), x), x, b*exp(-b*x/​S(2))*sqrt(exp(a + b*x))*Ei(b*x/​S(2))/​S(2) - sqrt(exp(a + b*x))/​x, expand=True, _diff=True, _numerical=True)116 assert rubi_test(rubi_integrate(sqrt(exp(a + b*x))/​x**S(3), x), x, b**S(2)*exp(-b*x/​S(2))*sqrt(exp(a + b*x))*Ei(b*x/​S(2))/​S(8) - b*sqrt(exp(a + b*x))/​(S(4)*x) - sqrt(exp(a + b*x))/​(S(2)*x**S(2)), expand=True, _diff=True, _numerical=True)117 assert rubi_test(rubi_integrate(sqrt(exp(a + b*x))/​x**S(4), x), x, b**S(3)*exp(-b*x/​S(2))*sqrt(exp(a + b*x))*Ei(b*x/​S(2))/​S(48) - b**S(2)*sqrt(exp(a + b*x))/​(S(24)*x) - b*sqrt(exp(a + b*x))/​(S(12)*x**S(2)) - sqrt(exp(a + b*x))/​(S(3)*x**S(3)), expand=True, _diff=True, _numerical=True)118def test_2():119 assert rubi_test(rubi_integrate(f**(c + d*x)*x**S(3)/​(a + b*f**(c + d*x)), x), x, x**S(3)*log(S(1) + b*f**(c + d*x)/​a)/​(b*d*log(f)) + S(3)*x**S(2)*polylog(S(2), -b*f**(c + d*x)/​a)/​(b*d**S(2)*log(f)**S(2)) - S(6)*x*polylog(S(3), -b*f**(c + d*x)/​a)/​(b*d**S(3)*log(f)**S(3)) + S(6)*polylog(S(4), -b*f**(c + d*x)/​a)/​(b*d**S(4)*log(f)**S(4)), expand=True, _diff=True, _numerical=True)120 assert rubi_test(rubi_integrate(f**(c + d*x)*x**S(2)/​(a + b*f**(c + d*x)), x), x, x**S(2)*log(S(1) + b*f**(c + d*x)/​a)/​(b*d*log(f)) + S(2)*x*polylog(S(2), -b*f**(c + d*x)/​a)/​(b*d**S(2)*log(f)**S(2)) - S(2)*polylog(S(3), -b*f**(c + d*x)/​a)/​(b*d**S(3)*log(f)**S(3)), expand=True, _diff=True, _numerical=True)121 assert rubi_test(rubi_integrate(f**(c + d*x)*x/​(a + b*f**(c + d*x)), x), x, x*log(S(1) + b*f**(c + d*x)/​a)/​(b*d*log(f)) + polylog(S(2), -b*f**(c + d*x)/​a)/​(b*d**S(2)*log(f)**S(2)), expand=True, _diff=True, _numerical=True)122 assert rubi_test(rubi_integrate(f**(c + d*x)/​(a + b*f**(c + d*x)), x), x, log(a + b*f**(c + d*x))/​(b*d*log(f)), expand=True, _diff=True, _numerical=True)123 assert rubi_test(rubi_integrate(f**(c + d*x)/​(x*(a + b*f**(c + d*x))), x), x, Integral(f**(c + d*x)/​(x*(a + b*f**(c + d*x))), x), expand=True, _diff=True, _numerical=True)124 assert rubi_test(rubi_integrate(f**(c + d*x)/​(x**S(2)*(a + b*f**(c + d*x))), x), x, Integral(f**(c + d*x)/​(x**S(2)*(a + b*f**(c + d*x))), x), expand=True, _diff=True, _numerical=True)125 assert rubi_test(rubi_integrate(f**(c + d*x)*x**S(3)/​(a + b*f**(c + d*x))**S(2), x), x, -x**S(3)/​(b*d*(a + b*f**(c + d*x))*log(f)) - S(3)*x**S(2)*log(a*f**(-c - d*x)/​b + S(1))/​(a*b*d**S(2)*log(f)**S(2)) + S(6)*x*polylog(S(2), -a*f**(-c - d*x)/​b)/​(a*b*d**S(3)*log(f)**S(3)) + S(6)*polylog(S(3), -a*f**(-c - d*x)/​b)/​(a*b*d**S(4)*log(f)**S(4)), expand=True, _diff=True, _numerical=True)126 assert rubi_test(rubi_integrate(f**(c + d*x)*x**S(2)/​(a + b*f**(c + d*x))**S(2), x), x, -x**S(2)/​(b*d*(a + b*f**(c + d*x))*log(f)) - S(2)*x*log(a*f**(-c - d*x)/​b + S(1))/​(a*b*d**S(2)*log(f)**S(2)) + S(2)*polylog(S(2), -a*f**(-c - d*x)/​b)/​(a*b*d**S(3)*log(f)**S(3)), expand=True, _diff=True, _numerical=True)127 assert rubi_test(rubi_integrate(f**(c + d*x)*x/​(a + b*f**(c + d*x))**S(2), x), x, -x/​(b*d*(a + b*f**(c + d*x))*log(f)) + x/​(a*b*d*log(f)) - log(a + b*f**(c + d*x))/​(a*b*d**S(2)*log(f)**S(2)), expand=True, _diff=True, _numerical=True)128 assert rubi_test(rubi_integrate(f**(c + d*x)/​(a + b*f**(c + d*x))**S(2), x), x, -S(1)/​(b*d*(a + b*f**(c + d*x))*log(f)), expand=True, _diff=True, _numerical=True)129 assert rubi_test(rubi_integrate(f**(c + d*x)/​(x*(a + b*f**(c + d*x))**S(2)), x), x, -Integral(S(1)/​(x**S(2)*(a + b*f**(c + d*x))), x)/​(b*d*log(f)) - S(1)/​(b*d*x*(a + b*f**(c + d*x))*log(f)), expand=True, _diff=True, _numerical=True)130 assert rubi_test(rubi_integrate(f**(c + d*x)/​(x**S(2)*(a + b*f**(c + d*x))**S(2)), x), x, -S(2)*Integral(S(1)/​(x**S(3)*(a + b*f**(c + d*x))), x)/​(b*d*log(f)) - S(1)/​(b*d*x**S(2)*(a + b*f**(c + d*x))*log(f)), expand=True, _diff=True, _numerical=True)131 # recursion assert rubi_test(rubi_integrate(f**(c + d*x)*x**S(3)/​(a + b*f**(c + d*x))**S(3), x), x, -x**S(3)/​(S(2)*b*d*(a + b*f**(c + d*x))**S(2)*log(f)) + S(3)*x**S(2)/​(S(2)*a*b*d**S(2)*(a + b*f**(c + d*x))*log(f)**S(2)) + x**S(3)/​(S(2)*a**S(2)*b*d*log(f)) - S(3)*x**S(2)*log(S(1) + b*f**(c + d*x)/​a)/​(S(2)*a**S(2)*b*d**S(2)*log(f)**S(2)) + S(3)*x*log(a*f**(-c - d*x)/​b + S(1))/​(a**S(2)*b*d**S(3)*log(f)**S(3)) - S(3)*x*polylog(S(2), -b*f**(c + d*x)/​a)/​(a**S(2)*b*d**S(3)*log(f)**S(3)) - S(3)*polylog(S(2), -a*f**(-c - d*x)/​b)/​(a**S(2)*b*d**S(4)*log(f)**S(4)) + S(3)*polylog(S(3), -b*f**(c + d*x)/​a)/​(a**S(2)*b*d**S(4)*log(f)**S(4)), expand=True, _diff=True, _numerical=True)132 assert rubi_test(rubi_integrate(f**(c + d*x)*x**S(2)/​(a + b*f**(c + d*x))**S(3), x), x, -x**S(2)/​(S(2)*b*d*(a + b*f**(c + d*x))**S(2)*log(f)) + x/​(a*b*d**S(2)*(a + b*f**(c + d*x))*log(f)**S(2)) + x**S(2)/​(S(2)*a**S(2)*b*d*log(f)) - x*log(S(1) + b*f**(c + d*x)/​a)/​(a**S(2)*b*d**S(2)*log(f)**S(2)) - x/​(a**S(2)*b*d**S(2)*log(f)**S(2)) + log(a + b*f**(c + d*x))/​(a**S(2)*b*d**S(3)*log(f)**S(3)) - polylog(S(2), -b*f**(c + d*x)/​a)/​(a**S(2)*b*d**S(3)*log(f)**S(3)), expand=True, _diff=True, _numerical=True)133 assert rubi_test(rubi_integrate(f**(c + d*x)*x/​(a + b*f**(c + d*x))**S(3), x), x, -x/​(S(2)*b*d*(a + b*f**(c + d*x))**S(2)*log(f)) + S(1)/​(S(2)*a*b*d**S(2)*(a + b*f**(c + d*x))*log(f)**S(2)) + x/​(S(2)*a**S(2)*b*d*log(f)) - log(a + b*f**(c + d*x))/​(S(2)*a**S(2)*b*d**S(2)*log(f)**S(2)), expand=True, _diff=True, _numerical=True)134 assert rubi_test(rubi_integrate(f**(c + d*x)/​(a + b*f**(c + d*x))**S(3), x), x, -S(1)/​(S(2)*b*d*(a + b*f**(c + d*x))**S(2)*log(f)), expand=True, _diff=True, _numerical=True)135 assert rubi_test(rubi_integrate(f**(c + d*x)/​(x*(a + b*f**(c + d*x))**S(3)), x), x, -Integral(S(1)/​(x**S(2)*(a + b*f**(c + d*x))**S(2)), x)/​(S(2)*b*d*log(f)) - S(1)/​(S(2)*b*d*x*(a + b*f**(c + d*x))**S(2)*log(f)), expand=True, _diff=True, _numerical=True)136 assert rubi_test(rubi_integrate(f**(c + d*x)/​(x**S(2)*(a + b*f**(c + d*x))**S(3)), x), x, -Integral(S(1)/​(x**S(3)*(a + b*f**(c + d*x))**S(2)), x)/​(b*d*log(f)) - S(1)/​(S(2)*b*d*x**S(2)*(a + b*f**(c + d*x))**S(2)*log(f)), expand=True, _diff=True, _numerical=True)137def test_3():138 assert rubi_test(rubi_integrate(exp(x)/​(S(6)*exp(x) + S(4)), x), x, log(S(3)*exp(x) + S(2))/​S(6), expand=True, _diff=True, _numerical=True)139 assert rubi_test(rubi_integrate(exp(x)/​(a + b*exp(x)), x), x, log(a + b*exp(x))/​b, expand=True, _diff=True, _numerical=True)140 assert rubi_test(rubi_integrate(exp(d*x)/​(a + b*exp(c + d*x)), x), x, exp(-c)*log(a + b*exp(c + d*x))/​(b*d), expand=True, _diff=True, _numerical=True)141 assert rubi_test(rubi_integrate(exp(c + d*x)/​(a + b*exp(c + d*x)), x), x, log(a + b*exp(c + d*x))/​(b*d), expand=True, _diff=True, _numerical=True)142 assert rubi_test(rubi_integrate((a + b*exp(x))**n*exp(x), x), x, (a + b*exp(x))**(n + S(1))/​(b*(n + S(1))), expand=True, _diff=True, _numerical=True)143 assert rubi_test(rubi_integrate((a + b*exp(c + d*x))**n*exp(d*x), x), x, (a + b*exp(c + d*x))**(n + S(1))*exp(-c)/​(b*d*(n + S(1))), expand=True, _diff=True, _numerical=True)144 assert rubi_test(rubi_integrate((a + b*exp(c + d*x))**n*exp(c + d*x), x), x, (a + b*exp(c + d*x))**(n + S(1))/​(b*d*(n + S(1))), expand=True, _diff=True, _numerical=True)145 assert rubi_test(rubi_integrate(F**x/​(F**x*b + a), x), x, log(F**x*b + a)/​(b*log(F)), expand=True, _diff=True, _numerical=True)146 assert rubi_test(rubi_integrate(F**(d*x)/​(F**(c + d*x)*b + a), x), x, F**(-c)*log(F**(c + d*x)*b + a)/​(b*d*log(F)), expand=True, _diff=True, _numerical=True)147 assert rubi_test(rubi_integrate(F**(c + d*x)/​(F**(c + d*x)*b + a), x), x, log(F**(c + d*x)*b + a)/​(b*d*log(F)), expand=True, _diff=True, _numerical=True)148 assert rubi_test(rubi_integrate(F**x*(F**x*b + a)**n, x), x, (F**x*b + a)**(n + S(1))/​(b*(n + S(1))*log(F)), expand=True, _diff=True, _numerical=True)149 assert rubi_test(rubi_integrate(F**(d*x)*(F**(c + d*x)*b + a)**n, x), x, F**(-c)*(F**(c + d*x)*b + a)**(n + S(1))/​(b*d*(n + S(1))*log(F)), expand=True, _diff=True, _numerical=True)150 assert rubi_test(rubi_integrate(F**(c + d*x)*(F**(c + d*x)*b + a)**n, x), x, (F**(c + d*x)*b + a)**(n + S(1))/​(b*d*(n + S(1))*log(F)), expand=True, _diff=True, _numerical=True)151 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))*x**m, x), x, -f**a*x**(m + S(1))*(-b*x**S(2)*log(f))**(-m/​S(2) + S(-1)/​2)*Gamma(m/​S(2) + S(1)/​2, -b*x**S(2)*log(f))/​S(2), expand=True, _diff=True, _numerical=True)152 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))*x**S(11), x), x, -f**a*Gamma(S(6), -b*x**S(2)*log(f))/​(S(2)*b**S(6)*log(f)**S(6)), expand=True, _diff=True, _numerical=True)153 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))*x**S(9), x), x, f**a*Gamma(S(5), -b*x**S(2)*log(f))/​(S(2)*b**S(5)*log(f)**S(5)), expand=True, _diff=True, _numerical=True)154 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))*x**S(7), x), x, f**(a + b*x**S(2))*x**S(6)/​(S(2)*b*log(f)) - S(3)*f**(a + b*x**S(2))*x**S(4)/​(S(2)*b**S(2)*log(f)**S(2)) + S(3)*f**(a + b*x**S(2))*x**S(2)/​(b**S(3)*log(f)**S(3)) - S(3)*f**(a + b*x**S(2))/​(b**S(4)*log(f)**S(4)), expand=True, _diff=True, _numerical=True)155 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))*x**S(5), x), x, f**(a + b*x**S(2))*x**S(4)/​(S(2)*b*log(f)) - f**(a + b*x**S(2))*x**S(2)/​(b**S(2)*log(f)**S(2)) + f**(a + b*x**S(2))/​(b**S(3)*log(f)**S(3)), expand=True, _diff=True, _numerical=True)156 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))*x**S(3), x), x, f**(a + b*x**S(2))*x**S(2)/​(S(2)*b*log(f)) - f**(a + b*x**S(2))/​(S(2)*b**S(2)*log(f)**S(2)), expand=True, _diff=True, _numerical=True)157 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))*x, x), x, f**(a + b*x**S(2))/​(S(2)*b*log(f)), expand=True, _diff=True, _numerical=True)158 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))/​x, x), x, f**a*Ei(b*x**S(2)*log(f))/​S(2), expand=True, _diff=True, _numerical=True)159 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))/​x**S(3), x), x, b*f**a*log(f)*Ei(b*x**S(2)*log(f))/​S(2) - f**(a + b*x**S(2))/​(S(2)*x**S(2)), expand=True, _diff=True, _numerical=True)160 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))/​x**S(5), x), x, b**S(2)*f**a*log(f)**S(2)*Ei(b*x**S(2)*log(f))/​S(4) - b*f**(a + b*x**S(2))*log(f)/​(S(4)*x**S(2)) - f**(a + b*x**S(2))/​(S(4)*x**S(4)), expand=True, _diff=True, _numerical=True)161 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))/​x**S(7), x), x, b**S(3)*f**a*log(f)**S(3)*Ei(b*x**S(2)*log(f))/​S(12) - b**S(2)*f**(a + b*x**S(2))*log(f)**S(2)/​(S(12)*x**S(2)) - b*f**(a + b*x**S(2))*log(f)/​(S(12)*x**S(4)) - f**(a + b*x**S(2))/​(S(6)*x**S(6)), expand=True, _diff=True, _numerical=True)162 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))/​x**S(9), x), x, -b**S(4)*f**a*Gamma(S(-4), -b*x**S(2)*log(f))*log(f)**S(4)/​S(2), expand=True, _diff=True, _numerical=True)163 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))/​x**S(11), x), x, b**S(5)*f**a*Gamma(S(-5), -b*x**S(2)*log(f))*log(f)**S(5)/​S(2), expand=True, _diff=True, _numerical=True)164 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))*x**S(12), x), x, -f**a*x**S(13)*Gamma(S(13)/​2, -b*x**S(2)*log(f))/​(S(2)*(-b*x**S(2)*log(f))**(S(13)/​2)), expand=True, _diff=True, _numerical=True)165 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))*x**S(10), x), x, -f**a*x**S(11)*Gamma(S(11)/​2, -b*x**S(2)*log(f))/​(S(2)*(-b*x**S(2)*log(f))**(S(11)/​2)), expand=True, _diff=True, _numerical=True)166 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))*x**S(8), x), x, f**(a + b*x**S(2))*x**S(7)/​(S(2)*b*log(f)) - S(7)*f**(a + b*x**S(2))*x**S(5)/​(S(4)*b**S(2)*log(f)**S(2)) + S(35)*f**(a + b*x**S(2))*x**S(3)/​(S(8)*b**S(3)*log(f)**S(3)) - S(105)*f**(a + b*x**S(2))*x/​(S(16)*b**S(4)*log(f)**S(4)) + S(105)*sqrt(pi)*f**a*erfi(sqrt(b)*x*sqrt(log(f)))/​(S(32)*b**(S(9)/​2)*log(f)**(S(9)/​2)), expand=True, _diff=True, _numerical=True)167 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))*x**S(6), x), x, f**(a + b*x**S(2))*x**S(5)/​(S(2)*b*log(f)) - S(5)*f**(a + b*x**S(2))*x**S(3)/​(S(4)*b**S(2)*log(f)**S(2)) + S(15)*f**(a + b*x**S(2))*x/​(S(8)*b**S(3)*log(f)**S(3)) - S(15)*sqrt(pi)*f**a*erfi(sqrt(b)*x*sqrt(log(f)))/​(S(16)*b**(S(7)/​2)*log(f)**(S(7)/​2)), expand=True, _diff=True, _numerical=True)168 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))*x**S(4), x), x, f**(a + b*x**S(2))*x**S(3)/​(S(2)*b*log(f)) - S(3)*f**(a + b*x**S(2))*x/​(S(4)*b**S(2)*log(f)**S(2)) + S(3)*sqrt(pi)*f**a*erfi(sqrt(b)*x*sqrt(log(f)))/​(S(8)*b**(S(5)/​2)*log(f)**(S(5)/​2)), expand=True, _diff=True, _numerical=True)169 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))*x**S(2), x), x, f**(a + b*x**S(2))*x/​(S(2)*b*log(f)) - sqrt(pi)*f**a*erfi(sqrt(b)*x*sqrt(log(f)))/​(S(4)*b**(S(3)/​2)*log(f)**(S(3)/​2)), expand=True, _diff=True, _numerical=True)170 assert rubi_test(rubi_integrate(f**(a + b*x**S(2)), x), x, sqrt(pi)*f**a*erfi(sqrt(b)*x*sqrt(log(f)))/​(S(2)*sqrt(b)*sqrt(log(f))), expand=True, _diff=True, _numerical=True)171 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))/​x**S(2), x), x, sqrt(pi)*sqrt(b)*f**a*sqrt(log(f))*erfi(sqrt(b)*x*sqrt(log(f))) - f**(a + b*x**S(2))/​x, expand=True, _diff=True, _numerical=True)172 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))/​x**S(4), x), x, S(2)*sqrt(pi)*b**(S(3)/​2)*f**a*log(f)**(S(3)/​2)*erfi(sqrt(b)*x*sqrt(log(f)))/​S(3) - S(2)*b*f**(a + b*x**S(2))*log(f)/​(S(3)*x) - f**(a + b*x**S(2))/​(S(3)*x**S(3)), expand=True, _diff=True, _numerical=True)173 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))/​x**S(6), x), x, S(4)*sqrt(pi)*b**(S(5)/​2)*f**a*log(f)**(S(5)/​2)*erfi(sqrt(b)*x*sqrt(log(f)))/​S(15) - S(4)*b**S(2)*f**(a + b*x**S(2))*log(f)**S(2)/​(S(15)*x) - S(2)*b*f**(a + b*x**S(2))*log(f)/​(S(15)*x**S(3)) - f**(a + b*x**S(2))/​(S(5)*x**S(5)), expand=True, _diff=True, _numerical=True)174 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))/​x**S(8), x), x, S(8)*sqrt(pi)*b**(S(7)/​2)*f**a*log(f)**(S(7)/​2)*erfi(sqrt(b)*x*sqrt(log(f)))/​S(105) - S(8)*b**S(3)*f**(a + b*x**S(2))*log(f)**S(3)/​(S(105)*x) - S(4)*b**S(2)*f**(a + b*x**S(2))*log(f)**S(2)/​(S(105)*x**S(3)) - S(2)*b*f**(a + b*x**S(2))*log(f)/​(S(35)*x**S(5)) - f**(a + b*x**S(2))/​(S(7)*x**S(7)), expand=True, _diff=True, _numerical=True)175 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))/​x**S(10), x), x, -f**a*(-b*x**S(2)*log(f))**(S(9)/​2)*Gamma(S(-9)/​2, -b*x**S(2)*log(f))/​(S(2)*x**S(9)), expand=True, _diff=True, _numerical=True)176 assert rubi_test(rubi_integrate(f**(a + b*x**S(2))/​x**S(12), x), x, -f**a*(-b*x**S(2)*log(f))**(S(11)/​2)*Gamma(S(-11)/​2, -b*x**S(2)*log(f))/​(S(2)*x**S(11)), expand=True, _diff=True, _numerical=True)177 assert rubi_test(rubi_integrate(f**(a + b*x**S(3))*x**m, x), x, -f**a*x**(m + S(1))*(-b*x**S(3)*log(f))**(-m/​S(3) + S(-1)/​3)*Gamma(m/​S(3) + S(1)/​3, -b*x**S(3)*log(f))/​S(3), expand=True, _diff=True, _numerical=True)178 assert rubi_test(rubi_integrate(f**(a + b*x**S(3))*x**S(17), x), x, -f**a*Gamma(S(6), -b*x**S(3)*log(f))/​(S(3)*b**S(6)*log(f)**S(6)), expand=True, _diff=True, _numerical=True)179 assert rubi_test(rubi_integrate(f**(a + b*x**S(3))*x**S(14), x), x, f**a*Gamma(S(5), -b*x**S(3)*log(f))/​(S(3)*b**S(5)*log(f)**S(5)), expand=True, _diff=True, _numerical=True)180 assert rubi_test(rubi_integrate(f**(a + b*x**S(3))*x**S(11), x), x, f**(a + b*x**S(3))*x**S(9)/​(S(3)*b*log(f)) - f**(a + b*x**S(3))*x**S(6)/​(b**S(2)*log(f)**S(2)) + S(2)*f**(a + b*x**S(3))*x**S(3)/​(b**S(3)*log(f)**S(3)) - S(2)*f**(a + b*x**S(3))/​(b**S(4)*log(f)**S(4)), expand=True, _diff=True, _numerical=True)181 assert rubi_test(rubi_integrate(f**(a + b*x**S(3))*x**S(8), x), x, f**(a + b*x**S(3))*x**S(6)/​(S(3)*b*log(f)) - S(2)*f**(a + b*x**S(3))*x**S(3)/​(S(3)*b**S(2)*log(f)**S(2)) + S(2)*f**(a + b*x**S(3))/​(S(3)*b**S(3)*log(f)**S(3)), expand=True, _diff=True, _numerical=True)182 assert rubi_test(rubi_integrate(f**(a + b*x**S(3))*x**S(5), x), x, f**(a + b*x**S(3))*x**S(3)/​(S(3)*b*log(f)) - f**(a + b*x**S(3))/​(S(3)*b**S(2)*log(f)**S(2)), expand=True, _diff=True, _numerical=True)183 assert rubi_test(rubi_integrate(f**(a + b*x**S(3))*x**S(2), x), x, f**(a + b*x**S(3))/​(S(3)*b*log(f)), expand=True, _diff=True, _numerical=True)184 assert rubi_test(rubi_integrate(f**(a + b*x**S(3))/​x, x), x, f**a*Ei(b*x**S(3)*log(f))/​S(3), expand=True, _diff=True, _numerical=True)185 assert rubi_test(rubi_integrate(f**(a + b*x**S(3))/​x**S(4), x), x, b*f**a*log(f)*Ei(b*x**S(3)*log(f))/​S(3) - f**(a + b*x**S(3))/​(S(3)*x**S(3)), expand=True, _diff=True, _numerical=True)186 assert rubi_test(rubi_integrate(f**(a + b*x**S(3))/​x**S(7), x), x, b**S(2)*f**a*log(f)**S(2)*Ei(b*x**S(3)*log(f))/​S(6) - b*f**(a + b*x**S(3))*log(f)/​(S(6)*x**S(3)) - f**(a + b*x**S(3))/​(S(6)*x**S(6)), expand=True, _diff=True, _numerical=True)187 assert rubi_test(rubi_integrate(f**(a + b*x**S(3))/​x**S(10), x), x, b**S(3)*f**a*log(f)**S(3)*Ei(b*x**S(3)*log(f))/​S(18) - b**S(2)*f**(a + b*x**S(3))*log(f)**S(2)/​(S(18)*x**S(3)) - b*f**(a + b*x**S(3))*log(f)/​(S(18)*x**S(6)) - f**(a + b*x**S(3))/​(S(9)*x**S(9)), expand=True, _diff=True, _numerical=True)188 assert rubi_test(rubi_integrate(f**(a + b*x**S(3))/​x**S(13), x), x, -b**S(4)*f**a*Gamma(S(-4), -b*x**S(3)*log(f))*log(f)**S(4)/​S(3), expand=True, _diff=True, _numerical=True)189 assert rubi_test(rubi_integrate(f**(a + b*x**S(3))/​x**S(16), x), x, b**S(5)*f**a*Gamma(S(-5), -b*x**S(3)*log(f))*log(f)**S(5)/​S(3), expand=True, _diff=True, _numerical=True)190 assert rubi_test(rubi_integrate(f**(a + b*x**S(3))*x**S(4), x), x, -f**a*x**S(5)*Gamma(S(5)/​3, -b*x**S(3)*log(f))/​(S(3)*(-b*x**S(3)*log(f))**(S(5)/​3)), expand=True, _diff=True, _numerical=True)191 assert rubi_test(rubi_integrate(f**(a + b*x**S(3))*x**S(3), x), x, -f**a*x**S(4)*Gamma(S(4)/​3, -b*x**S(3)*log(f))/​(S(3)*(-b*x**S(3)*log(f))**(S(4)/​3)), expand=True, _diff=True, _numerical=True)192 assert rubi_test(rubi_integrate(f**(a + b*x**S(3))*x, x), x, -f**a*x**S(2)*Gamma(S(2)/​3, -b*x**S(3)*log(f))/​(S(3)*(-b*x**S(3)*log(f))**(S(2)/​3)), expand=True, _diff=True, _numerical=True)193 assert rubi_test(rubi_integrate(f**(a + b*x**S(3)), x), x, -f**a*x*Gamma(S(1)/​3, -b*x**S(3)*log(f))/​(S(3)*(-b*x**S(3)*log(f))**(S(1)/​3)), expand=True, _diff=True, _numerical=True)194 assert rubi_test(rubi_integrate(f**(a + b*x**S(3))/​x**S(2), x), x, -f**a*(-b*x**S(3)*log(f))**(S(1)/​3)*Gamma(S(-1)/​3, -b*x**S(3)*log(f))/​(S(3)*x), expand=True, _diff=True, _numerical=True)195 assert rubi_test(rubi_integrate(f**(a + b*x**S(3))/​x**S(3), x), x, -f**a*(-b*x**S(3)*log(f))**(S(2)/​3)*Gamma(S(-2)/​3, -b*x**S(3)*log(f))/​(S(3)*x**S(2)), expand=True, _diff=True, _numerical=True)196 assert rubi_test(rubi_integrate(x**S(2)*exp(S(4)*x**S(3)), x), x, exp(S(4)*x**S(3))/​S(12), expand=True, _diff=True, _numerical=True)197 assert rubi_test(rubi_integrate(f**(a + b/​x)*x**m, x), x, f**a*x**(m + S(1))*(-b*log(f)/​x)**(m + S(1))*Gamma(-m + S(-1), -b*log(f)/​x), expand=True, _diff=True, _numerical=True)198 assert rubi_test(rubi_integrate(f**(a + b/​x)*x**S(4), x), x, -b**S(5)*f**a*Gamma(S(-5), -b*log(f)/​x)*log(f)**S(5), expand=True, _diff=True, _numerical=True)199 assert rubi_test(rubi_integrate(f**(a + b/​x)*x**S(3), x), x, b**S(4)*f**a*Gamma(S(-4), -b*log(f)/​x)*log(f)**S(4), expand=True, _diff=True, _numerical=True)200 assert rubi_test(rubi_integrate(f**(a + b/​x)*x**S(2), x), x, -b**S(3)*f**a*log(f)**S(3)*Ei(b*log(f)/​x)/​S(6) + b**S(2)*f**(a + b/​x)*x*log(f)**S(2)/​S(6) + b*f**(a + b/​x)*x**S(2)*log(f)/​S(6) + f**(a + b/​x)*x**S(3)/​S(3), expand=True, _diff=True, _numerical=True)201 assert rubi_test(rubi_integrate(f**(a + b/​x)*x, x), x, -b**S(2)*f**a*log(f)**S(2)*Ei(b*log(f)/​x)/​S(2) + b*f**(a + b/​x)*x*log(f)/​S(2) + f**(a + b/​x)*x**S(2)/​S(2), expand=True, _diff=True, _numerical=True)202 assert rubi_test(rubi_integrate(f**(a + b/​x), x), x, -b*f**a*log(f)*Ei(b*log(f)/​x) + f**(a + b/​x)*x, expand=True, _diff=True, _numerical=True)203 assert rubi_test(rubi_integrate(f**(a + b/​x)/​x, x), x, -f**a*Ei(b*log(f)/​x), expand=True, _diff=True, _numerical=True)204 assert rubi_test(rubi_integrate(f**(a + b/​x)/​x**S(2), x), x, -f**(a + b/​x)/​(b*log(f)), expand=True, _diff=True, _numerical=True)205 assert rubi_test(rubi_integrate(f**(a + b/​x)/​x**S(3), x), x, -f**(a + b/​x)/​(b*x*log(f)) + f**(a + b/​x)/​(b**S(2)*log(f)**S(2)), expand=True, _diff=True, _numerical=True)206 assert rubi_test(rubi_integrate(f**(a + b/​x)/​x**S(4), x), x, -f**(a + b/​x)/​(b*x**S(2)*log(f)) + S(2)*f**(a + b/​x)/​(b**S(2)*x*log(f)**S(2)) - S(2)*f**(a + b/​x)/​(b**S(3)*log(f)**S(3)), expand=True, _diff=True, _numerical=True)207 assert rubi_test(rubi_integrate(f**(a + b/​x)/​x**S(5), x), x, -f**(a + b/​x)/​(b*x**S(3)*log(f)) + S(3)*f**(a + b/​x)/​(b**S(2)*x**S(2)*log(f)**S(2)) - S(6)*f**(a + b/​x)/​(b**S(3)*x*log(f)**S(3)) + S(6)*f**(a + b/​x)/​(b**S(4)*log(f)**S(4)), expand=True, _diff=True, _numerical=True)208 assert rubi_test(rubi_integrate(f**(a + b/​x)/​x**S(6), x), x, -f**a*Gamma(S(5), -b*log(f)/​x)/​(b**S(5)*log(f)**S(5)), expand=True, _diff=True, _numerical=True)209 assert rubi_test(rubi_integrate(f**(a + b/​x)/​x**S(7), x), x, f**a*Gamma(S(6), -b*log(f)/​x)/​(b**S(6)*log(f)**S(6)), expand=True, _diff=True, _numerical=True)210 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))*x**m, x), x, f**a*x**(m + S(1))*(-b*log(f)/​x**S(2))**(m/​S(2) + S(1)/​2)*Gamma(-m/​S(2) + S(-1)/​2, -b*log(f)/​x**S(2))/​S(2), expand=True, _diff=True, _numerical=True)211 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))*x**S(9), x), x, -b**S(5)*f**a*Gamma(S(-5), -b*log(f)/​x**S(2))*log(f)**S(5)/​S(2), expand=True, _diff=True, _numerical=True)212 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))*x**S(7), x), x, b**S(4)*f**a*Gamma(S(-4), -b*log(f)/​x**S(2))*log(f)**S(4)/​S(2), expand=True, _diff=True, _numerical=True)213 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))*x**S(5), x), x, -b**S(3)*f**a*log(f)**S(3)*Ei(b*log(f)/​x**S(2))/​S(12) + b**S(2)*f**(a + b/​x**S(2))*x**S(2)*log(f)**S(2)/​S(12) + b*f**(a + b/​x**S(2))*x**S(4)*log(f)/​S(12) + f**(a + b/​x**S(2))*x**S(6)/​S(6), expand=True, _diff=True, _numerical=True)214 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))*x**S(3), x), x, -b**S(2)*f**a*log(f)**S(2)*Ei(b*log(f)/​x**S(2))/​S(4) + b*f**(a + b/​x**S(2))*x**S(2)*log(f)/​S(4) + f**(a + b/​x**S(2))*x**S(4)/​S(4), expand=True, _diff=True, _numerical=True)215 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))*x, x), x, -b*f**a*log(f)*Ei(b*log(f)/​x**S(2))/​S(2) + f**(a + b/​x**S(2))*x**S(2)/​S(2), expand=True, _diff=True, _numerical=True)216 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))/​x, x), x, -f**a*Ei(b*log(f)/​x**S(2))/​S(2), expand=True, _diff=True, _numerical=True)217 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))/​x**S(3), x), x, -f**(a + b/​x**S(2))/​(S(2)*b*log(f)), expand=True, _diff=True, _numerical=True)218 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))/​x**S(5), x), x, -f**(a + b/​x**S(2))/​(S(2)*b*x**S(2)*log(f)) + f**(a + b/​x**S(2))/​(S(2)*b**S(2)*log(f)**S(2)), expand=True, _diff=True, _numerical=True)219 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))/​x**S(7), x), x, -f**(a + b/​x**S(2))/​(S(2)*b*x**S(4)*log(f)) + f**(a + b/​x**S(2))/​(b**S(2)*x**S(2)*log(f)**S(2)) - f**(a + b/​x**S(2))/​(b**S(3)*log(f)**S(3)), expand=True, _diff=True, _numerical=True)220 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))/​x**S(9), x), x, -f**(a + b/​x**S(2))/​(S(2)*b*x**S(6)*log(f)) + S(3)*f**(a + b/​x**S(2))/​(S(2)*b**S(2)*x**S(4)*log(f)**S(2)) - S(3)*f**(a + b/​x**S(2))/​(b**S(3)*x**S(2)*log(f)**S(3)) + S(3)*f**(a + b/​x**S(2))/​(b**S(4)*log(f)**S(4)), expand=True, _diff=True, _numerical=True)221 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))/​x**S(11), x), x, -f**a*Gamma(S(5), -b*log(f)/​x**S(2))/​(S(2)*b**S(5)*log(f)**S(5)), expand=True, _diff=True, _numerical=True)222 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))/​x**S(13), x), x, f**a*Gamma(S(6), -b*log(f)/​x**S(2))/​(S(2)*b**S(6)*log(f)**S(6)), expand=True, _diff=True, _numerical=True)223 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))*x**S(10), x), x, f**a*x**S(11)*(-b*log(f)/​x**S(2))**(S(11)/​2)*Gamma(S(-11)/​2, -b*log(f)/​x**S(2))/​S(2), expand=True, _diff=True, _numerical=True)224 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))*x**S(8), x), x, f**a*x**S(9)*(-b*log(f)/​x**S(2))**(S(9)/​2)*Gamma(S(-9)/​2, -b*log(f)/​x**S(2))/​S(2), expand=True, _diff=True, _numerical=True)225 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))*x**S(6), x), x, -S(8)*sqrt(pi)*b**(S(7)/​2)*f**a*log(f)**(S(7)/​2)*erfi(sqrt(b)*sqrt(log(f))/​x)/​S(105) + S(8)*b**S(3)*f**(a + b/​x**S(2))*x*log(f)**S(3)/​S(105) + S(4)*b**S(2)*f**(a + b/​x**S(2))*x**S(3)*log(f)**S(2)/​S(105) + S(2)*b*f**(a + b/​x**S(2))*x**S(5)*log(f)/​S(35) + f**(a + b/​x**S(2))*x**S(7)/​S(7), expand=True, _diff=True, _numerical=True)226 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))*x**S(4), x), x, -S(4)*sqrt(pi)*b**(S(5)/​2)*f**a*log(f)**(S(5)/​2)*erfi(sqrt(b)*sqrt(log(f))/​x)/​S(15) + S(4)*b**S(2)*f**(a + b/​x**S(2))*x*log(f)**S(2)/​S(15) + S(2)*b*f**(a + b/​x**S(2))*x**S(3)*log(f)/​S(15) + f**(a + b/​x**S(2))*x**S(5)/​S(5), expand=True, _diff=True, _numerical=True)227 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))*x**S(2), x), x, -S(2)*sqrt(pi)*b**(S(3)/​2)*f**a*log(f)**(S(3)/​2)*erfi(sqrt(b)*sqrt(log(f))/​x)/​S(3) + S(2)*b*f**(a + b/​x**S(2))*x*log(f)/​S(3) + f**(a + b/​x**S(2))*x**S(3)/​S(3), expand=True, _diff=True, _numerical=True)228 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2)), x), x, -sqrt(pi)*sqrt(b)*f**a*sqrt(log(f))*erfi(sqrt(b)*sqrt(log(f))/​x) + f**(a + b/​x**S(2))*x, expand=True, _diff=True, _numerical=True)229 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))/​x**S(2), x), x, -sqrt(pi)*f**a*erfi(sqrt(b)*sqrt(log(f))/​x)/​(S(2)*sqrt(b)*sqrt(log(f))), expand=True, _diff=True, _numerical=True)230 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))/​x**S(4), x), x, -f**(a + b/​x**S(2))/​(S(2)*b*x*log(f)) + sqrt(pi)*f**a*erfi(sqrt(b)*sqrt(log(f))/​x)/​(S(4)*b**(S(3)/​2)*log(f)**(S(3)/​2)), expand=True, _diff=True, _numerical=True)231 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))/​x**S(6), x), x, -f**(a + b/​x**S(2))/​(S(2)*b*x**S(3)*log(f)) + S(3)*f**(a + b/​x**S(2))/​(S(4)*b**S(2)*x*log(f)**S(2)) - S(3)*sqrt(pi)*f**a*erfi(sqrt(b)*sqrt(log(f))/​x)/​(S(8)*b**(S(5)/​2)*log(f)**(S(5)/​2)), expand=True, _diff=True, _numerical=True)232 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))/​x**S(8), x), x, -f**(a + b/​x**S(2))/​(S(2)*b*x**S(5)*log(f)) + S(5)*f**(a + b/​x**S(2))/​(S(4)*b**S(2)*x**S(3)*log(f)**S(2)) - S(15)*f**(a + b/​x**S(2))/​(S(8)*b**S(3)*x*log(f)**S(3)) + S(15)*sqrt(pi)*f**a*erfi(sqrt(b)*sqrt(log(f))/​x)/​(S(16)*b**(S(7)/​2)*log(f)**(S(7)/​2)), expand=True, _diff=True, _numerical=True)233 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))/​x**S(10), x), x, -f**(a + b/​x**S(2))/​(S(2)*b*x**S(7)*log(f)) + S(7)*f**(a + b/​x**S(2))/​(S(4)*b**S(2)*x**S(5)*log(f)**S(2)) - S(35)*f**(a + b/​x**S(2))/​(S(8)*b**S(3)*x**S(3)*log(f)**S(3)) + S(105)*f**(a + b/​x**S(2))/​(S(16)*b**S(4)*x*log(f)**S(4)) - S(105)*sqrt(pi)*f**a*erfi(sqrt(b)*sqrt(log(f))/​x)/​(S(32)*b**(S(9)/​2)*log(f)**(S(9)/​2)), expand=True, _diff=True, _numerical=True)234 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))/​x**S(12), x), x, f**a*Gamma(S(11)/​2, -b*log(f)/​x**S(2))/​(S(2)*x**S(11)*(-b*log(f)/​x**S(2))**(S(11)/​2)), expand=True, _diff=True, _numerical=True)235 assert rubi_test(rubi_integrate(f**(a + b/​x**S(2))/​x**S(14), x), x, f**a*Gamma(S(13)/​2, -b*log(f)/​x**S(2))/​(S(2)*x**S(13)*(-b*log(f)/​x**S(2))**(S(13)/​2)), expand=True, _diff=True, _numerical=True)236 assert rubi_test(rubi_integrate(f**(a + b/​x**S(3))*x**m, x), x, f**a*x**(m + S(1))*(-b*log(f)/​x**S(3))**(m/​S(3) + S(1)/​3)*Gamma(-m/​S(3) + S(-1)/​3, -b*log(f)/​x**S(3))/​S(3), expand=True, _diff=True, _numerical=True)237 assert rubi_test(rubi_integrate(f**(a + b/​x**S(3))*x**S(14), x), x, -b**S(5)*f**a*Gamma(S(-5), -b*log(f)/​x**S(3))*log(f)**S(5)/​S(3), expand=True, _diff=True, _numerical=True)238 assert rubi_test(rubi_integrate(f**(a + b/​x**S(3))*x**S(11), x), x, b**S(4)*f**a*Gamma(S(-4), -b*log(f)/​x**S(3))*log(f)**S(4)/​S(3), expand=True, _diff=True, _numerical=True)239 assert rubi_test(rubi_integrate(f**(a + b/​x**S(3))*x**S(8), x), x, -b**S(3)*f**a*log(f)**S(3)*Ei(b*log(f)/​x**S(3))/​S(18) + b**S(2)*f**(a + b/​x**S(3))*x**S(3)*log(f)**S(2)/​S(18) + b*f**(a + b/​x**S(3))*x**S(6)*log(f)/​S(18) + f**(a + b/​x**S(3))*x**S(9)/​S(9), expand=True, _diff=True, _numerical=True)240 assert rubi_test(rubi_integrate(f**(a + b/​x**S(3))*x**S(5), x), x, -b**S(2)*f**a*log(f)**S(2)*Ei(b*log(f)/​x**S(3))/​S(6) + b*f**(a + b/​x**S(3))*x**S(3)*log(f)/​S(6) + f**(a + b/​x**S(3))*x**S(6)/​S(6), expand=True, _diff=True, _numerical=True)241 assert rubi_test(rubi_integrate(f**(a + b/​x**S(3))*x**S(2), x), x, -b*f**a*log(f)*Ei(b*log(f)/​x**S(3))/​S(3) + f**(a + b/​x**S(3))*x**S(3)/​S(3), expand=True, _diff=True, _numerical=True)242 assert rubi_test(rubi_integrate(f**(a + b/​x**S(3))/​x, x), x, -f**a*Ei(b*log(f)/​x**S(3))/​S(3), expand=True, _diff=True, _numerical=True)243 assert rubi_test(rubi_integrate(f**(a + b/​x**S(3))/​x**S(4), x), x, -f**(a + b/​x**S(3))/​(S(3)*b*log(f)), expand=True, _diff=True, _numerical=True)244 assert rubi_test(rubi_integrate(f**(a + b/​x**S(3))/​x**S(7), x), x, -f**(a + b/​x**S(3))/​(S(3)*b*x**S(3)*log(f)) + f**(a + b/​x**S(3))/​(S(3)*b**S(2)*log(f)**S(2)), expand=True, _diff=True, _numerical=True)245 assert rubi_test(rubi_integrate(f**(a + b/​x**S(3))/​x**S(10), x), x, -f**(a + b/​x**S(3))/​(S(3)*b*x**S(6)*log(f)) + S(2)*f**(a + b/​x**S(3))/​(S(3)*b**S(2)*x**S(3)*log(f)**S(2)) - S(2)*f**(a + b/​x**S(3))/​(S(3)*b**S(3)*log(f)**S(3)), expand=True, _diff=True, _numerical=True)246 assert rubi_test(rubi_integrate(f**(a + b/​x**S(3))/​x**S(13), x), x, -f**(a + b/​x**S(3))/​(S(3)*b*x**S(9)*log(f)) + f**(a + b/​x**S(3))/​(b**S(2)*x**S(6)*log(f)**S(2)) - S(2)*f**(a + b/​x**S(3))/​(b**S(3)*x**S(3)*log(f)**S(3)) + S(2)*f**(a + b/​x**S(3))/​(b**S(4)*log(f)**S(4)), expand=True, _diff=True, _numerical=True)247 assert rubi_test(rubi_integrate(f**(a + b/​x**S(3))/​x**S(16), x), x, -f**a*Gamma(S(5), -b*log(f)/​x**S(3))/​(S(3)*b**S(5)*log(f)**S(5)), expand=True, _diff=True, _numerical=True)248 assert rubi_test(rubi_integrate(f**(a + b/​x**S(3))/​x**S(19), x), x, f**a*Gamma(S(6), -b*log(f)/​x**S(3))/​(S(3)*b**S(6)*log(f)**S(6)), expand=True, _diff=True, _numerical=True)249 assert rubi_test(rubi_integrate(f**(a + b/​x**S(3))*x**S(4), x), x, f**a*x**S(5)*(-b*log(f)/​x**S(3))**(S(5)/​3)*Gamma(S(-5)/​3, -b*log(f)/​x**S(3))/​S(3), expand=True, _diff=True, _numerical=True)250 assert rubi_test(rubi_integrate(f**(a + b/​x**S(3))*x**S(3), x), x, f**a*x**S(4)*(-b*log(f)/​x**S(3))**(S(4)/​3)*Gamma(S(-4)/​3, -b*log(f)/​x**S(3))/​S(3), expand=True, _diff=True, _numerical=True)251 assert rubi_test(rubi_integrate(f**(a + b/​x**S(3))*x, x), x, f**a*x**S(2)*(-b*log(f)/​x**S(3))**(S(2)/​3)*Gamma(S(-2)/​3, -b*log(f)/​x**S(3))/​S(3), expand=True, _diff=True, _numerical=True)252 assert rubi_test(rubi_integrate(f**(a + b/​x**S(3)), x), x, f**a*x*(-b*log(f)/​x**S(3))**(S(1)/​3)*Gamma(S(-1)/​3, -b*log(f)/​x**S(3))/​S(3), expand=True, _diff=True, _numerical=True)253 assert rubi_test(rubi_integrate(f**(a + b/​x**S(3))/​x**S(2), x), x, f**a*Gamma(S(1)/​3, -b*log(f)/​x**S(3))/​(S(3)*x*(-b*log(f)/​x**S(3))**(S(1)/​3)), expand=True, _diff=True, _numerical=True)254 assert rubi_test(rubi_integrate(f**(a + b/​x**S(3))/​x**S(3), x), x, f**a*Gamma(S(2)/​3, -b*log(f)/​x**S(3))/​(S(3)*x**S(2)*(-b*log(f)/​x**S(3))**(S(2)/​3)), expand=True, _diff=True, _numerical=True)255 assert rubi_test(rubi_integrate(f**(a + b/​x**S(3))/​x**S(5), x), x, f**a*Gamma(S(4)/​3, -b*log(f)/​x**S(3))/​(S(3)*x**S(4)*(-b*log(f)/​x**S(3))**(S(4)/​3)), expand=True, _diff=True, _numerical=True)256 assert rubi_test(rubi_integrate(f**(a + b*x**n)*x**m, x), x, -f**a*x**(m + S(1))*(-b*x**n*log(f))**(-(m + S(1))/​n)*Gamma((m + S(1))/​n, -b*x**n*log(f))/​n, expand=True, _diff=True, _numerical=True)257 assert rubi_test(rubi_integrate(f**(a + b*x**n)*x**S(3), x), x, -f**a*x**S(4)*(-b*x**n*log(f))**(-S(4)/​n)*Gamma(S(4)/​n, -b*x**n*log(f))/​n, expand=True, _diff=True, _numerical=True)258 assert rubi_test(rubi_integrate(f**(a + b*x**n)*x**S(2), x), x, -f**a*x**S(3)*(-b*x**n*log(f))**(-S(3)/​n)*Gamma(S(3)/​n, -b*x**n*log(f))/​n, expand=True, _diff=True, _numerical=True)259 assert rubi_test(rubi_integrate(f**(a + b*x**n)*x, x), x, -f**a*x**S(2)*(-b*x**n*log(f))**(-S(2)/​n)*Gamma(S(2)/​n, -b*x**n*log(f))/​n, expand=True, _diff=True, _numerical=True)260 assert rubi_test(rubi_integrate(f**(a + b*x**n), x), x, -f**a*x*(-b*x**n*log(f))**(-S(1)/​n)*Gamma(S(1)/​n, -b*x**n*log(f))/​n, expand=True, _diff=True, _numerical=True)261 assert rubi_test(rubi_integrate(f**(a + b*x**n)/​x, x), x, f**a*Ei(b*x**n*log(f))/​n, expand=True, _diff=True, _numerical=True)262 assert rubi_test(rubi_integrate(f**(a + b*x**n)/​x**S(2), x), x, -f**a*(-b*x**n*log(f))**(S(1)/​n)*Gamma(-S(1)/​n, -b*x**n*log(f))/​(n*x), expand=True, _diff=True, _numerical=True)263 assert rubi_test(rubi_integrate(f**(a + b*x**n)/​x**S(3), x), x, -f**a*(-b*x**n*log(f))**(S(2)/​n)*Gamma(-S(2)/​n, -b*x**n*log(f))/​(n*x**S(2)), expand=True, _diff=True, _numerical=True)264 assert rubi_test(rubi_integrate(f**(a + b*x**n)/​x**S(4), x), x, -f**a*(-b*x**n*log(f))**(S(3)/​n)*Gamma(-S(3)/​n, -b*x**n*log(f))/​(n*x**S(3)), expand=True, _diff=True, _numerical=True)265 assert rubi_test(rubi_integrate(f**(a + b*x**n)*x**(S(3)*n + S(-1)), x), x, f**(a + b*x**n)*x**(S(2)*n)/​(b*n*log(f)) - S(2)*f**(a + b*x**n)*x**n/​(b**S(2)*n*log(f)**S(2)) + S(2)*f**(a + b*x**n)/​(b**S(3)*n*log(f)**S(3)), expand=True, _diff=True, _numerical=True)266 assert rubi_test(rubi_integrate(f**(a + b*x**n)*x**(S(2)*n + S(-1)), x), x, f**(a + b*x**n)*x**n/​(b*n*log(f)) - f**(a + b*x**n)/​(b**S(2)*n*log(f)**S(2)), expand=True, _diff=True, _numerical=True)267 assert rubi_test(rubi_integrate(f**(a + b*x**n)*x**(n + S(-1)), x), x, f**(a + b*x**n)/​(b*n*log(f)), expand=True, _diff=True, _numerical=True)268 assert rubi_test(rubi_integrate(f**(a + b*x**n)/​x, x), x, f**a*Ei(b*x**n*log(f))/​n, expand=True, _diff=True, _numerical=True)269 assert rubi_test(rubi_integrate(f**(a + b*x**n)*x**(-n + S(-1)), x), x, b*f**a*log(f)*Ei(b*x**n*log(f))/​n - f**(a + b*x**n)*x**(-n)/​n, expand=True, _diff=True, _numerical=True)270 assert rubi_test(rubi_integrate(f**(a + b*x**n)*x**(-S(2)*n + S(-1)), x), x, b**S(2)*f**a*log(f)**S(2)*Ei(b*x**n*log(f))/​(S(2)*n) - b*f**(a + b*x**n)*x**(-n)*log(f)/​(S(2)*n) - f**(a + b*x**n)*x**(-S(2)*n)/​(S(2)*n), expand=True, _diff=True, _numerical=True)271 assert rubi_test(rubi_integrate(f**(a + b*x**n)*x**(S(5)*n/​S(2) + S(-1)), x), x, f**(a + b*x**n)*x**(S(3)*n/​S(2))/​(b*n*log(f)) - S(3)*f**(a + b*x**n)*x**(n/​S(2))/​(S(2)*b**S(2)*n*log(f)**S(2)) + S(3)*sqrt(pi)*f**a*erfi(sqrt(b)*x**(n/​S(2))*sqrt(log(f)))/​(S(4)*b**(S(5)/​2)*n*log(f)**(S(5)/​2)), expand=True, _diff=True, _numerical=True)272 assert rubi_test(rubi_integrate(f**(a + b*x**n)*x**(S(3)*n/​S(2) + S(-1)), x), x, f**(a + b*x**n)*x**(n/​S(2))/​(b*n*log(f)) - sqrt(pi)*f**a*erfi(sqrt(b)*x**(n/​S(2))*sqrt(log(f)))/​(S(2)*b**(S(3)/​2)*n*log(f)**(S(3)/​2)), expand=True, _diff=True, _numerical=True)273 assert rubi_test(rubi_integrate(f**(a + b*x**n)*x**(n/​S(2) + S(-1)), x), x, sqrt(pi)*f**a*erfi(sqrt(b)*x**(n/​S(2))*sqrt(log(f)))/​(sqrt(b)*n*sqrt(log(f))), expand=True, _diff=True, _numerical=True)274 assert rubi_test(rubi_integrate(f**(a + b*x**n)*x**(-n/​S(2) + S(-1)), x), x, S(2)*sqrt(pi)*sqrt(b)*f**a*sqrt(log(f))*erfi(sqrt(b)*x**(n/​S(2))*sqrt(log(f)))/​n - S(2)*f**(a + b*x**n)*x**(-n/​S(2))/​n, expand=True, _diff=True, _numerical=True)275 assert rubi_test(rubi_integrate(f**(a + b*x**n)*x**(-S(3)*n/​S(2) + S(-1)), x), x, S(4)*sqrt(pi)*b**(S(3)/​2)*f**a*log(f)**(S(3)/​2)*erfi(sqrt(b)*x**(n/​S(2))*sqrt(log(f)))/​(S(3)*n) - S(4)*b*f**(a + b*x**n)*x**(-n/​S(2))*log(f)/​(S(3)*n) - S(2)*f**(a + b*x**n)*x**(-S(3)*n/​S(2))/​(S(3)*n), expand=True, _diff=True, _numerical=True)276 assert rubi_test(rubi_integrate(x*exp(-0.1*x), x), x, -10.0*x*exp(-0.1*x) - 100.0*exp(-0.1*x), expand=True, _diff=True, _numerical=True)277 assert rubi_test(rubi_integrate(f**(c*(a + b*x)**S(2))*x**m, x), x, Integral(f**(a**S(2)*c + S(2)*a*b*c*x + b**S(2)*c*x**S(2))*x**m, x), expand=True, _diff=True, _numerical=True)278 assert rubi_test(rubi_integrate(f**(c*(a + b*x)**S(2))*x**S(3), x), x, -sqrt(pi)*a**S(3)*erfi(sqrt(c)*(a + b*x)*sqrt(log(f)))/​(S(2)*b**S(4)*sqrt(c)*sqrt(log(f))) + S(3)*a**S(2)*f**(c*(a + b*x)**S(2))/​(S(2)*b**S(4)*c*log(f)) - S(3)*a*f**(c*(a + b*x)**S(2))*(a + b*x)/​(S(2)*b**S(4)*c*log(f)) + S(3)*sqrt(pi)*a*erfi(sqrt(c)*(a + b*x)*sqrt(log(f)))/​(S(4)*b**S(4)*c**(S(3)/​2)*log(f)**(S(3)/​2)) + f**(c*(a + b*x)**S(2))*(a + b*x)**S(2)/​(S(2)*b**S(4)*c*log(f)) - f**(c*(a + b*x)**S(2))/​(S(2)*b**S(4)*c**S(2)*log(f)**S(2)), expand=True, _diff=True, _numerical=True)279 assert rubi_test(rubi_integrate(f**(c*(a + b*x)**S(2))*x**S(2), x), x, sqrt(pi)*a**S(2)*erfi(sqrt(c)*(a + b*x)*sqrt(log(f)))/​(S(2)*b**S(3)*sqrt(c)*sqrt(log(f))) - a*f**(c*(a + b*x)**S(2))/​(b**S(3)*c*log(f)) + f**(c*(a + b*x)**S(2))*(a + b*x)/​(S(2)*b**S(3)*c*log(f)) - sqrt(pi)*erfi(sqrt(c)*(a + b*x)*sqrt(log(f)))/​(S(4)*b**S(3)*c**(S(3)/​2)*log(f)**(S(3)/​2)), expand=True, _diff=True, _numerical=True)280 assert rubi_test(rubi_integrate(f**(c*(a + b*x)**S(2))*x, x), x, -sqrt(pi)*a*erfi(sqrt(c)*(a + b*x)*sqrt(log(f)))/​(S(2)*b**S(2)*sqrt(c)*sqrt(log(f))) + f**(c*(a + b*x)**S(2))/​(S(2)*b**S(2)*c*log(f)), expand=True, _diff=True, _numerical=True)281 # long time in rubi_test(1940 is matched before 1909) assert rubi_test(rubi_integrate(f**(c*(a + b*x)**S(2)), x), x, sqrt(pi)*erfi(sqrt(c)*(a + b*x)*sqrt(log(f)))/​(S(2)*b*sqrt(c)*sqrt(log(f))), expand=True, _diff=True, _numerical=True)282 assert rubi_test(rubi_integrate(f**(c*(a + b*x)**S(2))/​x, x), x, Integral(f**(c*(a + b*x)**S(2))/​x, x), expand=True, _diff=True, _numerical=True)283 # long time in rubi_test(1940 is matched before 1909) assert rubi_test(rubi_integrate(f**(c*(a + b*x)**S(2))/​x**S(2), x), x, S(2)*a*b*c*log(f)*Integral(f**(c*(a + b*x)**S(2))/​x, x) + sqrt(pi)*b*sqrt(c)*sqrt(log(f))*erfi(sqrt(c)*(a + b*x)*sqrt(log(f))) - f**(c*(a + b*x)**S(2))/​x, expand=True, _diff=True, _numerical=True)284 assert rubi_test(rubi_integrate(f**(c*(a + b*x)**S(2))/​x**S(3), x), x, S(2)*a**S(2)*b**S(2)*c**S(2)*log(f)**S(2)*Integral(f**(c*(a + b*x)**S(2))/​x, x) + sqrt(pi)*a*b**S(2)*c**(S(3)/​2)*log(f)**(S(3)/​2)*erfi(sqrt(c)*(a + b*x)*sqrt(log(f))) - a*b*c*f**(c*(a + b*x)**S(2))*log(f)/​x + b**S(2)*c*log(f)*Integral(f**(c*(a + b*x)**S(2))/​x, x) - f**(c*(a + b*x)**S(2))/​(S(2)*x**S(2)), expand=True, _diff=True, _numerical=True)285 assert rubi_test(rubi_integrate(f**(c*(a + b*x)**S(3))*x**m, x), x, Integral(f**(c*(a + b*x)**S(3))*x**m, x), expand=True, _diff=True, _numerical=True)286 assert rubi_test(rubi_integrate(f**(c*(a + b*x)**S(3))*x**S(2), x), x, -a**S(2)*(a + b*x)*Gamma(S(1)/​3, -c*(a + b*x)**S(3)*log(f))/​(S(3)*b**S(3)*(-c*(a + b*x)**S(3)*log(f))**(S(1)/​3)) + S(2)*a*(a + b*x)**S(2)*Gamma(S(2)/​3, -c*(a + b*x)**S(3)*log(f))/​(S(3)*b**S(3)*(-c*(a + b*x)**S(3)*log(f))**(S(2)/​3)) + f**(c*(a + b*x)**S(3))/​(S(3)*b**S(3)*c*log(f)), expand=True, _diff=True, _numerical=True)287 assert rubi_test(rubi_integrate(f**(c*(a + b*x)**S(3))*x, x), x, a*(a + b*x)*Gamma(S(1)/​3, -c*(a + b*x)**S(3)*log(f))/​(S(3)*b**S(2)*(-c*(a + b*x)**S(3)*log(f))**(S(1)/​3)) - (a + b*x)**S(2)*Gamma(S(2)/​3, -c*(a + b*x)**S(3)*log(f))/​(S(3)*b**S(2)*(-c*(a + b*x)**S(3)*log(f))**(S(2)/​3)), expand=True, _diff=True, _numerical=True)288 assert rubi_test(rubi_integrate(f**(c*(a + b*x)**S(3)), x), x, (-a/​S(3) - b*x/​S(3))*Gamma(S(1)/​3, -c*(a + b*x)**S(3)*log(f))/​(b*(-c*(a + b*x)**S(3)*log(f))**(S(1)/​3)), expand=True, _diff=True, _numerical=True)289 assert rubi_test(rubi_integrate(f**(c*(a + b*x)**S(3))/​x, x), x, Integral(f**(c*(a + b*x)**S(3))/​x, x), expand=True, _diff=True, _numerical=True)290 assert rubi_test(rubi_integrate(f**(c*(a + b*x)**S(3))/​x**S(2), x), x, S(3)*a**S(2)*b*c*log(f)*Integral(f**(c*(a + b*x)**S(3))/​x, x) - a*b*c*(a + b*x)*Gamma(S(1)/​3, -c*(a + b*x)**S(3)*log(f))*log(f)/​(-c*(a + b*x)**S(3)*log(f))**(S(1)/​3) - b*c*(a + b*x)**S(2)*Gamma(S(2)/​3, -c*(a + b*x)**S(3)*log(f))*log(f)/​(-c*(a + b*x)**S(3)*log(f))**(S(2)/​3) - f**(c*(a + b*x)**S(3))/​x, expand=True, _diff=True, _numerical=True)291 # difference in simplify of sympy and mathematica assert rubi_test(rubi_integrate(f**(c*(a + b*x)**S(3))/​x**S(3), x), x, S(9)*a**S(4)*b**S(2)*c**S(2)*log(f)**S(2)*Integral(f**(c*(a + b*x)**S(3))/​x, x)/​S(2) - S(3)*a**S(3)*b**S(2)*c**S(2)*(a + b*x)*Gamma(S(1)/​3, -c*(a + b*x)**S(3)*log(f))*log(f)**S(2)/​(S(2)*(-c*(a + b*x)**S(3)*log(f))**(S(1)/​3)) - S(3)*a**S(2)*b**S(2)*c**S(2)*(a + b*x)**S(2)*Gamma(S(2)/​3, -c*(a + b*x)**S(3)*log(f))*log(f)**S(2)/​(S(2)*(-c*(a + b*x)**S(3)*log(f))**(S(2)/​3)) - S(3)*a**S(2)*b*c*f**(c*(a + b*x)**S(3))*log(f)/​(S(2)*x) + S(3)*a*b**S(2)*c*log(f)*Integral(f**(c*(a + b*x)**S(3))/​x, x) - b**S(2)*c*(a + b*x)*Gamma(S(1)/​3, -c*(a + b*x)**S(3)*log(f))*log(f)/​(S(2)*(-c*(a + b*x)**S(3)*log(f))**(S(1)/​3)) - f**(c*(a + b*x)**S(3))/​(S(2)*x**S(2)), expand=True, _diff=True, _numerical=True)292 assert rubi_test(rubi_integrate(x**m*exp(a**S(3) + S(3)*a**S(2)*b*x + S(3)*a*b**S(2)*x**S(2) + b**S(3)*x**S(3)), x), x, Integral(x**m*exp(a**S(3) + S(3)*a**S(2)*b*x + S(3)*a*b**S(2)*x**S(2) + b**S(3)*x**S(3)), x), expand=True, _diff=True, _numerical=True)293 assert rubi_test(rubi_integrate(x**S(4)*exp(a**S(3) + S(3)*a**S(2)*b*x + S(3)*a*b**S(2)*x**S(2) + b**S(3)*x**S(3)), x), x, -a**S(4)*(a + b*x)*Gamma(S(1)/​3, -(a + b*x)**S(3))/​(S(3)*b**S(5)*(-(a + b*x)**S(3))**(S(1)/​3)) + S(4)*a**S(3)*(a + b*x)**S(2)*Gamma(S(2)/​3, -(a + b*x)**S(3))/​(S(3)*b**S(5)*(-(a + b*x)**S(3))**(S(2)/​3)) + S(2)*a**S(2)*exp((a + b*x)**S(3))/​b**S(5) + S(4)*a*(a + b*x)**S(4)*Gamma(S(4)/​3, -(a + b*x)**S(3))/​(S(3)*b**S(5)*(-(a + b*x)**S(3))**(S(4)/​3)) - (a + b*x)**S(5)*Gamma(S(5)/​3, -(a + b*x)**S(3))/​(S(3)*b**S(5)*(-(a + b*x)**S(3))**(S(5)/​3)), expand=True, _diff=True, _numerical=True)294 assert rubi_test(rubi_integrate(x**S(3)*exp(a**S(3) + S(3)*a**S(2)*b*x + S(3)*a*b**S(2)*x**S(2) + b**S(3)*x**S(3)), x), x, a**S(3)*(a + b*x)*Gamma(S(1)/​3, -(a + b*x)**S(3))/​(S(3)*b**S(4)*(-(a + b*x)**S(3))**(S(1)/​3)) - a**S(2)*(a + b*x)**S(2)*Gamma(S(2)/​3, -(a + b*x)**S(3))/​(b**S(4)*(-(a + b*x)**S(3))**(S(2)/​3)) - a*exp((a + b*x)**S(3))/​b**S(4) - (a + b*x)**S(4)*Gamma(S(4)/​3, -(a + b*x)**S(3))/​(S(3)*b**S(4)*(-(a + b*x)**S(3))**(S(4)/​3)), expand=True, _diff=True, _numerical=True)295 assert rubi_test(rubi_integrate(x**S(2)*exp(a**S(3) + S(3)*a**S(2)*b*x + S(3)*a*b**S(2)*x**S(2) + b**S(3)*x**S(3)), x), x, -a**S(2)*(a + b*x)*Gamma(S(1)/​3, -(a + b*x)**S(3))/​(S(3)*b**S(3)*(-(a + b*x)**S(3))**(S(1)/​3)) + S(2)*a*(a + b*x)**S(2)*Gamma(S(2)/​3, -(a + b*x)**S(3))/​(S(3)*b**S(3)*(-(a + b*x)**S(3))**(S(2)/​3)) + exp((a + b*x)**S(3))/​(S(3)*b**S(3)), expand=True, _diff=True, _numerical=True)296 assert rubi_test(rubi_integrate(x*exp(a**S(3) + S(3)*a**S(2)*b*x + S(3)*a*b**S(2)*x**S(2) + b**S(3)*x**S(3)), x), x, a*(a + b*x)*Gamma(S(1)/​3, -(a + b*x)**S(3))/​(S(3)*b**S(2)*(-(a + b*x)**S(3))**(S(1)/​3)) - (a + b*x)**S(2)*Gamma(S(2)/​3, -(a + b*x)**S(3))/​(S(3)*b**S(2)*(-(a + b*x)**S(3))**(S(2)/​3)), expand=True, _diff=True, _numerical=True)297 assert rubi_test(rubi_integrate(exp(a**S(3) + S(3)*a**S(2)*b*x + S(3)*a*b**S(2)*x**S(2) + b**S(3)*x**S(3)), x), x, (-a/​S(3) - b*x/​S(3))*Gamma(S(1)/​3, -(a + b*x)**S(3))/​(b*(-(a + b*x)**S(3))**(S(1)/​3)), expand=True, _diff=True, _numerical=True)298 assert rubi_test(rubi_integrate(exp(a**S(3) + S(3)*a**S(2)*b*x + S(3)*a*b**S(2)*x**S(2) + b**S(3)*x**S(3))/​x, x), x, Integral(exp(a**S(3) + S(3)*a**S(2)*b*x + S(3)*a*b**S(2)*x**S(2) + b**S(3)*x**S(3))/​x, x), expand=True, _diff=True, _numerical=True)299 assert rubi_test(rubi_integrate(exp(sqrt(S(3)*x + S(5))), x), x, S(2)*sqrt(S(3)*x + S(5))*exp(sqrt(S(3)*x + S(5)))/​S(3) - S(2)*exp(sqrt(S(3)*x + S(5)))/​S(3), expand=True, _diff=True, _numerical=True)300 assert rubi_test(rubi_integrate(f**(c/​(a + b*x))*x**m, x), x, Integral(f**(c/​(a + b*x))*x**m, x), expand=True, _diff=True, _numerical=True)301 assert rubi_test(rubi_integrate(f**(c/​(a + b*x))*x**S(4), x), x, -a**S(4)*c*log(f)*Ei(c*log(f)/​(a + b*x))/​b**S(5) + a**S(4)*f**(c/​(a + b*x))*(a + b*x)/​b**S(5) + S(2)*a**S(3)*c**S(2)*log(f)**S(2)*Ei(c*log(f)/​(a + b*x))/​b**S(5) - S(2)*a**S(3)*c*f**(c/​(a + b*x))*(a + b*x)*log(f)/​b**S(5) - S(2)*a**S(3)*f**(c/​(a + b*x))*(a + b*x)**S(2)/​b**S(5) - a**S(2)*c**S(3)*log(f)**S(3)*Ei(c*log(f)/​(a + b*x))/​b**S(5) + a**S(2)*c**S(2)*f**(c/​(a + b*x))*(a + b*x)*log(f)**S(2)/​b**S(5) + a**S(2)*c*f**(c/​(a + b*x))*(a + b*x)**S(2)*log(f)/​b**S(5) + S(2)*a**S(2)*f**(c/​(a + b*x))*(a + b*x)**S(3)/​b**S(5) - S(4)*a*c**S(4)*Gamma(S(-4), -c*log(f)/​(a + b*x))*log(f)**S(4)/​b**S(5) - c**S(5)*Gamma(S(-5), -c*log(f)/​(a + b*x))*log(f)**S(5)/​b**S(5), expand=True, _diff=True, _numerical=True)302 assert rubi_test(rubi_integrate(f**(c/​(a + b*x))*x**S(3), x), x, a**S(3)*c*log(f)*Ei(c*log(f)/​(a + b*x))/​b**S(4) - a**S(3)*f**(c/​(a + b*x))*(a + b*x)/​b**S(4) - S(3)*a**S(2)*c**S(2)*log(f)**S(2)*Ei(c*log(f)/​(a + b*x))/​(S(2)*b**S(4)) + S(3)*a**S(2)*c*f**(c/​(a + b*x))*(a + b*x)*log(f)/​(S(2)*b**S(4)) + S(3)*a**S(2)*f**(c/​(a + b*x))*(a + b*x)**S(2)/​(S(2)*b**S(4)) + a*c**S(3)*log(f)**S(3)*Ei(c*log(f)/​(a + b*x))/​(S(2)*b**S(4)) - a*c**S(2)*f**(c/​(a + b*x))*(a + b*x)*log(f)**S(2)/​(S(2)*b**S(4)) - a*c*f**(c/​(a + b*x))*(a + b*x)**S(2)*log(f)/​(S(2)*b**S(4)) - a*f**(c/​(a + b*x))*(a + b*x)**S(3)/​b**S(4) + c**S(4)*Gamma(S(-4), -c*log(f)/​(a + b*x))*log(f)**S(4)/​b**S(4), expand=True, _diff=True, _numerical=True)303 assert rubi_test(rubi_integrate(f**(c/​(a + b*x))*x**S(2), x), x, -a**S(2)*c*log(f)*Ei(c*log(f)/​(a + b*x))/​b**S(3) + a**S(2)*f**(c/​(a + b*x))*(a + b*x)/​b**S(3) + a*c**S(2)*log(f)**S(2)*Ei(c*log(f)/​(a + b*x))/​b**S(3) - a*c*f**(c/​(a + b*x))*(a + b*x)*log(f)/​b**S(3) - a*f**(c/​(a + b*x))*(a + b*x)**S(2)/​b**S(3) - c**S(3)*log(f)**S(3)*Ei(c*log(f)/​(a + b*x))/​(S(6)*b**S(3)) + c**S(2)*f**(c/​(a + b*x))*(a + b*x)*log(f)**S(2)/​(S(6)*b**S(3)) + c*f**(c/​(a + b*x))*(a + b*x)**S(2)*log(f)/​(S(6)*b**S(3)) + f**(c/​(a + b*x))*(a + b*x)**S(3)/​(S(3)*b**S(3)), expand=True, _diff=True, _numerical=True)304 assert rubi_test(rubi_integrate(f**(c/​(a + b*x))*x, x), x, a*c*log(f)*Ei(c*log(f)/​(a + b*x))/​b**S(2) - a*f**(c/​(a + b*x))*(a + b*x)/​b**S(2) - c**S(2)*log(f)**S(2)*Ei(c*log(f)/​(a + b*x))/​(S(2)*b**S(2)) + c*f**(c/​(a + b*x))*(a + b*x)*log(f)/​(S(2)*b**S(2)) + f**(c/​(a + b*x))*(a + b*x)**S(2)/​(S(2)*b**S(2)), expand=True, _diff=True, _numerical=True)305 assert rubi_test(rubi_integrate(f**(c/​(a + b*x)), x), x, -c*log(f)*Ei(c*log(f)/​(a + b*x))/​b + f**(c/​(a + b*x))*(a + b*x)/​b, expand=True, _diff=True, _numerical=True)306 assert rubi_test(rubi_integrate(f**(c/​(a + b*x))/​x, x), x, f**(c/​a)*Ei(-b*c*x*log(f)/​(a*(a + b*x))) - Ei(c*log(f)/​(a + b*x)), expand=True, _diff=True, _numerical=True)307 assert rubi_test(rubi_integrate(f**(c/​(a + b*x))/​x**S(2), x), x, -f**(c/​(a + b*x))/​x - b*f**(c/​(a + b*x))/​a - b*c*f**(c/​a)*log(f)*Ei(-b*c*x*log(f)/​(a*(a + b*x)))/​a**S(2), expand=True, _diff=True, _numerical=True)308 assert rubi_test(rubi_integrate(f**(c/​(a + b*x))/​x**S(3), x), x, -f**(c/​(a + b*x))/​(S(2)*x**S(2)) + b**S(2)*f**(c/​(a + b*x))/​(S(2)*a**S(2)) + b*c*f**(c/​(a + b*x))*log(f)/​(S(2)*a**S(2)*x) + b**S(2)*c*f**(c/​a)*log(f)*Ei(-b*c*x*log(f)/​(a*(a + b*x)))/​a**S(3) + b**S(2)*c*f**(c/​(a + b*x))*log(f)/​(S(2)*a**S(3)) + b**S(2)*c**S(2)*f**(c/​a)*log(f)**S(2)*Ei(-b*c*x*log(f)/​(a*(a + b*x)))/​(S(2)*a**S(4)), expand=True, _diff=True, _numerical=True)309 assert rubi_test(rubi_integrate(f**(c/​(a + b*x)**S(2))*x**m, x), x, Integral(f**(c/​(a + b*x)**S(2))*x**m, x), expand=True, _diff=True, _numerical=True)310 assert rubi_test(rubi_integrate(f**(c/​(a + b*x)**S(2))*x**S(4), x), x, -sqrt(pi)*a**S(4)*sqrt(c)*sqrt(log(f))*erfi(sqrt(c)*sqrt(log(f))/​(a + b*x))/​b**S(5) + a**S(4)*f**(c/​(a + b*x)**S(2))*(a + b*x)/​b**S(5) + S(2)*a**S(3)*c*log(f)*Ei(c*log(f)/​(a + b*x)**S(2))/​b**S(5) - S(2)*a**S(3)*f**(c/​(a + b*x)**S(2))*(a + b*x)**S(2)/​b**S(5) - S(4)*sqrt(pi)*a**S(2)*c**(S(3)/​2)*log(f)**(S(3)/​2)*erfi(sqrt(c)*sqrt(log(f))/​(a + b*x))/​b**S(5) + S(4)*a**S(2)*c*f**(c/​(a + b*x)**S(2))*(a + b*x)*log(f)/​b**S(5) + S(2)*a**S(2)*f**(c/​(a + b*x)**S(2))*(a + b*x)**S(3)/​b**S(5) + a*c**S(2)*log(f)**S(2)*Ei(c*log(f)/​(a + b*x)**S(2))/​b**S(5) - a*c*f**(c/​(a + b*x)**S(2))*(a + b*x)**S(2)*log(f)/​b**S(5) - a*f**(c/​(a + b*x)**S(2))*(a + b*x)**S(4)/​b**S(5) - S(4)*sqrt(pi)*c**(S(5)/​2)*log(f)**(S(5)/​2)*erfi(sqrt(c)*sqrt(log(f))/​(a + b*x))/​(S(15)*b**S(5)) + S(4)*c**S(2)*f**(c/​(a + b*x)**S(2))*(a + b*x)*log(f)**S(2)/​(S(15)*b**S(5)) + S(2)*c*f**(c/​(a + b*x)**S(2))*(a + b*x)**S(3)*log(f)/​(S(15)*b**S(5)) + f**(c/​(a + b*x)**S(2))*(a + b*x)**S(5)/​(S(5)*b**S(5)), expand=True, _diff=True, _numerical=True)311 assert rubi_test(rubi_integrate(f**(c/​(a + b*x)**S(2))*x**S(3), x), x, sqrt(pi)*a**S(3)*sqrt(c)*sqrt(log(f))*erfi(sqrt(c)*sqrt(log(f))/​(a + b*x))/​b**S(4) - a**S(3)*f**(c/​(a + b*x)**S(2))*(a + b*x)/​b**S(4) - S(3)*a**S(2)*c*log(f)*Ei(c*log(f)/​(a + b*x)**S(2))/​(S(2)*b**S(4)) + S(3)*a**S(2)*f**(c/​(a + b*x)**S(2))*(a + b*x)**S(2)/​(S(2)*b**S(4)) + S(2)*sqrt(pi)*a*c**(S(3)/​2)*log(f)**(S(3)/​2)*erfi(sqrt(c)*sqrt(log(f))/​(a + b*x))/​b**S(4) - S(2)*a*c*f**(c/​(a + b*x)**S(2))*(a + b*x)*log(f)/​b**S(4) - a*f**(c/​(a + b*x)**S(2))*(a + b*x)**S(3)/​b**S(4) - c**S(2)*log(f)**S(2)*Ei(c*log(f)/​(a + b*x)**S(2))/​(S(4)*b**S(4)) + c*f**(c/​(a + b*x)**S(2))*(a + b*x)**S(2)*log(f)/​(S(4)*b**S(4)) + f**(c/​(a + b*x)**S(2))*(a + b*x)**S(4)/​(S(4)*b**S(4)), expand=True, _diff=True, _numerical=True)312 assert rubi_test(rubi_integrate(f**(c/​(a + b*x)**S(2))*x**S(2), x), x, -sqrt(pi)*a**S(2)*sqrt(c)*sqrt(log(f))*erfi(sqrt(c)*sqrt(log(f))/​(a + b*x))/​b**S(3) + a**S(2)*f**(c/​(a + b*x)**S(2))*(a + b*x)/​b**S(3) + a*c*log(f)*Ei(c*log(f)/​(a + b*x)**S(2))/​b**S(3) - a*f**(c/​(a + b*x)**S(2))*(a + b*x)**S(2)/​b**S(3) - S(2)*sqrt(pi)*c**(S(3)/​2)*log(f)**(S(3)/​2)*erfi(sqrt(c)*sqrt(log(f))/​(a + b*x))/​(S(3)*b**S(3)) + S(2)*c*f**(c/​(a + b*x)**S(2))*(a + b*x)*log(f)/​(S(3)*b**S(3)) + f**(c/​(a + b*x)**S(2))*(a + b*x)**S(3)/​(S(3)*b**S(3)), expand=True, _diff=True, _numerical=True)313 assert rubi_test(rubi_integrate(f**(c/​(a + b*x)**S(2))*x, x), x, sqrt(pi)*a*sqrt(c)*sqrt(log(f))*erfi(sqrt(c)*sqrt(log(f))/​(a + b*x))/​b**S(2) - a*f**(c/​(a + b*x)**S(2))*(a + b*x)/​b**S(2) - c*log(f)*Ei(c*log(f)/​(a + b*x)**S(2))/​(S(2)*b**S(2)) + f**(c/​(a + b*x)**S(2))*(a + b*x)**S(2)/​(S(2)*b**S(2)), expand=True, _diff=True, _numerical=True)314 assert rubi_test(rubi_integrate(f**(c/​(a + b*x)**S(2)), x), x, -sqrt(pi)*sqrt(c)*sqrt(log(f))*erfi(sqrt(c)*sqrt(log(f))/​(a + b*x))/​b + f**(c/​(a + b*x)**S(2))*(a + b*x)/​b, expand=True, _diff=True, _numerical=True)315 assert rubi_test(rubi_integrate(f**(c/​(a + b*x)**S(2))/​x, x), x, Integral(f**(c/​(a + b*x)**S(2))/​x, x), expand=True, _diff=True, _numerical=True)316 assert rubi_test(rubi_integrate(f**(c/​(a + b*x)**S(2))/​x**S(2), x), x, Integral(f**(c/​(a + b*x)**S(2))/​x**S(2), x), expand=True, _diff=True, _numerical=True)317 assert rubi_test(rubi_integrate(f**(c/​(a + b*x)**S(2))/​x**S(3), x), x, Integral(f**(c/​(a + b*x)**S(2))/​x**S(3), x), expand=True, _diff=True, _numerical=True)318 assert rubi_test(rubi_integrate(f**(c/​(a + b*x)**S(3))*x**m, x), x, Integral(f**(c/​(a + b*x)**S(3))*x**m, x), expand=True, _diff=True, _numerical=True)319 assert rubi_test(rubi_integrate(f**(c/​(a + b*x)**S(3))*x**S(4), x), x, a**S(4)*(-c*log(f)/​(a + b*x)**S(3))**(S(1)/​3)*(a + b*x)*Gamma(S(-1)/​3, -c*log(f)/​(a + b*x)**S(3))/​(S(3)*b**S(5)) - S(4)*a**S(3)*(-c*log(f)/​(a + b*x)**S(3))**(S(2)/​3)*(a + b*x)**S(2)*Gamma(S(-2)/​3, -c*log(f)/​(a + b*x)**S(3))/​(S(3)*b**S(5)) - S(2)*a**S(2)*c*log(f)*Ei(c*log(f)/​(a + b*x)**S(3))/​b**S(5) + S(2)*a**S(2)*f**(c/​(a + b*x)**S(3))*(a + b*x)**S(3)/​b**S(5) - S(4)*a*(-c*log(f)/​(a + b*x)**S(3))**(S(4)/​3)*(a + b*x)**S(4)*Gamma(S(-4)/​3, -c*log(f)/​(a + b*x)**S(3))/​(S(3)*b**S(5)) + (-c*log(f)/​(a + b*x)**S(3))**(S(5)/​3)*(a + b*x)**S(5)*Gamma(S(-5)/​3, -c*log(f)/​(a + b*x)**S(3))/​(S(3)*b**S(5)), expand=True, _diff=True, _numerical=True)320 assert rubi_test(rubi_integrate(f**(c/​(a + b*x)**S(3))*x**S(3), x), x, -a**S(3)*(-c*log(f)/​(a + b*x)**S(3))**(S(1)/​3)*(a + b*x)*Gamma(S(-1)/​3, -c*log(f)/​(a + b*x)**S(3))/​(S(3)*b**S(4)) + a**S(2)*(-c*log(f)/​(a + b*x)**S(3))**(S(2)/​3)*(a + b*x)**S(2)*Gamma(S(-2)/​3, -c*log(f)/​(a + b*x)**S(3))/​b**S(4) + a*c*log(f)*Ei(c*log(f)/​(a + b*x)**S(3))/​b**S(4) - a*f**(c/​(a + b*x)**S(3))*(a + b*x)**S(3)/​b**S(4) + (-c*log(f)/​(a + b*x)**S(3))**(S(4)/​3)*(a + b*x)**S(4)*Gamma(S(-4)/​3, -c*log(f)/​(a + b*x)**S(3))/​(S(3)*b**S(4)), expand=True, _diff=True, _numerical=True)321 assert rubi_test(rubi_integrate(f**(c/​(a + b*x)**S(3))*x**S(2), x), x, a**S(2)*(-c*log(f)/​(a + b*x)**S(3))**(S(1)/​3)*(a + b*x)*Gamma(S(-1)/​3, -c*log(f)/​(a + b*x)**S(3))/​(S(3)*b**S(3)) - S(2)*a*(-c*log(f)/​(a + b*x)**S(3))**(S(2)/​3)*(a + b*x)**S(2)*Gamma(S(-2)/​3, -c*log(f)/​(a + b*x)**S(3))/​(S(3)*b**S(3)) - c*log(f)*Ei(c*log(f)/​(a + b*x)**S(3))/​(S(3)*b**S(3)) + f**(c/​(a + b*x)**S(3))*(a + b*x)**S(3)/​(S(3)*b**S(3)), expand=True, _diff=True, _numerical=True)322 assert rubi_test(rubi_integrate(f**(c/​(a + b*x)**S(3))*x, x), x, -a*(-c*log(f)/​(a + b*x)**S(3))**(S(1)/​3)*(a + b*x)*Gamma(S(-1)/​3, -c*log(f)/​(a + b*x)**S(3))/​(S(3)*b**S(2)) + (-c*log(f)/​(a + b*x)**S(3))**(S(2)/​3)*(a + b*x)**S(2)*Gamma(S(-2)/​3, -c*log(f)/​(a + b*x)**S(3))/​(S(3)*b**S(2)), expand=True, _diff=True, _numerical=True)323 assert rubi_test(rubi_integrate(f**(c/​(a + b*x)**S(3)), x), x, (-c*log(f)/​(a + b*x)**S(3))**(S(1)/​3)*(a/​S(3) + b*x/​S(3))*Gamma(S(-1)/​3, -c*log(f)/​(a + b*x)**S(3))/​b, expand=True, _diff=True, _numerical=True)324 assert rubi_test(rubi_integrate(f**(c/​(a + b*x)**S(3))/​x, x), x, Integral(f**(c/​(a + b*x)**S(3))/​x, x), expand=True, _diff=True, _numerical=True)325 assert rubi_test(rubi_integrate(f**(c/​(a + b*x)**S(3))/​x**S(2), x), x, Integral(f**(c/​(a + b*x)**S(3))/​x**S(2), x), expand=True, _diff=True, _numerical=True)326 assert rubi_test(rubi_integrate(f**(c/​(a + b*x)**S(3))/​x**S(3), x), x, Integral(f**(c/​(a + b*x)**S(3))/​x**S(3), x), expand=True, _diff=True, _numerical=True)327 assert rubi_test(rubi_integrate(f**(c*(a + b*x)**n)*x**m, x), x, Integral(f**(c*(a + b*x)**n)*x**m, x), expand=True, _diff=True, _numerical=True)328 assert rubi_test(rubi_integrate(f**(c*(a + b*x)**n)*x**S(3), x), x, a**S(3)*(-c*(a + b*x)**n*log(f))**(-S(1)/​n)*(a + b*x)*Gamma(S(1)/​n, -c*(a + b*x)**n*log(f))/​(b**S(4)*n) - S(3)*a**S(2)*(-c*(a + b*x)**n*log(f))**(-S(2)/​n)*(a + b*x)**S(2)*Gamma(S(2)/​n, -c*(a + b*x)**n*log(f))/​(b**S(4)*n) + S(3)*a*(-c*(a + b*x)**n*log(f))**(-S(3)/​n)*(a + b*x)**S(3)*Gamma(S(3)/​n, -c*(a + b*x)**n*log(f))/​(b**S(4)*n) - (-c*(a + b*x)**n*log(f))**(-S(4)/​n)*(a + b*x)**S(4)*Gamma(S(4)/​n, -c*(a + b*x)**n*log(f))/​(b**S(4)*n), expand=True, _diff=True, _numerical=True)329 assert rubi_test(rubi_integrate(f**(c*(a + b*x)**n)*x**S(2), x), x, -a**S(2)*(-c*(a + b*x)**n*log(f))**(-S(1)/​n)*(a + b*x)*Gamma(S(1)/​n, -c*(a + b*x)**n*log(f))/​(b**S(3)*n) + S(2)*a*(-c*(a + b*x)**n*log(f))**(-S(2)/​n)*(a + b*x)**S(2)*Gamma(S(2)/​n, -c*(a + b*x)**n*log(f))/​(b**S(3)*n) - (-c*(a + b*x)**n*log(f))**(-S(3)/​n)*(a + b*x)**S(3)*Gamma(S(3)/​n, -c*(a + b*x)**n*log(f))/​(b**S(3)*n), expand=True, _diff=True, _numerical=True)330 assert rubi_test(rubi_integrate(f**(c*(a + b*x)**n)*x, x), x, a*(-c*(a + b*x)**n*log(f))**(-S(1)/​n)*(a + b*x)*Gamma(S(1)/​n, -c*(a + b*x)**n*log(f))/​(b**S(2)*n) - (-c*(a + b*x)**n*log(f))**(-S(2)/​n)*(a + b*x)**S(2)*Gamma(S(2)/​n, -c*(a + b*x)**n*log(f))/​(b**S(2)*n), expand=True, _diff=True, _numerical=True)331 assert rubi_test(rubi_integrate(f**(c*(a + b*x)**n), x), x, (-c*(a + b*x)**n*log(f))**(-S(1)/​n)*(-a - b*x)*Gamma(S(1)/​n, -c*(a + b*x)**n*log(f))/​(b*n), expand=True, _diff=True, _numerical=True)332 assert rubi_test(rubi_integrate(f**(c*(a + b*x)**n)/​x, x), x, Integral(f**(c*(a + b*x)**n)/​x, x), expand=True, _diff=True, _numerical=True)333 assert rubi_test(rubi_integrate(f**(c*(a + b*x)**n)/​x**S(2), x), x, Integral(f**(c*(a + b*x)**n)/​x**S(2), x), expand=True, _diff=True, _numerical=True)334 assert rubi_test(rubi_integrate(f**(c*(a + b*x)**n)/​x**S(3), x), x, Integral(f**(c*(a + b*x)**n)/​x**S(3), x), expand=True, _diff=True, _numerical=True)335 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))*(c + d*x)**m, x), x, -F**a*(-b*(c + d*x)**S(2)*log(F))**(-m/​S(2) + S(-1)/​2)*(c + d*x)**(m + S(1))*Gamma(m/​S(2) + S(1)/​2, -b*(c + d*x)**S(2)*log(F))/​(S(2)*d), expand=True, _diff=True, _numerical=True)336 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(11), x), x, -F**a*Gamma(S(6), -b*(c + d*x)**S(2)*log(F))/​(S(2)*b**S(6)*d*log(F)**S(6)), expand=True, _diff=True, _numerical=True)337 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(9), x), x, F**a*Gamma(S(5), -b*(c + d*x)**S(2)*log(F))/​(S(2)*b**S(5)*d*log(F)**S(5)), expand=True, _diff=True, _numerical=True)338 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(7), x), x, F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(6)/​(S(2)*b*d*log(F)) - S(3)*F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(4)/​(S(2)*b**S(2)*d*log(F)**S(2)) + S(3)*F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(2)/​(b**S(3)*d*log(F)**S(3)) - S(3)*F**(a + b*(c + d*x)**S(2))/​(b**S(4)*d*log(F)**S(4)), expand=True, _diff=True, _numerical=True)339 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(5), x), x, F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(4)/​(S(2)*b*d*log(F)) - F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(2)/​(b**S(2)*d*log(F)**S(2)) + F**(a + b*(c + d*x)**S(2))/​(b**S(3)*d*log(F)**S(3)), expand=True, _diff=True, _numerical=True)340 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(3), x), x, F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(2)/​(S(2)*b*d*log(F)) - F**(a + b*(c + d*x)**S(2))/​(S(2)*b**S(2)*d*log(F)**S(2)), expand=True, _diff=True, _numerical=True)341 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))*(c + d*x), x), x, F**(a + b*(c + d*x)**S(2))/​(S(2)*b*d*log(F)), expand=True, _diff=True, _numerical=True)342 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))/​(c + d*x), x), x, F**a*Ei(b*(c + d*x)**S(2)*log(F))/​(S(2)*d), expand=True, _diff=True, _numerical=True)343 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))/​(c + d*x)**S(3), x), x, F**a*b*log(F)*Ei(b*(c + d*x)**S(2)*log(F))/​(S(2)*d) - F**(a + b*(c + d*x)**S(2))/​(S(2)*d*(c + d*x)**S(2)), expand=True, _diff=True, _numerical=True)344 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))/​(c + d*x)**S(5), x), x, F**a*b**S(2)*log(F)**S(2)*Ei(b*(c + d*x)**S(2)*log(F))/​(S(4)*d) - F**(a + b*(c + d*x)**S(2))*b*log(F)/​(S(4)*d*(c + d*x)**S(2)) - F**(a + b*(c + d*x)**S(2))/​(S(4)*d*(c + d*x)**S(4)), expand=True, _diff=True, _numerical=True)345 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))/​(c + d*x)**S(7), x), x, F**a*b**S(3)*log(F)**S(3)*Ei(b*(c + d*x)**S(2)*log(F))/​(S(12)*d) - F**(a + b*(c + d*x)**S(2))*b**S(2)*log(F)**S(2)/​(S(12)*d*(c + d*x)**S(2)) - F**(a + b*(c + d*x)**S(2))*b*log(F)/​(S(12)*d*(c + d*x)**S(4)) - F**(a + b*(c + d*x)**S(2))/​(S(6)*d*(c + d*x)**S(6)), expand=True, _diff=True, _numerical=True)346 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))/​(c + d*x)**S(9), x), x, -F**a*b**S(4)*Gamma(S(-4), -b*(c + d*x)**S(2)*log(F))*log(F)**S(4)/​(S(2)*d), expand=True, _diff=True, _numerical=True)347 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))/​(c + d*x)**S(11), x), x, F**a*b**S(5)*Gamma(S(-5), -b*(c + d*x)**S(2)*log(F))*log(F)**S(5)/​(S(2)*d), expand=True, _diff=True, _numerical=True)348 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(12), x), x, -F**a*(c + d*x)**S(13)*Gamma(S(13)/​2, -b*(c + d*x)**S(2)*log(F))/​(S(2)*d*(-b*(c + d*x)**S(2)*log(F))**(S(13)/​2)), expand=True, _diff=True, _numerical=True)349 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(10), x), x, -F**a*(c + d*x)**S(11)*Gamma(S(11)/​2, -b*(c + d*x)**S(2)*log(F))/​(S(2)*d*(-b*(c + d*x)**S(2)*log(F))**(S(11)/​2)), expand=True, _diff=True, _numerical=True)350 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(8), x), x, S(105)*sqrt(pi)*F**a*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​(S(32)*b**(S(9)/​2)*d*log(F)**(S(9)/​2)) + F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(7)/​(S(2)*b*d*log(F)) - S(7)*F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(5)/​(S(4)*b**S(2)*d*log(F)**S(2)) + S(35)*F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(3)/​(S(8)*b**S(3)*d*log(F)**S(3)) - S(105)*F**(a + b*(c + d*x)**S(2))*(c + d*x)/​(S(16)*b**S(4)*d*log(F)**S(4)), expand=True, _diff=True, _numerical=True)351 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(6), x), x, -S(15)*sqrt(pi)*F**a*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​(S(16)*b**(S(7)/​2)*d*log(F)**(S(7)/​2)) + F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(5)/​(S(2)*b*d*log(F)) - S(5)*F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(3)/​(S(4)*b**S(2)*d*log(F)**S(2)) + S(15)*F**(a + b*(c + d*x)**S(2))*(c + d*x)/​(S(8)*b**S(3)*d*log(F)**S(3)), expand=True, _diff=True, _numerical=True)352 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(4), x), x, S(3)*sqrt(pi)*F**a*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​(S(8)*b**(S(5)/​2)*d*log(F)**(S(5)/​2)) + F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(3)/​(S(2)*b*d*log(F)) - S(3)*F**(a + b*(c + d*x)**S(2))*(c + d*x)/​(S(4)*b**S(2)*d*log(F)**S(2)), expand=True, _diff=True, _numerical=True)353 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))*(c + d*x)**S(2), x), x, -sqrt(pi)*F**a*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​(S(4)*b**(S(3)/​2)*d*log(F)**(S(3)/​2)) + F**(a + b*(c + d*x)**S(2))*(c + d*x)/​(S(2)*b*d*log(F)), expand=True, _diff=True, _numerical=True)354 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2)), x), x, sqrt(pi)*F**a*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​(S(2)*sqrt(b)*d*sqrt(log(F))), expand=True, _diff=True, _numerical=True)355 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))/​(c + d*x)**S(2), x), x, sqrt(pi)*F**a*sqrt(b)*sqrt(log(F))*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​d - F**(a + b*(c + d*x)**S(2))/​(d*(c + d*x)), expand=True, _diff=True, _numerical=True)356 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))/​(c + d*x)**S(4), x), x, S(2)*sqrt(pi)*F**a*b**(S(3)/​2)*log(F)**(S(3)/​2)*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​(S(3)*d) - S(2)*F**(a + b*(c + d*x)**S(2))*b*log(F)/​(S(3)*d*(c + d*x)) - F**(a + b*(c + d*x)**S(2))/​(S(3)*d*(c + d*x)**S(3)), expand=True, _diff=True, _numerical=True)357 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))/​(c + d*x)**S(6), x), x, S(4)*sqrt(pi)*F**a*b**(S(5)/​2)*log(F)**(S(5)/​2)*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​(S(15)*d) - S(4)*F**(a + b*(c + d*x)**S(2))*b**S(2)*log(F)**S(2)/​(S(15)*d*(c + d*x)) - S(2)*F**(a + b*(c + d*x)**S(2))*b*log(F)/​(S(15)*d*(c + d*x)**S(3)) - F**(a + b*(c + d*x)**S(2))/​(S(5)*d*(c + d*x)**S(5)), expand=True, _diff=True, _numerical=True)358 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))/​(c + d*x)**S(8), x), x, S(8)*sqrt(pi)*F**a*b**(S(7)/​2)*log(F)**(S(7)/​2)*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​(S(105)*d) - S(8)*F**(a + b*(c + d*x)**S(2))*b**S(3)*log(F)**S(3)/​(S(105)*d*(c + d*x)) - S(4)*F**(a + b*(c + d*x)**S(2))*b**S(2)*log(F)**S(2)/​(S(105)*d*(c + d*x)**S(3)) - S(2)*F**(a + b*(c + d*x)**S(2))*b*log(F)/​(S(35)*d*(c + d*x)**S(5)) - F**(a + b*(c + d*x)**S(2))/​(S(7)*d*(c + d*x)**S(7)), expand=True, _diff=True, _numerical=True)359 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))/​(c + d*x)**S(10), x), x, -F**a*(-b*(c + d*x)**S(2)*log(F))**(S(9)/​2)*Gamma(S(-9)/​2, -b*(c + d*x)**S(2)*log(F))/​(S(2)*d*(c + d*x)**S(9)), expand=True, _diff=True, _numerical=True)360 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))/​(c + d*x)**S(12), x), x, -F**a*(-b*(c + d*x)**S(2)*log(F))**(S(11)/​2)*Gamma(S(-11)/​2, -b*(c + d*x)**S(2)*log(F))/​(S(2)*d*(c + d*x)**S(11)), expand=True, _diff=True, _numerical=True)361 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(3))*(c + d*x)**m, x), x, -F**a*(-b*(c + d*x)**S(3)*log(F))**(-m/​S(3) + S(-1)/​3)*(c + d*x)**(m + S(1))*Gamma(m/​S(3) + S(1)/​3, -b*(c + d*x)**S(3)*log(F))/​(S(3)*d), expand=True, _diff=True, _numerical=True)362 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(3))*(c + d*x)**S(17), x), x, -F**a*Gamma(S(6), -b*(c + d*x)**S(3)*log(F))/​(S(3)*b**S(6)*d*log(F)**S(6)), expand=True, _diff=True, _numerical=True)363 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(3))*(c + d*x)**S(14), x), x, F**a*Gamma(S(5), -b*(c + d*x)**S(3)*log(F))/​(S(3)*b**S(5)*d*log(F)**S(5)), expand=True, _diff=True, _numerical=True)364 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(3))*(c + d*x)**S(11), x), x, F**(a + b*(c + d*x)**S(3))*(c + d*x)**S(9)/​(S(3)*b*d*log(F)) - F**(a + b*(c + d*x)**S(3))*(c + d*x)**S(6)/​(b**S(2)*d*log(F)**S(2)) + S(2)*F**(a + b*(c + d*x)**S(3))*(c + d*x)**S(3)/​(b**S(3)*d*log(F)**S(3)) - S(2)*F**(a + b*(c + d*x)**S(3))/​(b**S(4)*d*log(F)**S(4)), expand=True, _diff=True, _numerical=True)365 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(3))*(c + d*x)**S(8), x), x, F**(a + b*(c + d*x)**S(3))*(c + d*x)**S(6)/​(S(3)*b*d*log(F)) - S(2)*F**(a + b*(c + d*x)**S(3))*(c + d*x)**S(3)/​(S(3)*b**S(2)*d*log(F)**S(2)) + S(2)*F**(a + b*(c + d*x)**S(3))/​(S(3)*b**S(3)*d*log(F)**S(3)), expand=True, _diff=True, _numerical=True)366 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(3))*(c + d*x)**S(5), x), x, F**(a + b*(c + d*x)**S(3))*(c + d*x)**S(3)/​(S(3)*b*d*log(F)) - F**(a + b*(c + d*x)**S(3))/​(S(3)*b**S(2)*d*log(F)**S(2)), expand=True, _diff=True, _numerical=True)367 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(3))*(c + d*x)**S(2), x), x, F**(a + b*(c + d*x)**S(3))/​(S(3)*b*d*log(F)), expand=True, _diff=True, _numerical=True)368 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(3))/​(c + d*x), x), x, F**a*Ei(b*(c + d*x)**S(3)*log(F))/​(S(3)*d), expand=True, _diff=True, _numerical=True)369 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(3))/​(c + d*x)**S(4), x), x, F**a*b*log(F)*Ei(b*(c + d*x)**S(3)*log(F))/​(S(3)*d) - F**(a + b*(c + d*x)**S(3))/​(S(3)*d*(c + d*x)**S(3)), expand=True, _diff=True, _numerical=True)370 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(3))/​(c + d*x)**S(7), x), x, F**a*b**S(2)*log(F)**S(2)*Ei(b*(c + d*x)**S(3)*log(F))/​(S(6)*d) - F**(a + b*(c + d*x)**S(3))*b*log(F)/​(S(6)*d*(c + d*x)**S(3)) - F**(a + b*(c + d*x)**S(3))/​(S(6)*d*(c + d*x)**S(6)), expand=True, _diff=True, _numerical=True)371 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(3))/​(c + d*x)**S(10), x), x, F**a*b**S(3)*log(F)**S(3)*Ei(b*(c + d*x)**S(3)*log(F))/​(S(18)*d) - F**(a + b*(c + d*x)**S(3))*b**S(2)*log(F)**S(2)/​(S(18)*d*(c + d*x)**S(3)) - F**(a + b*(c + d*x)**S(3))*b*log(F)/​(S(18)*d*(c + d*x)**S(6)) - F**(a + b*(c + d*x)**S(3))/​(S(9)*d*(c + d*x)**S(9)), expand=True, _diff=True, _numerical=True)372 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(3))/​(c + d*x)**S(13), x), x, -F**a*b**S(4)*Gamma(S(-4), -b*(c + d*x)**S(3)*log(F))*log(F)**S(4)/​(S(3)*d), expand=True, _diff=True, _numerical=True)373 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(3))/​(c + d*x)**S(16), x), x, F**a*b**S(5)*Gamma(S(-5), -b*(c + d*x)**S(3)*log(F))*log(F)**S(5)/​(S(3)*d), expand=True, _diff=True, _numerical=True)374 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(3))*(c + d*x)**S(3), x), x, -F**a*(c + d*x)**S(4)*Gamma(S(4)/​3, -b*(c + d*x)**S(3)*log(F))/​(S(3)*d*(-b*(c + d*x)**S(3)*log(F))**(S(4)/​3)), expand=True, _diff=True, _numerical=True)375 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(3))*(c + d*x), x), x, -F**a*(c + d*x)**S(2)*Gamma(S(2)/​3, -b*(c + d*x)**S(3)*log(F))/​(S(3)*d*(-b*(c + d*x)**S(3)*log(F))**(S(2)/​3)), expand=True, _diff=True, _numerical=True)376 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(3)), x), x, -F**a*(c + d*x)*Gamma(S(1)/​3, -b*(c + d*x)**S(3)*log(F))/​(S(3)*d*(-b*(c + d*x)**S(3)*log(F))**(S(1)/​3)), expand=True, _diff=True, _numerical=True)377 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(3))/​(c + d*x)**S(2), x), x, -F**a*(-b*(c + d*x)**S(3)*log(F))**(S(1)/​3)*Gamma(S(-1)/​3, -b*(c + d*x)**S(3)*log(F))/​(S(3)*d*(c + d*x)), expand=True, _diff=True, _numerical=True)378 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(3))/​(c + d*x)**S(3), x), x, -F**a*(-b*(c + d*x)**S(3)*log(F))**(S(2)/​3)*Gamma(S(-2)/​3, -b*(c + d*x)**S(3)*log(F))/​(S(3)*d*(c + d*x)**S(2)), expand=True, _diff=True, _numerical=True)379 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(3))/​(c + d*x)**S(5), x), x, -F**a*(-b*(c + d*x)**S(3)*log(F))**(S(4)/​3)*Gamma(S(-4)/​3, -b*(c + d*x)**S(3)*log(F))/​(S(3)*d*(c + d*x)**S(4)), expand=True, _diff=True, _numerical=True)380 assert rubi_test(rubi_integrate(f**(a + b*sqrt(c + d*x)), x), x, S(2)*f**(a + b*sqrt(c + d*x))*sqrt(c + d*x)/​(b*d*log(f)) - S(2)*f**(a + b*sqrt(c + d*x))/​(b**S(2)*d*log(f)**S(2)), expand=True, _diff=True, _numerical=True)381 assert rubi_test(rubi_integrate(f**(a + b*(c + d*x)**(S(1)/​3)), x), x, S(3)*f**(a + b*(c + d*x)**(S(1)/​3))*(c + d*x)**(S(2)/​3)/​(b*d*log(f)) - S(6)*f**(a + b*(c + d*x)**(S(1)/​3))*(c + d*x)**(S(1)/​3)/​(b**S(2)*d*log(f)**S(2)) + S(6)*f**(a + b*(c + d*x)**(S(1)/​3))/​(b**S(3)*d*log(f)**S(3)), expand=True, _diff=True, _numerical=True)382 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x))*(c + d*x)**m, x), x, F**a*(-b*log(F)/​(c + d*x))**(m + S(1))*(c + d*x)**(m + S(1))*Gamma(-m + S(-1), -b*log(F)/​(c + d*x))/​d, expand=True, _diff=True, _numerical=True)383 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x))*(c + d*x)**S(4), x), x, -F**a*b**S(5)*Gamma(S(-5), -b*log(F)/​(c + d*x))*log(F)**S(5)/​d, expand=True, _diff=True, _numerical=True)384 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x))*(c + d*x)**S(3), x), x, F**a*b**S(4)*Gamma(S(-4), -b*log(F)/​(c + d*x))*log(F)**S(4)/​d, expand=True, _diff=True, _numerical=True)385 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x))*(c + d*x)**S(2), x), x, -F**a*b**S(3)*log(F)**S(3)*Ei(b*log(F)/​(c + d*x))/​(S(6)*d) + F**(a + b/​(c + d*x))*b**S(2)*(c + d*x)*log(F)**S(2)/​(S(6)*d) + F**(a + b/​(c + d*x))*b*(c + d*x)**S(2)*log(F)/​(S(6)*d) + F**(a + b/​(c + d*x))*(c + d*x)**S(3)/​(S(3)*d), expand=True, _diff=True, _numerical=True)386 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x))*(c + d*x), x), x, -F**a*b**S(2)*log(F)**S(2)*Ei(b*log(F)/​(c + d*x))/​(S(2)*d) + F**(a + b/​(c + d*x))*b*(c + d*x)*log(F)/​(S(2)*d) + F**(a + b/​(c + d*x))*(c + d*x)**S(2)/​(S(2)*d), expand=True, _diff=True, _numerical=True)387 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)), x), x, -F**a*b*log(F)*Ei(b*log(F)/​(c + d*x))/​d + F**(a + b/​(c + d*x))*(c + d*x)/​d, expand=True, _diff=True, _numerical=True)388 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x))/​(c + d*x), x), x, -F**a*Ei(b*log(F)/​(c + d*x))/​d, expand=True, _diff=True, _numerical=True)389 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x))/​(c + d*x)**S(2), x), x, -F**(a + b/​(c + d*x))/​(b*d*log(F)), expand=True, _diff=True, _numerical=True)390 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x))/​(c + d*x)**S(3), x), x, -F**(a + b/​(c + d*x))/​(b*d*(c + d*x)*log(F)) + F**(a + b/​(c + d*x))/​(b**S(2)*d*log(F)**S(2)), expand=True, _diff=True, _numerical=True)391 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x))/​(c + d*x)**S(4), x), x, -F**(a + b/​(c + d*x))/​(b*d*(c + d*x)**S(2)*log(F)) + S(2)*F**(a + b/​(c + d*x))/​(b**S(2)*d*(c + d*x)*log(F)**S(2)) - S(2)*F**(a + b/​(c + d*x))/​(b**S(3)*d*log(F)**S(3)), expand=True, _diff=True, _numerical=True)392 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x))/​(c + d*x)**S(5), x), x, -F**(a + b/​(c + d*x))/​(b*d*(c + d*x)**S(3)*log(F)) + S(3)*F**(a + b/​(c + d*x))/​(b**S(2)*d*(c + d*x)**S(2)*log(F)**S(2)) - S(6)*F**(a + b/​(c + d*x))/​(b**S(3)*d*(c + d*x)*log(F)**S(3)) + S(6)*F**(a + b/​(c + d*x))/​(b**S(4)*d*log(F)**S(4)), expand=True, _diff=True, _numerical=True)393 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x))/​(c + d*x)**S(6), x), x, -F**a*Gamma(S(5), -b*log(F)/​(c + d*x))/​(b**S(5)*d*log(F)**S(5)), expand=True, _diff=True, _numerical=True)394 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x))/​(c + d*x)**S(7), x), x, F**a*Gamma(S(6), -b*log(F)/​(c + d*x))/​(b**S(6)*d*log(F)**S(6)), expand=True, _diff=True, _numerical=True)395 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))*(c + d*x)**m, x), x, F**a*(-b*log(F)/​(c + d*x)**S(2))**(m/​S(2) + S(1)/​2)*(c + d*x)**(m + S(1))*Gamma(-m/​S(2) + S(-1)/​2, -b*log(F)/​(c + d*x)**S(2))/​(S(2)*d), expand=True, _diff=True, _numerical=True)396 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))*(c + d*x)**S(9), x), x, -F**a*b**S(5)*Gamma(S(-5), -b*log(F)/​(c + d*x)**S(2))*log(F)**S(5)/​(S(2)*d), expand=True, _diff=True, _numerical=True)397 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))*(c + d*x)**S(7), x), x, F**a*b**S(4)*Gamma(S(-4), -b*log(F)/​(c + d*x)**S(2))*log(F)**S(4)/​(S(2)*d), expand=True, _diff=True, _numerical=True)398 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))*(c + d*x)**S(5), x), x, -F**a*b**S(3)*log(F)**S(3)*Ei(b*log(F)/​(c + d*x)**S(2))/​(S(12)*d) + F**(a + b/​(c + d*x)**S(2))*b**S(2)*(c + d*x)**S(2)*log(F)**S(2)/​(S(12)*d) + F**(a + b/​(c + d*x)**S(2))*b*(c + d*x)**S(4)*log(F)/​(S(12)*d) + F**(a + b/​(c + d*x)**S(2))*(c + d*x)**S(6)/​(S(6)*d), expand=True, _diff=True, _numerical=True)399 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))*(c + d*x)**S(3), x), x, -F**a*b**S(2)*log(F)**S(2)*Ei(b*log(F)/​(c + d*x)**S(2))/​(S(4)*d) + F**(a + b/​(c + d*x)**S(2))*b*(c + d*x)**S(2)*log(F)/​(S(4)*d) + F**(a + b/​(c + d*x)**S(2))*(c + d*x)**S(4)/​(S(4)*d), expand=True, _diff=True, _numerical=True)400 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))*(c + d*x), x), x, -F**a*b*log(F)*Ei(b*log(F)/​(c + d*x)**S(2))/​(S(2)*d) + F**(a + b/​(c + d*x)**S(2))*(c + d*x)**S(2)/​(S(2)*d), expand=True, _diff=True, _numerical=True)401 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))/​(c + d*x), x), x, -F**a*Ei(b*log(F)/​(c + d*x)**S(2))/​(S(2)*d), expand=True, _diff=True, _numerical=True)402 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))/​(c + d*x)**S(3), x), x, -F**(a + b/​(c + d*x)**S(2))/​(S(2)*b*d*log(F)), expand=True, _diff=True, _numerical=True)403 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))/​(c + d*x)**S(5), x), x, -F**(a + b/​(c + d*x)**S(2))/​(S(2)*b*d*(c + d*x)**S(2)*log(F)) + F**(a + b/​(c + d*x)**S(2))/​(S(2)*b**S(2)*d*log(F)**S(2)), expand=True, _diff=True, _numerical=True)404 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))/​(c + d*x)**S(7), x), x, -F**(a + b/​(c + d*x)**S(2))/​(S(2)*b*d*(c + d*x)**S(4)*log(F)) + F**(a + b/​(c + d*x)**S(2))/​(b**S(2)*d*(c + d*x)**S(2)*log(F)**S(2)) - F**(a + b/​(c + d*x)**S(2))/​(b**S(3)*d*log(F)**S(3)), expand=True, _diff=True, _numerical=True)405 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))/​(c + d*x)**S(9), x), x, -F**(a + b/​(c + d*x)**S(2))/​(S(2)*b*d*(c + d*x)**S(6)*log(F)) + S(3)*F**(a + b/​(c + d*x)**S(2))/​(S(2)*b**S(2)*d*(c + d*x)**S(4)*log(F)**S(2)) - S(3)*F**(a + b/​(c + d*x)**S(2))/​(b**S(3)*d*(c + d*x)**S(2)*log(F)**S(3)) + S(3)*F**(a + b/​(c + d*x)**S(2))/​(b**S(4)*d*log(F)**S(4)), expand=True, _diff=True, _numerical=True)406 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))/​(c + d*x)**S(11), x), x, -F**a*Gamma(S(5), -b*log(F)/​(c + d*x)**S(2))/​(S(2)*b**S(5)*d*log(F)**S(5)), expand=True, _diff=True, _numerical=True)407 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))/​(c + d*x)**S(13), x), x, F**a*Gamma(S(6), -b*log(F)/​(c + d*x)**S(2))/​(S(2)*b**S(6)*d*log(F)**S(6)), expand=True, _diff=True, _numerical=True)408 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))*(c + d*x)**S(10), x), x, F**a*(-b*log(F)/​(c + d*x)**S(2))**(S(11)/​2)*(c + d*x)**S(11)*Gamma(S(-11)/​2, -b*log(F)/​(c + d*x)**S(2))/​(S(2)*d), expand=True, _diff=True, _numerical=True)409 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))*(c + d*x)**S(8), x), x, F**a*(-b*log(F)/​(c + d*x)**S(2))**(S(9)/​2)*(c + d*x)**S(9)*Gamma(S(-9)/​2, -b*log(F)/​(c + d*x)**S(2))/​(S(2)*d), expand=True, _diff=True, _numerical=True)410 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))*(c + d*x)**S(6), x), x, -S(8)*sqrt(pi)*F**a*b**(S(7)/​2)*log(F)**(S(7)/​2)*erfi(sqrt(b)*sqrt(log(F))/​(c + d*x))/​(S(105)*d) + S(8)*F**(a + b/​(c + d*x)**S(2))*b**S(3)*(c + d*x)*log(F)**S(3)/​(S(105)*d) + S(4)*F**(a + b/​(c + d*x)**S(2))*b**S(2)*(c + d*x)**S(3)*log(F)**S(2)/​(S(105)*d) + S(2)*F**(a + b/​(c + d*x)**S(2))*b*(c + d*x)**S(5)*log(F)/​(S(35)*d) + F**(a + b/​(c + d*x)**S(2))*(c + d*x)**S(7)/​(S(7)*d), expand=True, _diff=True, _numerical=True)411 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))*(c + d*x)**S(4), x), x, -S(4)*sqrt(pi)*F**a*b**(S(5)/​2)*log(F)**(S(5)/​2)*erfi(sqrt(b)*sqrt(log(F))/​(c + d*x))/​(S(15)*d) + S(4)*F**(a + b/​(c + d*x)**S(2))*b**S(2)*(c + d*x)*log(F)**S(2)/​(S(15)*d) + S(2)*F**(a + b/​(c + d*x)**S(2))*b*(c + d*x)**S(3)*log(F)/​(S(15)*d) + F**(a + b/​(c + d*x)**S(2))*(c + d*x)**S(5)/​(S(5)*d), expand=True, _diff=True, _numerical=True)412 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))*(c + d*x)**S(2), x), x, -S(2)*sqrt(pi)*F**a*b**(S(3)/​2)*log(F)**(S(3)/​2)*erfi(sqrt(b)*sqrt(log(F))/​(c + d*x))/​(S(3)*d) + S(2)*F**(a + b/​(c + d*x)**S(2))*b*(c + d*x)*log(F)/​(S(3)*d) + F**(a + b/​(c + d*x)**S(2))*(c + d*x)**S(3)/​(S(3)*d), expand=True, _diff=True, _numerical=True)413 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2)), x), x, -sqrt(pi)*F**a*sqrt(b)*sqrt(log(F))*erfi(sqrt(b)*sqrt(log(F))/​(c + d*x))/​d + F**(a + b/​(c + d*x)**S(2))*(c + d*x)/​d, expand=True, _diff=True, _numerical=True)414 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))/​(c + d*x)**S(2), x), x, -sqrt(pi)*F**a*erfi(sqrt(b)*sqrt(log(F))/​(c + d*x))/​(S(2)*sqrt(b)*d*sqrt(log(F))), expand=True, _diff=True, _numerical=True)415 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))/​(c + d*x)**S(4), x), x, sqrt(pi)*F**a*erfi(sqrt(b)*sqrt(log(F))/​(c + d*x))/​(S(4)*b**(S(3)/​2)*d*log(F)**(S(3)/​2)) - F**(a + b/​(c + d*x)**S(2))/​(S(2)*b*d*(c + d*x)*log(F)), expand=True, _diff=True, _numerical=True)416 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))/​(c + d*x)**S(6), x), x, -S(3)*sqrt(pi)*F**a*erfi(sqrt(b)*sqrt(log(F))/​(c + d*x))/​(S(8)*b**(S(5)/​2)*d*log(F)**(S(5)/​2)) - F**(a + b/​(c + d*x)**S(2))/​(S(2)*b*d*(c + d*x)**S(3)*log(F)) + S(3)*F**(a + b/​(c + d*x)**S(2))/​(S(4)*b**S(2)*d*(c + d*x)*log(F)**S(2)), expand=True, _diff=True, _numerical=True)417 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))/​(c + d*x)**S(8), x), x, S(15)*sqrt(pi)*F**a*erfi(sqrt(b)*sqrt(log(F))/​(c + d*x))/​(S(16)*b**(S(7)/​2)*d*log(F)**(S(7)/​2)) - F**(a + b/​(c + d*x)**S(2))/​(S(2)*b*d*(c + d*x)**S(5)*log(F)) + S(5)*F**(a + b/​(c + d*x)**S(2))/​(S(4)*b**S(2)*d*(c + d*x)**S(3)*log(F)**S(2)) - S(15)*F**(a + b/​(c + d*x)**S(2))/​(S(8)*b**S(3)*d*(c + d*x)*log(F)**S(3)), expand=True, _diff=True, _numerical=True)418 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))/​(c + d*x)**S(10), x), x, -S(105)*sqrt(pi)*F**a*erfi(sqrt(b)*sqrt(log(F))/​(c + d*x))/​(S(32)*b**(S(9)/​2)*d*log(F)**(S(9)/​2)) - F**(a + b/​(c + d*x)**S(2))/​(S(2)*b*d*(c + d*x)**S(7)*log(F)) + S(7)*F**(a + b/​(c + d*x)**S(2))/​(S(4)*b**S(2)*d*(c + d*x)**S(5)*log(F)**S(2)) - S(35)*F**(a + b/​(c + d*x)**S(2))/​(S(8)*b**S(3)*d*(c + d*x)**S(3)*log(F)**S(3)) + S(105)*F**(a + b/​(c + d*x)**S(2))/​(S(16)*b**S(4)*d*(c + d*x)*log(F)**S(4)), expand=True, _diff=True, _numerical=True)419 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))/​(c + d*x)**S(12), x), x, F**a*Gamma(S(11)/​2, -b*log(F)/​(c + d*x)**S(2))/​(S(2)*d*(-b*log(F)/​(c + d*x)**S(2))**(S(11)/​2)*(c + d*x)**S(11)), expand=True, _diff=True, _numerical=True)420 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(2))/​(c + d*x)**S(14), x), x, F**a*Gamma(S(13)/​2, -b*log(F)/​(c + d*x)**S(2))/​(S(2)*d*(-b*log(F)/​(c + d*x)**S(2))**(S(13)/​2)*(c + d*x)**S(13)), expand=True, _diff=True, _numerical=True)421 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(3))*(c + d*x)**m, x), x, F**a*(-b*log(F)/​(c + d*x)**S(3))**(m/​S(3) + S(1)/​3)*(c + d*x)**(m + S(1))*Gamma(-m/​S(3) + S(-1)/​3, -b*log(F)/​(c + d*x)**S(3))/​(S(3)*d), expand=True, _diff=True, _numerical=True)422 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(3))*(c + d*x)**S(14), x), x, -F**a*b**S(5)*Gamma(S(-5), -b*log(F)/​(c + d*x)**S(3))*log(F)**S(5)/​(S(3)*d), expand=True, _diff=True, _numerical=True)423 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(3))*(c + d*x)**S(11), x), x, F**a*b**S(4)*Gamma(S(-4), -b*log(F)/​(c + d*x)**S(3))*log(F)**S(4)/​(S(3)*d), expand=True, _diff=True, _numerical=True)424 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(3))*(c + d*x)**S(8), x), x, -F**a*b**S(3)*log(F)**S(3)*Ei(b*log(F)/​(c + d*x)**S(3))/​(S(18)*d) + F**(a + b/​(c + d*x)**S(3))*b**S(2)*(c + d*x)**S(3)*log(F)**S(2)/​(S(18)*d) + F**(a + b/​(c + d*x)**S(3))*b*(c + d*x)**S(6)*log(F)/​(S(18)*d) + F**(a + b/​(c + d*x)**S(3))*(c + d*x)**S(9)/​(S(9)*d), expand=True, _diff=True, _numerical=True)425 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(3))*(c + d*x)**S(5), x), x, -F**a*b**S(2)*log(F)**S(2)*Ei(b*log(F)/​(c + d*x)**S(3))/​(S(6)*d) + F**(a + b/​(c + d*x)**S(3))*b*(c + d*x)**S(3)*log(F)/​(S(6)*d) + F**(a + b/​(c + d*x)**S(3))*(c + d*x)**S(6)/​(S(6)*d), expand=True, _diff=True, _numerical=True)426 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(3))*(c + d*x)**S(2), x), x, -F**a*b*log(F)*Ei(b*log(F)/​(c + d*x)**S(3))/​(S(3)*d) + F**(a + b/​(c + d*x)**S(3))*(c + d*x)**S(3)/​(S(3)*d), expand=True, _diff=True, _numerical=True)427 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(3))/​(c + d*x), x), x, -F**a*Ei(b*log(F)/​(c + d*x)**S(3))/​(S(3)*d), expand=True, _diff=True, _numerical=True)428 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(3))/​(c + d*x)**S(4), x), x, -F**(a + b/​(c + d*x)**S(3))/​(S(3)*b*d*log(F)), expand=True, _diff=True, _numerical=True)429 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(3))/​(c + d*x)**S(7), x), x, -F**(a + b/​(c + d*x)**S(3))/​(S(3)*b*d*(c + d*x)**S(3)*log(F)) + F**(a + b/​(c + d*x)**S(3))/​(S(3)*b**S(2)*d*log(F)**S(2)), expand=True, _diff=True, _numerical=True)430 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(3))/​(c + d*x)**S(10), x), x, -F**(a + b/​(c + d*x)**S(3))/​(S(3)*b*d*(c + d*x)**S(6)*log(F)) + S(2)*F**(a + b/​(c + d*x)**S(3))/​(S(3)*b**S(2)*d*(c + d*x)**S(3)*log(F)**S(2)) - S(2)*F**(a + b/​(c + d*x)**S(3))/​(S(3)*b**S(3)*d*log(F)**S(3)), expand=True, _diff=True, _numerical=True)431 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(3))/​(c + d*x)**S(13), x), x, -F**(a + b/​(c + d*x)**S(3))/​(S(3)*b*d*(c + d*x)**S(9)*log(F)) + F**(a + b/​(c + d*x)**S(3))/​(b**S(2)*d*(c + d*x)**S(6)*log(F)**S(2)) - S(2)*F**(a + b/​(c + d*x)**S(3))/​(b**S(3)*d*(c + d*x)**S(3)*log(F)**S(3)) + S(2)*F**(a + b/​(c + d*x)**S(3))/​(b**S(4)*d*log(F)**S(4)), expand=True, _diff=True, _numerical=True)432 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(3))/​(c + d*x)**S(16), x), x, -F**a*Gamma(S(5), -b*log(F)/​(c + d*x)**S(3))/​(S(3)*b**S(5)*d*log(F)**S(5)), expand=True, _diff=True, _numerical=True)433 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(3))/​(c + d*x)**S(19), x), x, F**a*Gamma(S(6), -b*log(F)/​(c + d*x)**S(3))/​(S(3)*b**S(6)*d*log(F)**S(6)), expand=True, _diff=True, _numerical=True)434 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(3))*(c + d*x)**S(3), x), x, F**a*(-b*log(F)/​(c + d*x)**S(3))**(S(4)/​3)*(c + d*x)**S(4)*Gamma(S(-4)/​3, -b*log(F)/​(c + d*x)**S(3))/​(S(3)*d), expand=True, _diff=True, _numerical=True)435 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(3))*(c + d*x), x), x, F**a*(-b*log(F)/​(c + d*x)**S(3))**(S(2)/​3)*(c + d*x)**S(2)*Gamma(S(-2)/​3, -b*log(F)/​(c + d*x)**S(3))/​(S(3)*d), expand=True, _diff=True, _numerical=True)436 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(3)), x), x, F**a*(-b*log(F)/​(c + d*x)**S(3))**(S(1)/​3)*(c + d*x)*Gamma(S(-1)/​3, -b*log(F)/​(c + d*x)**S(3))/​(S(3)*d), expand=True, _diff=True, _numerical=True)437 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(3))/​(c + d*x)**S(2), x), x, F**a*Gamma(S(1)/​3, -b*log(F)/​(c + d*x)**S(3))/​(S(3)*d*(-b*log(F)/​(c + d*x)**S(3))**(S(1)/​3)*(c + d*x)), expand=True, _diff=True, _numerical=True)438 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(3))/​(c + d*x)**S(3), x), x, F**a*Gamma(S(2)/​3, -b*log(F)/​(c + d*x)**S(3))/​(S(3)*d*(-b*log(F)/​(c + d*x)**S(3))**(S(2)/​3)*(c + d*x)**S(2)), expand=True, _diff=True, _numerical=True)439 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x)**S(3))/​(c + d*x)**S(5), x), x, F**a*Gamma(S(4)/​3, -b*log(F)/​(c + d*x)**S(3))/​(S(3)*d*(-b*log(F)/​(c + d*x)**S(3))**(S(4)/​3)*(c + d*x)**S(4)), expand=True, _diff=True, _numerical=True)440 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**n)*(c + d*x)**m, x), x, -F**a*(-b*(c + d*x)**n*log(F))**(-(m + S(1))/​n)*(c + d*x)**(m + S(1))*Gamma((m + S(1))/​n, -b*(c + d*x)**n*log(F))/​(d*n), expand=True, _diff=True, _numerical=True)441 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**n)*(c + d*x)**S(3), x), x, -F**a*(-b*(c + d*x)**n*log(F))**(-S(4)/​n)*(c + d*x)**S(4)*Gamma(S(4)/​n, -b*(c + d*x)**n*log(F))/​(d*n), expand=True, _diff=True, _numerical=True)442 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**n)*(c + d*x)**S(2), x), x, -F**a*(-b*(c + d*x)**n*log(F))**(-S(3)/​n)*(c + d*x)**S(3)*Gamma(S(3)/​n, -b*(c + d*x)**n*log(F))/​(d*n), expand=True, _diff=True, _numerical=True)443 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**n)*(c + d*x), x), x, -F**a*(-b*(c + d*x)**n*log(F))**(-S(2)/​n)*(c + d*x)**S(2)*Gamma(S(2)/​n, -b*(c + d*x)**n*log(F))/​(d*n), expand=True, _diff=True, _numerical=True)444 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**n), x), x, -F**a*(-b*(c + d*x)**n*log(F))**(-S(1)/​n)*(c + d*x)*Gamma(S(1)/​n, -b*(c + d*x)**n*log(F))/​(d*n), expand=True, _diff=True, _numerical=True)445 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**n)/​(c + d*x), x), x, F**a*Ei(b*(c + d*x)**n*log(F))/​(d*n), expand=True, _diff=True, _numerical=True)446 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**n)/​(c + d*x)**S(2), x), x, -F**a*(-b*(c + d*x)**n*log(F))**(S(1)/​n)*Gamma(-S(1)/​n, -b*(c + d*x)**n*log(F))/​(d*n*(c + d*x)), expand=True, _diff=True, _numerical=True)447 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**n)/​(c + d*x)**S(3), x), x, -F**a*(-b*(c + d*x)**n*log(F))**(S(2)/​n)*Gamma(-S(2)/​n, -b*(c + d*x)**n*log(F))/​(d*n*(c + d*x)**S(2)), expand=True, _diff=True, _numerical=True)448 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**n)/​(c + d*x)**S(4), x), x, -F**a*(-b*(c + d*x)**n*log(F))**(S(3)/​n)*Gamma(-S(3)/​n, -b*(c + d*x)**n*log(F))/​(d*n*(c + d*x)**S(3)), expand=True, _diff=True, _numerical=True)449 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**n)*(c + d*x)**(S(6)*n + S(-1)), x), x, -F**a*Gamma(S(6), -b*(c + d*x)**n*log(F))/​(b**S(6)*d*n*log(F)**S(6)), expand=True, _diff=True, _numerical=True)450 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**n)*(c + d*x)**(S(5)*n + S(-1)), x), x, F**a*Gamma(S(5), -b*(c + d*x)**n*log(F))/​(b**S(5)*d*n*log(F)**S(5)), expand=True, _diff=True, _numerical=True)451 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**n)*(c + d*x)**(S(4)*n + S(-1)), x), x, F**(a + b*(c + d*x)**n)*(c + d*x)**(S(3)*n)/​(b*d*n*log(F)) - S(3)*F**(a + b*(c + d*x)**n)*(c + d*x)**(S(2)*n)/​(b**S(2)*d*n*log(F)**S(2)) + S(6)*F**(a + b*(c + d*x)**n)*(c + d*x)**n/​(b**S(3)*d*n*log(F)**S(3)) - S(6)*F**(a + b*(c + d*x)**n)/​(b**S(4)*d*n*log(F)**S(4)), expand=True, _diff=True, _numerical=True)452 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**n)*(c + d*x)**(S(3)*n + S(-1)), x), x, F**(a + b*(c + d*x)**n)*(c + d*x)**(S(2)*n)/​(b*d*n*log(F)) - S(2)*F**(a + b*(c + d*x)**n)*(c + d*x)**n/​(b**S(2)*d*n*log(F)**S(2)) + S(2)*F**(a + b*(c + d*x)**n)/​(b**S(3)*d*n*log(F)**S(3)), expand=True, _diff=True, _numerical=True)453 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**n)*(c + d*x)**(S(2)*n + S(-1)), x), x, F**(a + b*(c + d*x)**n)*(c + d*x)**n/​(b*d*n*log(F)) - F**(a + b*(c + d*x)**n)/​(b**S(2)*d*n*log(F)**S(2)), expand=True, _diff=True, _numerical=True)454 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**n)*(c + d*x)**(n + S(-1)), x), x, F**(a + b*(c + d*x)**n)/​(b*d*n*log(F)), expand=True, _diff=True, _numerical=True)455 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**n)/​(c + d*x), x), x, F**a*Ei(b*(c + d*x)**n*log(F))/​(d*n), expand=True, _diff=True, _numerical=True)456 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**n)*(c + d*x)**(-n + S(-1)), x), x, F**a*b*log(F)*Ei(b*(c + d*x)**n*log(F))/​(d*n) - F**(a + b*(c + d*x)**n)*(c + d*x)**(-n)/​(d*n), expand=True, _diff=True, _numerical=True)457 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**n)*(c + d*x)**(-S(2)*n + S(-1)), x), x, F**a*b**S(2)*log(F)**S(2)*Ei(b*(c + d*x)**n*log(F))/​(S(2)*d*n) - F**(a + b*(c + d*x)**n)*b*(c + d*x)**(-n)*log(F)/​(S(2)*d*n) - F**(a + b*(c + d*x)**n)*(c + d*x)**(-S(2)*n)/​(S(2)*d*n), expand=True, _diff=True, _numerical=True)458 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**n)*(c + d*x)**(-S(3)*n + S(-1)), x), x, F**a*b**S(3)*log(F)**S(3)*Ei(b*(c + d*x)**n*log(F))/​(S(6)*d*n) - F**(a + b*(c + d*x)**n)*b**S(2)*(c + d*x)**(-n)*log(F)**S(2)/​(S(6)*d*n) - F**(a + b*(c + d*x)**n)*b*(c + d*x)**(-S(2)*n)*log(F)/​(S(6)*d*n) - F**(a + b*(c + d*x)**n)*(c + d*x)**(-S(3)*n)/​(S(3)*d*n), expand=True, _diff=True, _numerical=True)459 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**n)*(c + d*x)**(-S(4)*n + S(-1)), x), x, -F**a*b**S(4)*Gamma(S(-4), -b*(c + d*x)**n*log(F))*log(F)**S(4)/​(d*n), expand=True, _diff=True, _numerical=True)460 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**n)*(c + d*x)**(-S(5)*n + S(-1)), x), x, F**a*b**S(5)*Gamma(S(-5), -b*(c + d*x)**n*log(F))*log(F)**S(5)/​(d*n), expand=True, _diff=True, _numerical=True)461 assert rubi_test(rubi_integrate(F**(c*(a + b*x)**n)*(a + b*x)**(n/​S(2) + S(-1)), x), x, sqrt(pi)*erfi(sqrt(c)*(a + b*x)**(n/​S(2))*sqrt(log(F)))/​(b*sqrt(c)*n*sqrt(log(F))), expand=True, _diff=True, _numerical=True)462 assert rubi_test(rubi_integrate(F**(-c*(a + b*x)**n)*(a + b*x)**(n/​S(2) + S(-1)), x), x, sqrt(pi)*erf(sqrt(c)*(a + b*x)**(n/​S(2))*sqrt(log(F)))/​(b*sqrt(c)*n*sqrt(log(F))), expand=True, _diff=True, _numerical=True)463 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))*(e + f*x)**S(5), x), x, sqrt(pi)*F**a*(-c*f + d*e)**S(5)*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​(S(2)*sqrt(b)*d**S(6)*sqrt(log(F))) - S(5)*sqrt(pi)*F**a*f**S(2)*(-c*f + d*e)**S(3)*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​(S(2)*b**(S(3)/​2)*d**S(6)*log(F)**(S(3)/​2)) + S(15)*sqrt(pi)*F**a*f**S(4)*(-c*f + d*e)*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​(S(8)*b**(S(5)/​2)*d**S(6)*log(F)**(S(5)/​2)) + F**(a + b*(c + d*x)**S(2))*f**S(5)*(c + d*x)**S(4)/​(S(2)*b*d**S(6)*log(F)) + S(5)*F**(a + b*(c + d*x)**S(2))*f**S(4)*(c + d*x)**S(3)*(-c*f + d*e)/​(S(2)*b*d**S(6)*log(F)) + S(5)*F**(a + b*(c + d*x)**S(2))*f**S(3)*(c + d*x)**S(2)*(-c*f + d*e)**S(2)/​(b*d**S(6)*log(F)) + S(5)*F**(a + b*(c + d*x)**S(2))*f**S(2)*(c + d*x)*(-c*f + d*e)**S(3)/​(b*d**S(6)*log(F)) + S(5)*F**(a + b*(c + d*x)**S(2))*f*(-c*f + d*e)**S(4)/​(S(2)*b*d**S(6)*log(F)) - F**(a + b*(c + d*x)**S(2))*f**S(5)*(c + d*x)**S(2)/​(b**S(2)*d**S(6)*log(F)**S(2)) - S(15)*F**(a + b*(c + d*x)**S(2))*f**S(4)*(c + d*x)*(-c*f + d*e)/​(S(4)*b**S(2)*d**S(6)*log(F)**S(2)) - S(5)*F**(a + b*(c + d*x)**S(2))*f**S(3)*(-c*f + d*e)**S(2)/​(b**S(2)*d**S(6)*log(F)**S(2)) + F**(a + b*(c + d*x)**S(2))*f**S(5)/​(b**S(3)*d**S(6)*log(F)**S(3)), expand=True, _diff=True, _numerical=True)464 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))*(e + f*x)**S(4), x), x, sqrt(pi)*F**a*(-c*f + d*e)**S(4)*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​(S(2)*sqrt(b)*d**S(5)*sqrt(log(F))) - S(3)*sqrt(pi)*F**a*f**S(2)*(-c*f + d*e)**S(2)*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​(S(2)*b**(S(3)/​2)*d**S(5)*log(F)**(S(3)/​2)) + S(3)*sqrt(pi)*F**a*f**S(4)*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​(S(8)*b**(S(5)/​2)*d**S(5)*log(F)**(S(5)/​2)) + F**(a + b*(c + d*x)**S(2))*f**S(4)*(c + d*x)**S(3)/​(S(2)*b*d**S(5)*log(F)) + S(2)*F**(a + b*(c + d*x)**S(2))*f**S(3)*(c + d*x)**S(2)*(-c*f + d*e)/​(b*d**S(5)*log(F)) + S(3)*F**(a + b*(c + d*x)**S(2))*f**S(2)*(c + d*x)*(-c*f + d*e)**S(2)/​(b*d**S(5)*log(F)) + S(2)*F**(a + b*(c + d*x)**S(2))*f*(-c*f + d*e)**S(3)/​(b*d**S(5)*log(F)) - S(3)*F**(a + b*(c + d*x)**S(2))*f**S(4)*(c + d*x)/​(S(4)*b**S(2)*d**S(5)*log(F)**S(2)) - S(2)*F**(a + b*(c + d*x)**S(2))*f**S(3)*(-c*f + d*e)/​(b**S(2)*d**S(5)*log(F)**S(2)), expand=True, _diff=True, _numerical=True)465 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))*(e + f*x)**S(3), x), x, sqrt(pi)*F**a*(-c*f + d*e)**S(3)*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​(S(2)*sqrt(b)*d**S(4)*sqrt(log(F))) - S(3)*sqrt(pi)*F**a*f**S(2)*(-c*f + d*e)*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​(S(4)*b**(S(3)/​2)*d**S(4)*log(F)**(S(3)/​2)) + F**(a + b*(c + d*x)**S(2))*f**S(3)*(c + d*x)**S(2)/​(S(2)*b*d**S(4)*log(F)) + S(3)*F**(a + b*(c + d*x)**S(2))*f**S(2)*(c + d*x)*(-c*f + d*e)/​(S(2)*b*d**S(4)*log(F)) + S(3)*F**(a + b*(c + d*x)**S(2))*f*(-c*f + d*e)**S(2)/​(S(2)*b*d**S(4)*log(F)) - F**(a + b*(c + d*x)**S(2))*f**S(3)/​(S(2)*b**S(2)*d**S(4)*log(F)**S(2)), expand=True, _diff=True, _numerical=True)466 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))*(e + f*x)**S(2), x), x, sqrt(pi)*F**a*(-c*f + d*e)**S(2)*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​(S(2)*sqrt(b)*d**S(3)*sqrt(log(F))) - sqrt(pi)*F**a*f**S(2)*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​(S(4)*b**(S(3)/​2)*d**S(3)*log(F)**(S(3)/​2)) + F**(a + b*(c + d*x)**S(2))*f**S(2)*(c + d*x)/​(S(2)*b*d**S(3)*log(F)) + F**(a + b*(c + d*x)**S(2))*f*(-c*f + d*e)/​(b*d**S(3)*log(F)), expand=True, _diff=True, _numerical=True)467 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))*(e + f*x), x), x, sqrt(pi)*F**a*(-c*f/​S(2) + d*e/​S(2))*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​(sqrt(b)*d**S(2)*sqrt(log(F))) + F**(a + b*(c + d*x)**S(2))*f/​(S(2)*b*d**S(2)*log(F)), expand=True, _diff=True, _numerical=True)468 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2)), x), x, sqrt(pi)*F**a*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​(S(2)*sqrt(b)*d*sqrt(log(F))), expand=True, _diff=True, _numerical=True)469 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))/​(e + f*x), x), x, Integral(F**(a + b*(c + d*x)**S(2))/​(e + f*x), x), expand=True, _diff=True, _numerical=True)470 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))/​(e + f*x)**S(2), x), x, sqrt(pi)*F**a*sqrt(b)*d*sqrt(log(F))*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​f**S(2) - F**(a + b*(c + d*x)**S(2))/​(f*(e + f*x)) - S(2)*b*d*(-c*f + d*e)*log(F)*Integral(F**(a + b*(c + d*x)**S(2))/​(e + f*x), x)/​f**S(2), expand=True, _diff=True, _numerical=True)471 assert rubi_test(rubi_integrate(F**(a + b*(c + d*x)**S(2))/​(e + f*x)**S(3), x), x, -sqrt(pi)*F**a*b**(S(3)/​2)*d**S(2)*(-c*f + d*e)*log(F)**(S(3)/​2)*erfi(sqrt(b)*(c + d*x)*sqrt(log(F)))/​f**S(4) + F**(a + b*(c + d*x)**S(2))*b*d*(-c*f + d*e)*log(F)/​(f**S(3)*(e + f*x)) - F**(a + b*(c + d*x)**S(2))/​(S(2)*f*(e + f*x)**S(2)) + S(2)*b**S(2)*d**S(2)*(-c*f + d*e)**S(2)*log(F)**S(2)*Integral(F**(a + b*(c + d*x)**S(2))/​(e + f*x), x)/​f**S(4) + b*d**S(2)*log(F)*Integral(F**(a + b*(c + d*x)**S(2))/​(e + f*x), x)/​f**S(2), expand=True, _diff=True, _numerical=True)472 assert rubi_test(rubi_integrate((a + b*x)**S(3)*exp(e*(c + d*x)**S(3)), x), x, -b**S(3)*(c + d*x)**S(4)*Gamma(S(4)/​3, -e*(c + d*x)**S(3))/​(S(3)*d**S(4)*(-e*(c + d*x)**S(3))**(S(4)/​3)) - b**S(2)*(-a*d + b*c)*exp(e*(c + d*x)**S(3))/​(d**S(4)*e) - b*(c + d*x)**S(2)*(-a*d + b*c)**S(2)*Gamma(S(2)/​3, -e*(c + d*x)**S(3))/​(d**S(4)*(-e*(c + d*x)**S(3))**(S(2)/​3)) + (c + d*x)*(-a*d + b*c)**S(3)*Gamma(S(1)/​3, -e*(c + d*x)**S(3))/​(S(3)*d**S(4)*(-e*(c + d*x)**S(3))**(S(1)/​3)), expand=True, _diff=True, _numerical=True)473 assert rubi_test(rubi_integrate((a + b*x)**S(2)*exp(e*(c + d*x)**S(3)), x), x, b**S(2)*exp(e*(c + d*x)**S(3))/​(S(3)*d**S(3)*e) + S(2)*b*(c + d*x)**S(2)*(-a*d + b*c)*Gamma(S(2)/​3, -e*(c + d*x)**S(3))/​(S(3)*d**S(3)*(-e*(c + d*x)**S(3))**(S(2)/​3)) - (c + d*x)*(-a*d + b*c)**S(2)*Gamma(S(1)/​3, -e*(c + d*x)**S(3))/​(S(3)*d**S(3)*(-e*(c + d*x)**S(3))**(S(1)/​3)), expand=True, _diff=True, _numerical=True)474 assert rubi_test(rubi_integrate((a + b*x)*exp(e*(c + d*x)**S(3)), x), x, -b*(c + d*x)**S(2)*Gamma(S(2)/​3, -e*(c + d*x)**S(3))/​(S(3)*d**S(2)*(-e*(c + d*x)**S(3))**(S(2)/​3)) + (c + d*x)*(-a*d/​S(3) + b*c/​S(3))*Gamma(S(1)/​3, -e*(c + d*x)**S(3))/​(d**S(2)*(-e*(c + d*x)**S(3))**(S(1)/​3)), expand=True, _diff=True, _numerical=True)475 assert rubi_test(rubi_integrate(exp(e*(c + d*x)**S(3)), x), x, (-c/​S(3) - d*x/​S(3))*Gamma(S(1)/​3, -e*(c + d*x)**S(3))/​(d*(-e*(c + d*x)**S(3))**(S(1)/​3)), expand=True, _diff=True, _numerical=True)476 assert rubi_test(rubi_integrate(exp(e*(c + d*x)**S(3))/​(a + b*x), x), x, Integral(exp(e*(c + d*x)**S(3))/​(a + b*x), x), expand=True, _diff=True, _numerical=True)477 assert rubi_test(rubi_integrate(exp(e*(c + d*x)**S(3))/​(a + b*x)**S(2), x), x, -exp(e*(c + d*x)**S(3))/​(b*(a + b*x)) - d*e*(c + d*x)**S(2)*Gamma(S(2)/​3, -e*(c + d*x)**S(3))/​(b**S(2)*(-e*(c + d*x)**S(3))**(S(2)/​3)) + S(3)*d*e*(-a*d + b*c)**S(2)*Integral(exp(e*(c + d*x)**S(3))/​(a + b*x), x)/​b**S(3) - d*e*(c + d*x)*(-a*d + b*c)*Gamma(S(1)/​3, -e*(c + d*x)**S(3))/​(b**S(3)*(-e*(c + d*x)**S(3))**(S(1)/​3)), expand=True, _diff=True, _numerical=True)478 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x))/​(e + f*x), x), x, -F**a*Ei(b*log(F)/​(c + d*x))/​f + F**(a - b*f/​(-c*f + d*e))*Ei(b*d*(e + f*x)*log(F)/​((c + d*x)*(-c*f + d*e)))/​f, expand=True, _diff=True, _numerical=True)479 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x))/​(e + f*x)**S(2), x), x, F**(a + b/​(c + d*x))*d/​(f*(-c*f + d*e)) - F**(a + b/​(c + d*x))/​(f*(e + f*x)) - F**(a - b*f/​(-c*f + d*e))*b*d*log(F)*Ei(b*d*(e + f*x)*log(F)/​((c + d*x)*(-c*f + d*e)))/​(-c*f + d*e)**S(2), expand=True, _diff=True, _numerical=True)480 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x))/​(e + f*x)**S(3), x), x, -F**(a + b/​(c + d*x))*b*d**S(2)*log(F)/​(S(2)*(-c*f + d*e)**S(3)) + F**(a + b/​(c + d*x))*b*d*log(F)/​(S(2)*(e + f*x)*(-c*f + d*e)**S(2)) + F**(a + b/​(c + d*x))*d**S(2)/​(S(2)*f*(-c*f + d*e)**S(2)) - F**(a + b/​(c + d*x))/​(S(2)*f*(e + f*x)**S(2)) + F**(a - b*f/​(-c*f + d*e))*b**S(2)*d**S(2)*f*log(F)**S(2)*Ei(b*d*(e + f*x)*log(F)/​((c + d*x)*(-c*f + d*e)))/​(S(2)*(-c*f + d*e)**S(4)) - F**(a - b*f/​(-c*f + d*e))*b*d**S(2)*log(F)*Ei(b*d*(e + f*x)*log(F)/​((c + d*x)*(-c*f + d*e)))/​(-c*f + d*e)**S(3), expand=True, _diff=True, _numerical=True)481 assert rubi_test(rubi_integrate(F**(a + b/​(c + d*x))/​(e + f*x)**S(4), x), x, F**(a + b/​(c + d*x))*b**S(2)*d**S(3)*f*log(F)**S(2)/​(S(6)*(-c*f + d*e)**S(5)) - F**(a + b/​(c + d*x))*b**S(2)*d**S(2)*f*log(F)**S(2)/​(S(6)*(e + f*x)*(-c*f + d*e)**S(4)) - S(5)*F**(a + b/​(c + d*x))*b*d**S(3)*log(F)/​(S(6)*(-c*f + d*e)**S(4)) + S(2)*F**(a + b/​(c + d*x))*b*d**S(2)*log(F)/​(S(3)*(e + f*x)*(-c*f + d*e)**S(3)) + F**(a + b/​(c + d*x))*b*d*log(F)/​(S(6)*(e + f*x)**S(2)*(-c*f + d*e)**S(2)) + F**(a + b/​(c + d*x))*d**S(3)/​(S(3)*f*(-c*f + d*e)**S(3)) - F**(a + b/​(c + d*x))/​(S(3)*f*(e + f*x)**S(3)) - F**(a - b*f/​(-c*f + d*e))*b**S(3)*d**S(3)*f**S(2)*log(F)**S(3)*Ei(b*d*(e + f*x)*log(F)/​((c + d*x)*(-c*f + d*e)))/​(S(6)*(-c*f + d*e)**S(6)) + F**(a - b*f/​(-c*f + d*e))*b**S(2)*d**S(3)*f*log(F)**S(2)*Ei(b*d*(e + f*x)*log(F)/​((c + d*x)*(-c*f + d*e)))/​(-c*f + d*e)**S(5) - F**(a - b*f/​(-c*f + d*e))*b*d**S(3)*log(F)*Ei(b*d*(e + f*x)*log(F)/​((c + d*x)*(-c*f + d*e)))/​(-c*f + d*e)**S(4), expand=True, _diff=True, _numerical=True)482 assert rubi_test(rubi_integrate((a + b*x)**S(4)*exp(e/​(c + d*x)), x), x, -b**S(4)*e**S(5)*Gamma(S(-5), -e/​(c + d*x))/​d**S(5) - S(4)*b**S(3)*e**S(4)*(-a*d + b*c)*Gamma(S(-4), -e/​(c + d*x))/​d**S(5) - b**S(2)*e**S(3)*(-a*d + b*c)**S(2)*Ei(e/​(c + d*x))/​d**S(5) + b**S(2)*e**S(2)*(c + d*x)*(-a*d + b*c)**S(2)*exp(e/​(c + d*x))/​d**S(5) + b**S(2)*e*(c + d*x)**S(2)*(-a*d + b*c)**S(2)*exp(e/​(c + d*x))/​d**S(5) + S(2)*b**S(2)*(c + d*x)**S(3)*(-a*d + b*c)**S(2)*exp(e/​(c + d*x))/​d**S(5) + S(2)*b*e**S(2)*(-a*d + b*c)**S(3)*Ei(e/​(c + d*x))/​d**S(5) - S(2)*b*e*(c + d*x)*(-a*d + b*c)**S(3)*exp(e/​(c + d*x))/​d**S(5) - S(2)*b*(c + d*x)**S(2)*(-a*d + b*c)**S(3)*exp(e/​(c + d*x))/​d**S(5) - e*(-a*d + b*c)**S(4)*Ei(e/​(c + d*x))/​d**S(5) + (c + d*x)*(-a*d + b*c)**S(4)*exp(e/​(c + d*x))/​d**S(5), expand=True, _diff=True, _numerical=True)483 assert rubi_test(rubi_integrate((a + b*x)**S(3)*exp(e/​(c + d*x)), x), x, b**S(3)*e**S(4)*Gamma(S(-4), -e/​(c + d*x))/​d**S(4) + b**S(2)*e**S(3)*(-a*d + b*c)*Ei(e/​(c + d*x))/​(S(2)*d**S(4)) - b**S(2)*e**S(2)*(c + d*x)*(-a*d + b*c)*exp(e/​(c + d*x))/​(S(2)*d**S(4)) - b**S(2)*e*(c + d*x)**S(2)*(-a*d + b*c)*exp(e/​(c + d*x))/​(S(2)*d**S(4)) - b**S(2)*(c + d*x)**S(3)*(-a*d + b*c)*exp(e/​(c + d*x))/​d**S(4) - S(3)*b*e**S(2)*(-a*d + b*c)**S(2)*Ei(e/​(c + d*x))/​(S(2)*d**S(4)) + S(3)*b*e*(c + d*x)*(-a*d + b*c)**S(2)*exp(e/​(c + d*x))/​(S(2)*d**S(4)) + S(3)*b*(c + d*x)**S(2)*(-a*d + b*c)**S(2)*exp(e/​(c + d*x))/​(S(2)*d**S(4)) + e*(-a*d + b*c)**S(3)*Ei(e/​(c + d*x))/​d**S(4) - (c + d*x)*(-a*d + b*c)**S(3)*exp(e/​(c + d*x))/​d**S(4), expand=True, _diff=True, _numerical=True)484 assert rubi_test(rubi_integrate((a + b*x)**S(2)*exp(e/​(c + d*x)), x), x, -b**S(2)*e**S(3)*Ei(e/​(c + d*x))/​(S(6)*d**S(3)) + b**S(2)*e**S(2)*(c + d*x)*exp(e/​(c + d*x))/​(S(6)*d**S(3)) + b**S(2)*e*(c + d*x)**S(2)*exp(e/​(c + d*x))/​(S(6)*d**S(3)) + b**S(2)*(c + d*x)**S(3)*exp(e/​(c + d*x))/​(S(3)*d**S(3)) + b*e**S(2)*(-a*d + b*c)*Ei(e/​(c + d*x))/​d**S(3) - b*e*(c + d*x)*(-a*d + b*c)*exp(e/​(c + d*x))/​d**S(3) - b*(c + d*x)**S(2)*(-a*d + b*c)*exp(e/​(c + d*x))/​d**S(3) - e*(-a*d + b*c)**S(2)*Ei(e/​(c + d*x))/​d**S(3) + (c + d*x)*(-a*d + b*c)**S(2)*exp(e/​(c + d*x))/​d**S(3), expand=True, _diff=True, _numerical=True)485 assert rubi_test(rubi_integrate((a + b*x)*exp(e/​(c + d*x)), x), x, -b*e**S(2)*Ei(e/​(c + d*x))/​(S(2)*d**S(2)) + b*e*(c + d*x)*exp(e/​(c + d*x))/​(S(2)*d**S(2)) + b*(c + d*x)**S(2)*exp(e/​(c + d*x))/​(S(2)*d**S(2)) + e*(-a*d + b*c)*Ei(e/​(c + d*x))/​d**S(2) + (c + d*x)*(a*d - b*c)*exp(e/​(c + d*x))/​d**S(2), expand=True, _diff=True, _numerical=True)486 assert rubi_test(rubi_integrate(exp(e/​(c + d*x)), x), x, -e*Ei(e/​(c + d*x))/​d + (c + d*x)*exp(e/​(c + d*x))/​d, expand=True, _diff=True, _numerical=True)487 assert rubi_test(rubi_integrate(exp(e/​(c + d*x))/​(a + b*x), x), x, exp(b*e/​(-a*d + b*c))*Ei(-d*e*(a + b*x)/​((c + d*x)*(-a*d + b*c)))/​b - Ei(e/​(c + d*x))/​b, expand=True, _diff=True, _numerical=True)488 assert rubi_test(rubi_integrate(exp(e/​(c + d*x))/​(a + b*x)**S(2), x), x, -d*e*exp(b*e/​(-a*d + b*c))*Ei(-d*e*(a + b*x)/​((c + d*x)*(-a*d + b*c)))/​(-a*d + b*c)**S(2) - d*exp(e/​(c + d*x))/​(b*(-a*d + b*c)) - exp(e/​(c + d*x))/​(b*(a + b*x)), expand=True, _diff=True, _numerical=True)489 assert rubi_test(rubi_integrate(exp(e/​(c + d*x))/​(a + b*x)**S(3), x), x, b*d**S(2)*e**S(2)*exp(b*e/​(-a*d + b*c))*Ei(-d*e*(a + b*x)/​((c + d*x)*(-a*d + b*c)))/​(S(2)*(-a*d + b*c)**S(4)) + d**S(2)*e*exp(e/​(c + d*x))/​(S(2)*(-a*d + b*c)**S(3)) + d**S(2)*e*exp(b*e/​(-a*d + b*c))*Ei(-d*e*(a + b*x)/​((c + d*x)*(-a*d + b*c)))/​(-a*d + b*c)**S(3) + d*e*exp(e/​(c + d*x))/​(S(2)*(a + b*x)*(-a*d + b*c)**S(2)) + d**S(2)*exp(e/​(c + d*x))/​(S(2)*b*(-a*d + b*c)**S(2)) - exp(e/​(c + d*x))/​(S(2)*b*(a + b*x)**S(2)), expand=True, _diff=True, _numerical=True)490 assert rubi_test(rubi_integrate((a + b*x)**S(3)*exp(e/​(c + d*x)**S(2)), x), x, -b**S(3)*e**S(2)*Ei(e/​(c + d*x)**S(2))/​(S(4)*d**S(4)) + b**S(3)*e*(c + d*x)**S(2)*exp(e/​(c + d*x)**S(2))/​(S(4)*d**S(4)) + b**S(3)*(c + d*x)**S(4)*exp(e/​(c + d*x)**S(2))/​(S(4)*d**S(4)) + S(2)*sqrt(pi)*b**S(2)*e**(S(3)/​2)*(-a*d + b*c)*erfi(sqrt(e)/​(c + d*x))/​d**S(4) - S(2)*b**S(2)*e*(c + d*x)*(-a*d + b*c)*exp(e/​(c + d*x)**S(2))/​d**S(4) - b**S(2)*(c + d*x)**S(3)*(-a*d + b*c)*exp(e/​(c + d*x)**S(2))/​d**S(4) - S(3)*b*e*(-a*d + b*c)**S(2)*Ei(e/​(c + d*x)**S(2))/​(S(2)*d**S(4)) + S(3)*b*(c + d*x)**S(2)*(-a*d + b*c)**S(2)*exp(e/​(c + d*x)**S(2))/​(S(2)*d**S(4)) + sqrt(pi)*sqrt(e)*(-a*d + b*c)**S(3)*erfi(sqrt(e)/​(c + d*x))/​d**S(4) - (c + d*x)*(-a*d + b*c)**S(3)*exp(e/​(c + d*x)**S(2))/​d**S(4), expand=True, _diff=True, _numerical=True)491 assert rubi_test(rubi_integrate((a + b*x)**S(2)*exp(e/​(c + d*x)**S(2)), x), x, -S(2)*sqrt(pi)*b**S(2)*e**(S(3)/​2)*erfi(sqrt(e)/​(c + d*x))/​(S(3)*d**S(3)) + S(2)*b**S(2)*e*(c + d*x)*exp(e/​(c + d*x)**S(2))/​(S(3)*d**S(3)) + b**S(2)*(c + d*x)**S(3)*exp(e/​(c + d*x)**S(2))/​(S(3)*d**S(3)) + b*e*(-a*d + b*c)*Ei(e/​(c + d*x)**S(2))/​d**S(3) - b*(c + d*x)**S(2)*(-a*d + b*c)*exp(e/​(c + d*x)**S(2))/​d**S(3) - sqrt(pi)*sqrt(e)*(-a*d + b*c)**S(2)*erfi(sqrt(e)/​(c + d*x))/​d**S(3) + (c + d*x)*(-a*d + b*c)**S(2)*exp(e/​(c + d*x)**S(2))/​d**S(3), expand=True, _diff=True, _numerical=True)492 assert rubi_test(rubi_integrate((a + b*x)*exp(e/​(c + d*x)**S(2)), x), x, -b*e*Ei(e/​(c + d*x)**S(2))/​(S(2)*d**S(2)) + b*(c + d*x)**S(2)*exp(e/​(c + d*x)**S(2))/​(S(2)*d**S(2)) + sqrt(pi)*sqrt(e)*(-a*d + b*c)*erfi(sqrt(e)/​(c + d*x))/​d**S(2) + (c + d*x)*(a*d - b*c)*exp(e/​(c + d*x)**S(2))/​d**S(2), expand=True, _diff=True, _numerical=True)493 assert rubi_test(rubi_integrate(exp(e/​(c + d*x)**S(2)), x), x, -sqrt(pi)*sqrt(e)*erfi(sqrt(e)/​(c + d*x))/​d + (c + d*x)*exp(e/​(c + d*x)**S(2))/​d, expand=True, _diff=True, _numerical=True)494 assert rubi_test(rubi_integrate(exp(e/​(c + d*x)**S(2))/​(a + b*x), x), x, Integral(exp(e/​(c + d*x)**S(2))/​(a + b*x), x), expand=True, _diff=True, _numerical=True)495 assert rubi_test(rubi_integrate(exp(e/​(c + d*x)**S(2))/​(a + b*x)**S(2), x), x, Integral(exp(e/​(c + d*x)**S(2))/​(a + b*x)**S(2), x), expand=True, _diff=True, _numerical=True)496 assert rubi_test(rubi_integrate(exp(e/​(c + d*x)**S(2))/​(a + b*x)**S(3), x), x, Integral(exp(e/​(c + d*x)**S(2))/​(a + b*x)**S(3), x), expand=True, _diff=True, _numerical=True)497 assert rubi_test(rubi_integrate((a + b*x)**S(3)*exp(e/​(c + d*x)**S(3)), x), x, b**S(3)*(-e/​(c + d*x)**S(3))**(S(4)/​3)*(c + d*x)**S(4)*Gamma(S(-4)/​3, -e/​(c + d*x)**S(3))/​(S(3)*d**S(4)) + b**S(2)*e*(-a*d + b*c)*Ei(e/​(c + d*x)**S(3))/​d**S(4) - b**S(2)*(c + d*x)**S(3)*(-a*d + b*c)*exp(e/​(c + d*x)**S(3))/​d**S(4) + b*(-e/​(c + d*x)**S(3))**(S(2)/​3)*(c + d*x)**S(2)*(-a*d + b*c)**S(2)*Gamma(S(-2)/​3, -e/​(c + d*x)**S(3))/​d**S(4) - (-e/​(c + d*x)**S(3))**(S(1)/​3)*(c + d*x)*(-a*d + b*c)**S(3)*Gamma(S(-1)/​3, -e/​(c + d*x)**S(3))/​(S(3)*d**S(4)), expand=True, _diff=True, _numerical=True)498 assert rubi_test(rubi_integrate((a + b*x)**S(2)*exp(e/​(c + d*x)**S(3)), x), x, -b**S(2)*e*Ei(e/​(c + d*x)**S(3))/​(S(3)*d**S(3)) + b**S(2)*(c + d*x)**S(3)*exp(e/​(c + d*x)**S(3))/​(S(3)*d**S(3)) - S(2)*b*(-e/​(c + d*x)**S(3))**(S(2)/​3)*(c + d*x)**S(2)*(-a*d + b*c)*Gamma(S(-2)/​3, -e/​(c + d*x)**S(3))/​(S(3)*d**S(3)) + (-e/​(c + d*x)**S(3))**(S(1)/​3)*(c + d*x)*(-a*d + b*c)**S(2)*Gamma(S(-1)/​3, -e/​(c + d*x)**S(3))/​(S(3)*d**S(3)), expand=True, _diff=True, _numerical=True)499 assert rubi_test(rubi_integrate((a + b*x)*exp(e/​(c + d*x)**S(3)), x), x, b*(-e/​(c + d*x)**S(3))**(S(2)/​3)*(c + d*x)**S(2)*Gamma(S(-2)/​3, -e/​(c + d*x)**S(3))/​(S(3)*d**S(2)) - (-e/​(c + d*x)**S(3))**(S(1)/​3)*(c + d*x)*(-a*d/​S(3) + b*c/​S(3))*Gamma(S(-1)/​3, -e/​(c + d*x)**S(3))/​d**S(2), expand=True, _diff=True, _numerical=True)500 assert rubi_test(rubi_integrate(exp(e/​(c + d*x)**S(3)), x), x, (-e/​(c + d*x)**S(3))**(S(1)/​3)*(c + d*x)*Gamma(S(-1)/​3, -e/​(c + d*x)**S(3))/​(S(3)*d), expand=True, _diff=True, _numerical=True)501 assert rubi_test(rubi_integrate(exp(e/​(c + d*x)**S(3))/​(a + b*x), x), x, Integral(exp(e/​(c + d*x)**S(3))/​(a + b*x), x), expand=True, _diff=True, _numerical=True)502 assert rubi_test(rubi_integrate(exp(e/​(c + d*x)**S(3))/​(a + b*x)**S(2), x), x, Integral(exp(e/​(c + d*x)**S(3))/​(a + b*x)**S(2), x), expand=True, _diff=True, _numerical=True)503 assert rubi_test(rubi_integrate(F**(e + f*(a + b*x)/​(c + d*x))/​(g + h*x), x), x, F**(e + f*(-a*h + b*g)/​(-c*h + d*g))*Ei(f*(g + h*x)*(a*d - b*c)*log(F)/​((c + d*x)*(-c*h + d*g)))/​h - F**(b*f/​d + e)*Ei(f*(a*d - b*c)*log(F)/​(d*(c + d*x)))/​h, expand=True, _diff=True, _numerical=True)504 assert rubi_test(rubi_integrate(F**(e + f*(a + b*x)/​(c + d*x))/​(g + h*x)**S(2), x), x, -F**(e + f*(a + b*x)/​(c + d*x))/​(h*(g + h*x)) + F**(e + f*(-a*h + b*g)/​(-c*h + d*g))*f*(-a*d + b*c)*log(F)*Ei(f*(g + h*x)*(a*d - b*c)*log(F)/​((c + d*x)*(-c*h + d*g)))/​(-c*h + d*g)**S(2) + F**(b*f/​d + e - f*(-a*d + b*c)/​(d*(c + d*x)))*d/​(h*(-c*h + d*g)), expand=True, _diff=True, _numerical=True)505 assert rubi_test(rubi_integrate(F**(e + f*(a + b*x)/​(c + d*x))/​(g + h*x)**S(3), x), x, -F**(e + f*(a + b*x)/​(c + d*x))*f*(-a*d/​S(2) + b*c/​S(2))*log(F)/​((g + h*x)*(-c*h + d*g)**S(2)) - F**(e + f*(a + b*x)/​(c + d*x))/​(S(2)*h*(g + h*x)**S(2)) + F**(e + f*(-a*h + b*g)/​(-c*h + d*g))*d*f*(-a*d + b*c)*log(F)*Ei(f*(g + h*x)*(a*d - b*c)*log(F)/​((c + d*x)*(-c*h + d*g)))/​(-c*h + d*g)**S(3) + F**(e + f*(-a*h + b*g)/​(-c*h + d*g))*f**S(2)*h*(-a*d + b*c)**S(2)*log(F)**S(2)*Ei(f*(g + h*x)*(a*d - b*c)*log(F)/​((c + d*x)*(-c*h + d*g)))/​(S(2)*(-c*h + d*g)**S(4)) + F**(b*f/​d + e - f*(-a*d + b*c)/​(d*(c + d*x)))*d**S(2)/​(S(2)*h*(-c*h + d*g)**S(2)) + F**(b*f/​d + e - f*(-a*d + b*c)/​(d*(c + d*x)))*d*f*(-a*d + b*c)*log(F)/​(S(2)*(-c*h + d*g)**S(3)), expand=True, _diff=True, _numerical=True)506 assert rubi_test(rubi_integrate(F**(e + f*(a + b*x)/​(c + d*x))/​(g + h*x)**S(4), x), x, -S(2)*F**(e + f*(a + b*x)/​(c + d*x))*d*f*(-a*d + b*c)*log(F)/​(S(3)*(g + h*x)*(-c*h + d*g)**S(3)) - F**(e + f*(a + b*x)/​(c + d*x))*f**S(2)*h*(-a*d + b*c)**S(2)*log(F)**S(2)/​(S(6)*(g + h*x)*(-c*h + d*g)**S(4)) - F**(e + f*(a + b*x)/​(c + d*x))*f*(-a*d/​S(6) + b*c/​S(6))*log(F)/​((g + h*x)**S(2)*(-c*h + d*g)**S(2)) - F**(e + f*(a + b*x)/​(c + d*x))/​(S(3)*h*(g + h*x)**S(3)) + F**(e + f*(-a*h + b*g)/​(-c*h + d*g))*d**S(2)*f*(-a*d + b*c)*log(F)*Ei(f*(g + h*x)*(a*d - b*c)*log(F)/​((c + d*x)*(-c*h + d*g)))/​(-c*h + d*g)**S(4) + F**(e + f*(-a*h + b*g)/​(-c*h + d*g))*d*f**S(2)*h*(-a*d + b*c)**S(2)*log(F)**S(2)*Ei(f*(g + h*x)*(a*d - b*c)*log(F)/​((c + d*x)*(-c*h + d*g)))/​(-c*h + d*g)**S(5) + F**(e + f*(-a*h + b*g)/​(-c*h + d*g))*f**S(3)*h**S(2)*(-a*d + b*c)**S(3)*log(F)**S(3)*Ei(f*(g + h*x)*(a*d - b*c)*log(F)/​((c + d*x)*(-c*h + d*g)))/​(S(6)*(-c*h + d*g)**S(6)) + F**(b*f/​d + e - f*(-a*d + b*c)/​(d*(c + d*x)))*d**S(3)/​(S(3)*h*(-c*h + d*g)**S(3)) + S(5)*F**(b*f/​d + e - f*(-a*d + b*c)/​(d*(c + d*x)))*d**S(2)*f*(-a*d + b*c)*log(F)/​(S(6)*(-c*h + d*g)**S(4)) + F**(b*f/​d + e - f*(-a*d + b*c)/​(d*(c + d*x)))*d*f**S(2)*h*(-a*d + b*c)**S(2)*log(F)**S(2)/​(S(6)*(-c*h + d*g)**S(5)), expand=True, _diff=True, _numerical=True)507 # fails 1940 and 1939 recursion assert rubi_test(rubi_integrate(f**(a + b*x + c*x**S(2))*x**S(3), x), x, -sqrt(pi)*b**S(3)*f**(a - b**S(2)/​(S(4)*c))*erfi((b/​S(2) + c*x)*sqrt(log(f))/​sqrt(c))/​(S(16)*c**(S(7)/​2)*sqrt(log(f))) + b**S(2)*f**(a + b*x + c*x**S(2))/​(S(8)*c**S(3)*log(f)) - b*f**(a + b*x + c*x**S(2))*x/​(S(4)*c**S(2)*log(f)) + S(3)*sqrt(pi)*b*f**(a - b**S(2)/​(S(4)*c))*erfi((b/​S(2) + c*x)*sqrt(log(f))/​sqrt(c))/​(S(8)*c**(S(5)/​2)*log(f)**(S(3)/​2)) + f**(a + b*x + c*x**S(2))*x**S(2)/​(S(2)*c*log(f)) - f**(a + b*x + c*x**S(2))/​(S(2)*c**S(2)*log(f)**S(2)), expand=True, _diff=True, _numerical=True)508 assert rubi_test(rubi_integrate(f**(a + b*x + c*x**S(2))*x**S(2), x), x, sqrt(pi)*b**S(2)*f**(a - b**S(2)/​(S(4)*c))*erfi((b/​S(2) + c*x)*sqrt(log(f))/​sqrt(c))/​(S(8)*c**(S(5)/​2)*sqrt(log(f))) - b*f**(a + b*x + c*x**S(2))/​(S(4)*c**S(2)*log(f)) + f**(a + b*x + c*x**S(2))*x/​(S(2)*c*log(f)) - sqrt(pi)*f**(a - b**S(2)/​(S(4)*c))*erfi((b/​S(2) + c*x)*sqrt(log(f))/​sqrt(c))/​(S(4)*c**(S(3)/​2)*log(f)**(S(3)/​2)), expand=True, _diff=True, _numerical=True)509 assert rubi_test(rubi_integrate(f**(a + b*x + c*x**S(2))*x, x), x, -sqrt(pi)*b*f**(a - b**S(2)/​(S(4)*c))*erfi((b/​S(2) + c*x)*sqrt(log(f))/​sqrt(c))/​(S(4)*c**(S(3)/​2)*sqrt(log(f))) + f**(a + b*x + c*x**S(2))/​(S(2)*c*log(f)), expand=True, _diff=True, _numerical=True)510 assert rubi_test(rubi_integrate(f**(a + b*x + c*x**S(2)), x), x, sqrt(pi)*f**(a - b**S(2)/​(S(4)*c))*erfi((b/​S(2) + c*x)*sqrt(log(f))/​sqrt(c))/​(S(2)*sqrt(c)*sqrt(log(f))), expand=True, _diff=True, _numerical=True)511 assert rubi_test(rubi_integrate(f**(a + b*x + c*x**S(2))/​x, x), x, Integral(f**(a + b*x + c*x**S(2))/​x, x), expand=True, _diff=True, _numerical=True)512 assert rubi_test(rubi_integrate(f**(a + b*x + c*x**S(2))/​x**S(2), x), x, b*log(f)*Integral(f**(a + b*x + c*x**S(2))/​x, x) + sqrt(pi)*sqrt(c)*f**(a - b**S(2)/​(S(4)*c))*sqrt(log(f))*erfi((b/​S(2) + c*x)*sqrt(log(f))/​sqrt(c)) - f**(a + b*x + c*x**S(2))/​x, expand=True, _diff=True, _numerical=True)513 assert rubi_test(rubi_integrate(x**S(3)*exp(a + b*x - c*x**S(2)), x), x, -sqrt(pi)*b**S(3)*exp(a + b**S(2)/​(S(4)*c))*erf((b/​S(2) - c*x)/​sqrt(c))/​(S(16)*c**(S(7)/​2)) - b**S(2)*exp(a + b*x - c*x**S(2))/​(S(8)*c**S(3)) - b*x*exp(a + b*x - c*x**S(2))/​(S(4)*c**S(2)) - S(3)*sqrt(pi)*b*exp(a + b**S(2)/​(S(4)*c))*erf((b/​S(2) - c*x)/​sqrt(c))/​(S(8)*c**(S(5)/​2)) - x**S(2)*exp(a + b*x - c*x**S(2))/​(S(2)*c) - exp(a + b*x - c*x**S(2))/​(S(2)*c**S(2)), expand=True, _diff=True, _numerical=True)514 assert rubi_test(rubi_integrate(x**S(2)*exp(a + b*x - c*x**S(2)), x), x, -sqrt(pi)*b**S(2)*exp(a + b**S(2)/​(S(4)*c))*erf((b/​S(2) - c*x)/​sqrt(c))/​(S(8)*c**(S(5)/​2)) - b*exp(a + b*x - c*x**S(2))/​(S(4)*c**S(2)) - x*exp(a + b*x - c*x**S(2))/​(S(2)*c) - sqrt(pi)*exp(a + b**S(2)/​(S(4)*c))*erf((b/​S(2) - c*x)/​sqrt(c))/​(S(4)*c**(S(3)/​2)), expand=True, _diff=True, _numerical=True)515 assert rubi_test(rubi_integrate(x*exp(a + b*x - c*x**S(2)), x), x, -sqrt(pi)*b*exp(a + b**S(2)/​(S(4)*c))*erf((b/​S(2) - c*x)/​sqrt(c))/​(S(4)*c**(S(3)/​2)) - exp(a + b*x - c*x**S(2))/​(S(2)*c), expand=True, _diff=True, _numerical=True)516 assert rubi_test(rubi_integrate(exp(a + b*x - c*x**S(2)), x), x, -sqrt(pi)*exp(a + b**S(2)/​(S(4)*c))*erf((b/​S(2) - c*x)/​sqrt(c))/​(S(2)*sqrt(c)), expand=True, _diff=True, _numerical=True)517 assert rubi_test(rubi_integrate(exp(a + b*x - c*x**S(2))/​x, x), x, Integral(exp(a + b*x - c*x**S(2))/​x, x), expand=True, _diff=True, _numerical=True)518 assert rubi_test(rubi_integrate(exp(a + b*x - c*x**S(2))/​x**S(2), x), x, b*Integral(exp(a + b*x - c*x**S(2))/​x, x) + sqrt(pi)*sqrt(c)*exp(a + b**S(2)/​(S(4)*c))*erf((b/​S(2) - c*x)/​sqrt(c)) - exp(a + b*x - c*x**S(2))/​x, expand=True, _diff=True, _numerical=True)519 assert rubi_test(rubi_integrate(x**S(3)*exp((a + b*x)*(c + d*x)), x), x, x**S(2)*exp(a*c + b*d*x**S(2) + x*(a*d + b*c))/​(S(2)*b*d) - x*(a*d/​S(4) + b*c/​S(4))*exp(a*c + b*d*x**S(2) + x*(a*d + b*c))/​(b**S(2)*d**S(2)) - exp(a*c + b*d*x**S(2) + x*(a*d + b*c))/​(S(2)*b**S(2)*d**S(2)) + (a*d + b*c)**S(2)*exp(a*c + b*d*x**S(2) + x*(a*d + b*c))/​(S(8)*b**S(3)*d**S(3)) + sqrt(pi)*(S(3)*a*d/​S(8) + S(3)*b*c/​S(8))*exp(-(-a*d + b*c)**S(2)/​(S(4)*b*d))*erfi((a*d/​S(2) + b*c/​S(2) + b*d*x)/​(sqrt(b)*sqrt(d)))/​(b**(S(5)/​2)*d**(S(5)/​2)) - sqrt(pi)*(a*d + b*c)**S(3)*exp(-(-a*d + b*c)**S(2)/​(S(4)*b*d))*erfi((a*d/​S(2) + b*c/​S(2) + b*d*x)/​(sqrt(b)*sqrt(d)))/​(S(16)*b**(S(7)/​2)*d**(S(7)/​2)), expand=True, _diff=True, _numerical=True)520 assert rubi_test(rubi_integrate(x**S(2)*exp((a + b*x)*(c + d*x)), x), x, x*exp(a*c + b*d*x**S(2) + x*(a*d + b*c))/​(S(2)*b*d) + (-a*d/​S(4) - b*c/​S(4))*exp(a*c + b*d*x**S(2) + x*(a*d + b*c))/​(b**S(2)*d**S(2)) - sqrt(pi)*exp(-(-a*d + b*c)**S(2)/​(S(4)*b*d))*erfi((a*d/​S(2) + b*c/​S(2) + b*d*x)/​(sqrt(b)*sqrt(d)))/​(S(4)*b**(S(3)/​2)*d**(S(3)/​2)) + sqrt(pi)*(a*d + b*c)**S(2)*exp(-(-a*d + b*c)**S(2)/​(S(4)*b*d))*erfi((a*d/​S(2) + b*c/​S(2) + b*d*x)/​(sqrt(b)*sqrt(d)))/​(S(8)*b**(S(5)/​2)*d**(S(5)/​2)), expand=True, _diff=True, _numerical=True)521 assert rubi_test(rubi_integrate(x*exp((a + b*x)*(c + d*x)), x), x, exp(a*c + b*d*x**S(2) + x*(a*d + b*c))/​(S(2)*b*d) - sqrt(pi)*(a*d/​S(4) + b*c/​S(4))*exp(-(-a*d + b*c)**S(2)/​(S(4)*b*d))*erfi((a*d/​S(2) + b*c/​S(2) + b*d*x)/​(sqrt(b)*sqrt(d)))/​(b**(S(3)/​2)*d**(S(3)/​2)), expand=True, _diff=True, _numerical=True)522 assert rubi_test(rubi_integrate(exp((a + b*x)*(c + d*x)), x), x, sqrt(pi)*exp(-(-a*d + b*c)**S(2)/​(S(4)*b*d))*erfi((a*d/​S(2) + b*c/​S(2) + b*d*x)/​(sqrt(b)*sqrt(d)))/​(S(2)*sqrt(b)*sqrt(d)), expand=True, _diff=True, _numerical=True)523 assert rubi_test(rubi_integrate(exp((a + b*x)*(c + d*x))/​x, x), x, Integral(exp(a*c + b*d*x**S(2) + x*(a*d + b*c))/​x, x), expand=True, _diff=True, _numerical=True)524 assert rubi_test(rubi_integrate(exp((a + b*x)*(c + d*x))/​x**S(2), x), x, sqrt(pi)*sqrt(b)*sqrt(d)*exp(-(-a*d + b*c)**S(2)/​(S(4)*b*d))*erfi((a*d/​S(2) + b*c/​S(2) + b*d*x)/​(sqrt(b)*sqrt(d))) + (a*d + b*c)*Integral(exp(a*c + b*d*x**S(2) + x*(a*d + b*c))/​x, x) - exp(a*c + b*d*x**S(2) + x*(a*d + b*c))/​x, expand=True, _diff=True, _numerical=True)525 assert rubi_test(rubi_integrate(f**(a + b*x + c*x**S(2))*(d + e*x)**S(3), x), x, e*f**(a + b*x + c*x**S(2))*(d + e*x)**S(2)/​(S(2)*c*log(f)) - e**S(3)*f**(a + b*x + c*x**S(2))/​(S(2)*c**S(2)*log(f)**S(2)) + e*f**(a + b*x + c*x**S(2))*(d + e*x)*(-b*e + S(2)*c*d)/​(S(4)*c**S(2)*log(f)) + e*f**(a + b*x + c*x**S(2))*(-b*e + S(2)*c*d)**S(2)/​(S(8)*c**S(3)*log(f)) - S(3)*sqrt(pi)*e**S(2)*f**(a - b**S(2)/​(S(4)*c))*(-b*e + S(2)*c*d)*erfi((b/​S(2) + c*x)*sqrt(log(f))/​sqrt(c))/​(S(8)*c**(S(5)/​2)*log(f)**(S(3)/​2)) + sqrt(pi)*f**(a - b**S(2)/​(S(4)*c))*(-b*e + S(2)*c*d)**S(3)*erfi((b/​S(2) + c*x)*sqrt(log(f))/​sqrt(c))/​(S(16)*c**(S(7)/​2)*sqrt(log(f))), expand=True, _diff=True, _numerical=True)526 assert rubi_test(rubi_integrate(f**(a + b*x + c*x**S(2))*(d + e*x)**S(2), x), x, e*f**(a + b*x + c*x**S(2))*(d + e*x)/​(S(2)*c*log(f)) + e*f**(a + b*x + c*x**S(2))*(-b*e + S(2)*c*d)/​(S(4)*c**S(2)*log(f)) - sqrt(pi)*e**S(2)*f**(a - b**S(2)/​(S(4)*c))*erfi((b/​S(2) + c*x)*sqrt(log(f))/​sqrt(c))/​(S(4)*c**(S(3)/​2)*log(f)**(S(3)/​2)) + sqrt(pi)*f**(a - b**S(2)/​(S(4)*c))*(-b*e + S(2)*c*d)**S(2)*erfi((b/​S(2) + c*x)*sqrt(log(f))/​sqrt(c))/​(S(8)*c**(S(5)/​2)*sqrt(log(f))), expand=True, _diff=True, _numerical=True)527 assert rubi_test(rubi_integrate(f**(a + b*x + c*x**S(2))*(d + e*x), x), x, e*f**(a + b*x + c*x**S(2))/​(S(2)*c*log(f)) + sqrt(pi)*f**(a - b**S(2)/​(S(4)*c))*(-b*e/​S(4) + c*d/​S(2))*erfi((b/​S(2) + c*x)*sqrt(log(f))/​sqrt(c))/​(c**(S(3)/​2)*sqrt(log(f))), expand=True, _diff=True, _numerical=True)528 assert rubi_test(rubi_integrate(f**(a + b*x + c*x**S(2))/​(d + e*x), x), x, Integral(f**(a + b*x + c*x**S(2))/​(d + e*x), x), expand=True, _diff=True, _numerical=True)529 assert rubi_test(rubi_integrate(f**(a + b*x + c*x**S(2))/​(d + e*x)**S(2), x), x, sqrt(pi)*sqrt(c)*f**(a - b**S(2)/​(S(4)*c))*sqrt(log(f))*erfi((b/​S(2) + c*x)*sqrt(log(f))/​sqrt(c))/​e**S(2) - f**(a + b*x + c*x**S(2))/​(e*(d + e*x)) - (-b*e + S(2)*c*d)*log(f)*Integral(f**(a + b*x + c*x**S(2))/​(d + e*x), x)/​e**S(2), expand=True, _diff=True, _numerical=True)530 assert rubi_test(rubi_integrate(f**(a + b*x + c*x**S(2))/​(d + e*x)**S(3), x), x, -sqrt(pi)*sqrt(c)*f**(a - b**S(2)/​(S(4)*c))*(-b*e/​S(2) + c*d)*log(f)**(S(3)/​2)*erfi((b/​S(2) + c*x)*sqrt(log(f))/​sqrt(c))/​e**S(4) + c*log(f)*Integral(f**(a + b*x + c*x**S(2))/​(d + e*x), x)/​e**S(2) - f**(a + b*x + c*x**S(2))/​(S(2)*e*(d + e*x)**S(2)) + f**(a + b*x + c*x**S(2))*(-b*e/​S(2) + c*d)*log(f)/​(e**S(3)*(d + e*x)) + (-b*e + S(2)*c*d)**S(2)*log(f)**S(2)*Integral(f**(a + b*x + c*x**S(2))/​(d + e*x), x)/​(S(2)*e**S(4)), expand=True, _diff=True, _numerical=True)531 assert rubi_test(rubi_integrate(f**(a + b*x + c*x**S(2))*(b + S(2)*c*x)**S(3), x), x, -S(4)*c*f**(a + b*x + c*x**S(2))/​log(f)**S(2) + f**(a + b*x + c*x**S(2))*(b + S(2)*c*x)**S(2)/​log(f), expand=True, _diff=True, _numerical=True)532 assert rubi_test(rubi_integrate(f**(a + b*x + c*x**S(2))*(b + S(2)*c*x)**S(2), x), x, -sqrt(pi)*sqrt(c)*f**(a - b**S(2)/​(S(4)*c))*erfi((b/​S(2) + c*x)*sqrt(log(f))/​sqrt(c))/​log(f)**(S(3)/​2) + f**(a + b*x + c*x**S(2))*(b + S(2)*c*x)/​log(f), expand=True, _diff=True, _numerical=True)533 assert rubi_test(rubi_integrate(f**(a + b*x + c*x**S(2))*(b + S(2)*c*x), x), x, f**(a + b*x + c*x**S(2))/​log(f), expand=True, _diff=True, _numerical=True)534 assert rubi_test(rubi_integrate(f**(a + b*x + c*x**S(2))/​(b + S(2)*c*x), x), x, f**(a - b**S(2)/​(S(4)*c))*Ei((b + S(2)*c*x)**S(2)*log(f)/​(S(4)*c))/​(S(4)*c), expand=True, _diff=True, _numerical=True)535 assert rubi_test(rubi_integrate(f**(a + b*x + c*x**S(2))/​(b + S(2)*c*x)**S(2), x), x, -f**(a + b*x + c*x**S(2))/​(S(2)*c*(b + S(2)*c*x)) + sqrt(pi)*f**(a - b**S(2)/​(S(4)*c))*sqrt(log(f))*erfi((b/​S(2) + c*x)*sqrt(log(f))/​sqrt(c))/​(S(4)*c**(S(3)/​2)), expand=True, _diff=True, _numerical=True)536 assert rubi_test(rubi_integrate(f**(a + b*x + c*x**S(2))/​(b + S(2)*c*x)**S(3), x), x, -f**(a + b*x + c*x**S(2))/​(S(4)*c*(b + S(2)*c*x)**S(2)) + f**(a - b**S(2)/​(S(4)*c))*log(f)*Ei((b + S(2)*c*x)**S(2)*log(f)/​(S(4)*c))/​(S(16)*c**S(2)), expand=True, _diff=True, _numerical=True)537 assert rubi_test(rubi_integrate(f**(b*x + c*x**S(2))*(b + S(2)*c*x)**S(3), x), x, -S(4)*c*f**(b*x + c*x**S(2))/​log(f)**S(2) + f**(b*x + c*x**S(2))*(b + S(2)*c*x)**S(2)/​log(f), expand=True, _diff=True, _numerical=True)538 assert rubi_test(rubi_integrate(f**(b*x + c*x**S(2))*(b + S(2)*c*x)**S(2), x), x, -sqrt(pi)*sqrt(c)*f**(-b**S(2)/​(S(4)*c))*erfi((b/​S(2) + c*x)*sqrt(log(f))/​sqrt(c))/​log(f)**(S(3)/​2) + f**(b*x + c*x**S(2))*(b + S(2)*c*x)/​log(f), expand=True, _diff=True, _numerical=True)539 assert rubi_test(rubi_integrate(f**(b*x + c*x**S(2))*(b + S(2)*c*x), x), x, f**(b*x + c*x**S(2))/​log(f), expand=True, _diff=True, _numerical=True)540 assert rubi_test(rubi_integrate(f**(b*x + c*x**S(2))/​(b + S(2)*c*x), x), x, f**(-b**S(2)/​(S(4)*c))*Ei((b + S(2)*c*x)**S(2)*log(f)/​(S(4)*c))/​(S(4)*c), expand=True, _diff=True, _numerical=True)541 assert rubi_test(rubi_integrate(f**(b*x + c*x**S(2))/​(b + S(2)*c*x)**S(2), x), x, -f**(b*x + c*x**S(2))/​(S(2)*c*(b + S(2)*c*x)) + sqrt(pi)*f**(-b**S(2)/​(S(4)*c))*sqrt(log(f))*erfi((b/​S(2) + c*x)*sqrt(log(f))/​sqrt(c))/​(S(4)*c**(S(3)/​2)), expand=True, _diff=True, _numerical=True)542 assert rubi_test(rubi_integrate(f**(b*x + c*x**S(2))/​(b + S(2)*c*x)**S(3), x), x, -f**(b*x + c*x**S(2))/​(S(4)*c*(b + S(2)*c*x)**S(2)) + f**(-b**S(2)/​(S(4)*c))*log(f)*Ei((b + S(2)*c*x)**S(2)*log(f)/​(S(4)*c))/​(S(16)*c**S(2)), expand=True, _diff=True, _numerical=True)543 assert rubi_test(rubi_integrate(S(1)/​(x*(a + b*exp(c + d*x))), x), x, Integral(S(1)/​(x*(a + b*exp(c + d*x))), x), expand=True, _diff=True, _numerical=True)544 assert rubi_test(rubi_integrate(S(1)/​(a + b*exp(c + d*x)), x), x, x/​a - log(a + b*exp(c + d*x))/​(a*d), expand=True, _diff=True, _numerical=True)545 assert rubi_test(rubi_integrate(x/​(a + b*exp(c + d*x)), x), x, -x*log(a*exp(-c - d*x)/​b + S(1))/​(a*d) + polylog(S(2), -a*exp(-c - d*x)/​b)/​(a*d**S(2)), expand=True, _diff=True, _numerical=True)546 assert rubi_test(rubi_integrate(x**S(2)/​(a + b*exp(c + d*x)), x), x, -x**S(2)*log(a*exp(-c - d*x)/​b + S(1))/​(a*d) + S(2)*x*polylog(S(2), -a*exp(-c - d*x)/​b)/​(a*d**S(2)) + S(2)*polylog(S(3), -a*exp(-c - d*x)/​b)/​(a*d**S(3)), expand=True, _diff=True, _numerical=True)547 assert rubi_test(rubi_integrate(x**S(3)/​(a + b*exp(c + d*x)), x), x, -x**S(3)*log(a*exp(-c - d*x)/​b + S(1))/​(a*d) + S(3)*x**S(2)*polylog(S(2), -a*exp(-c - d*x)/​b)/​(a*d**S(2)) + S(6)*x*polylog(S(3), -a*exp(-c - d*x)/​b)/​(a*d**S(3)) + S(6)*polylog(S(4), -a*exp(-c - d*x)/​b)/​(a*d**S(4)), expand=True, _diff=True, _numerical=True)548 assert rubi_test(rubi_integrate(S(1)/​(a + b*exp(c - d*x)), x), x, x/​a + log(a + b*exp(c - d*x))/​(a*d), expand=True, _diff=True, _numerical=True)549 assert rubi_test(rubi_integrate(S(1)/​(a + b*exp(-c - d*x)), x), x, x/​a + log(a + b*exp(-c - d*x))/​(a*d), expand=True, _diff=True, _numerical=True)550 assert rubi_test(rubi_integrate(S(1)/​(x*(a + b*exp(c + d*x))**S(2)), x), x, Integral(S(1)/​(x*(a + b*exp(c + d*x))**S(2)), x), expand=True, _diff=True, _numerical=True)551 assert rubi_test(rubi_integrate((a + b*exp(c + d*x))**(S(-2)), x), x, S(1)/​(a*d*(a + b*exp(c + d*x))) + x/​a**S(2) - log(a + b*exp(c + d*x))/​(a**S(2)*d), expand=True, _diff=True, _numerical=True)552 assert rubi_test(rubi_integrate(x/​(a + b*exp(c + d*x))**S(2), x), x, x/​(a*d*(a + b*exp(c + d*x))) + x**S(2)/​(S(2)*a**S(2)) - x*log(S(1) + b*exp(c + d*x)/​a)/​(a**S(2)*d) - x/​(a**S(2)*d) + log(a + b*exp(c + d*x))/​(a**S(2)*d**S(2)) - polylog(S(2), -b*exp(c + d*x)/​a)/​(a**S(2)*d**S(2)), expand=True, _diff=True, _numerical=True)553 assert rubi_test(rubi_integrate(x**S(2)/​(a + b*exp(c + d*x))**S(2), x), x, x**S(2)/​(a*d*(a + b*exp(c + d*x))) + x**S(3)/​(S(3)*a**S(2)) - x**S(2)*log(S(1) + b*exp(c + d*x)/​a)/​(a**S(2)*d) + S(2)*x*log(a*exp(-c - d*x)/​b + S(1))/​(a**S(2)*d**S(2)) - S(2)*x*polylog(S(2), -b*exp(c + d*x)/​a)/​(a**S(2)*d**S(2)) - S(2)*polylog(S(2), -a*exp(-c - d*x)/​b)/​(a**S(2)*d**S(3)) + S(2)*polylog(S(3), -b*exp(c + d*x)/​a)/​(a**S(2)*d**S(3)), expand=True, _diff=True, _numerical=True)554 assert rubi_test(rubi_integrate(x**S(3)/​(a + b*exp(c + d*x))**S(2), x), x, x**S(3)/​(a*d*(a + b*exp(c + d*x))) + x**S(4)/​(S(4)*a**S(2)) - x**S(3)*log(S(1) + b*exp(c + d*x)/​a)/​(a**S(2)*d) + S(3)*x**S(2)*log(a*exp(-c - d*x)/​b + S(1))/​(a**S(2)*d**S(2)) - S(3)*x**S(2)*polylog(S(2), -b*exp(c + d*x)/​a)/​(a**S(2)*d**S(2)) - S(6)*x*polylog(S(2), -a*exp(-c - d*x)/​b)/​(a**S(2)*d**S(3)) + S(6)*x*polylog(S(3), -b*exp(c + d*x)/​a)/​(a**S(2)*d**S(3)) - S(6)*polylog(S(3), -a*exp(-c - d*x)/​b)/​(a**S(2)*d**S(4)) - S(6)*polylog(S(4), -b*exp(c + d*x)/​a)/​(a**S(2)*d**S(4)), expand=True, _diff=True, _numerical=True)555 assert rubi_test(rubi_integrate((a + b*exp(c - d*x))**(S(-2)), x), x, -S(1)/​(a*d*(a + b*exp(c - d*x))) + x/​a**S(2) + log(a + b*exp(c - d*x))/​(a**S(2)*d), expand=True, _diff=True, _numerical=True)556 assert rubi_test(rubi_integrate((a + b*exp(-c - d*x))**(S(-2)), x), x, -S(1)/​(a*d*(a + b*exp(-c - d*x))) + x/​a**S(2) + log(a + b*exp(-c - d*x))/​(a**S(2)*d), expand=True, _diff=True, _numerical=True)557 assert rubi_test(rubi_integrate(S(1)/​(x*(a + b*exp(c + d*x))**S(3)), x), x, Integral(S(1)/​(x*(a + b*exp(c + d*x))**S(3)), x), expand=True, _diff=True, _numerical=True)558 assert rubi_test(rubi_integrate((a + b*exp(c + d*x))**(S(-3)), x), x, S(1)/​(S(2)*a*d*(a + b*exp(c + d*x))**S(2)) + S(1)/​(a**S(2)*d*(a + b*exp(c + d*x))) + x/​a**S(3) - log(a + b*exp(c + d*x))/​(a**S(3)*d), expand=True, _diff=True, _numerical=True)559 assert rubi_test(rubi_integrate(x/​(a + b*exp(c + d*x))**S(3), x), x, x/​(S(2)*a*d*(a + b*exp(c + d*x))**S(2)) + x/​(a**S(2)*d*(a + b*exp(c + d*x))) - S(1)/​(S(2)*a**S(2)*d**S(2)*(a + b*exp(c + d*x))) + x**S(2)/​(S(2)*a**S(3)) - x*log(S(1) + b*exp(c + d*x)/​a)/​(a**S(3)*d) - S(3)*x/​(S(2)*a**S(3)*d) + S(3)*log(a + b*exp(c + d*x))/​(S(2)*a**S(3)*d**S(2)) - polylog(S(2), -b*exp(c + d*x)/​a)/​(a**S(3)*d**S(2)), expand=True, _diff=True, _numerical=True)560 # recursion assert rubi_test(rubi_integrate(x**S(2)/​(a + b*exp(c + d*x))**S(3), x), x, x**S(2)/​(S(2)*a*d*(a + b*exp(c + d*x))**S(2)) + x**S(2)/​(a**S(2)*d*(a + b*exp(c + d*x))) - x/​(a**S(2)*d**S(2)*(a + b*exp(c + d*x))) + x**S(3)/​(S(3)*a**S(3)) - x**S(2)*log(S(1) + b*exp(c + d*x)/​a)/​(a**S(3)*d) - S(3)*x**S(2)/​(S(2)*a**S(3)*d) + S(3)*x*log(S(1) + b*exp(c + d*x)/​a)/​(a**S(3)*d**S(2)) - S(2)*x*polylog(S(2), -b*exp(c + d*x)/​a)/​(a**S(3)*d**S(2)) + x/​(a**S(3)*d**S(2)) - log(a + b*exp(c + d*x))/​(a**S(3)*d**S(3)) + S(3)*polylog(S(2), -b*exp(c + d*x)/​a)/​(a**S(3)*d**S(3)) + S(2)*polylog(S(3), -b*exp(c + d*x)/​a)/​(a**S(3)*d**S(3)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate(x**S(2)/​(a + b*exp(c + d*x))**S(3), x), x, x**S(2)/​(S(2)*a*d*(a + b*exp(c + d*x))**S(2)) + x**S(2)/​(a**S(2)*d*(a + b*exp(c + d*x))) - x/​(a**S(2)*d**S(2)*(a + b*exp(c + d*x))) + x**S(3)/​(S(3)*a**S(3)) - x**S(2)*log(S(1) + b*exp(c + d*x)/​a)/​(a**S(3)*d) - x**S(2)/​(S(2)*a**S(3)*d) + x*log(S(1) + b*exp(c + d*x)/​a)/​(a**S(3)*d**S(2)) + S(2)*x*log(a*exp(-c - d*x)/​b + S(1))/​(a**S(3)*d**S(2)) - S(2)*x*polylog(S(2), -b*exp(c + d*x)/​a)/​(a**S(3)*d**S(2)) + x/​(a**S(3)*d**S(2)) - log(a + b*exp(c + d*x))/​(a**S(3)*d**S(3)) + polylog(S(2), -b*exp(c + d*x)/​a)/​(a**S(3)*d**S(3)) - S(2)*polylog(S(2), -a*exp(-c - d*x)/​b)/​(a**S(3)*d**S(3)) + S(2)*polylog(S(3), -b*exp(c + d*x)/​a)/​(a**S(3)*d**S(3)), expand=True, _diff=True, _numerical=True)561 assert rubi_test(rubi_integrate((a + b*exp(c - d*x))**(S(-3)), x), x, -S(1)/​(S(2)*a*d*(a + b*exp(c - d*x))**S(2)) - S(1)/​(a**S(2)*d*(a + b*exp(c - d*x))) + x/​a**S(3) + log(a + b*exp(c - d*x))/​(a**S(3)*d), expand=True, _diff=True, _numerical=True)562 assert rubi_test(rubi_integrate((a + b*exp(-c - d*x))**(S(-3)), x), x, -S(1)/​(S(2)*a*d*(a + b*exp(-c - d*x))**S(2)) - S(1)/​(a**S(2)*d*(a + b*exp(-c - d*x))) + x/​a**S(3) + log(a + b*exp(-c - d*x))/​(a**S(3)*d), expand=True, _diff=True, _numerical=True)563 assert rubi_test(rubi_integrate(exp(a + b*x)/​(x**S(2)*(c + d*x**S(2))), x), x, b*exp(a)*Ei(b*x)/​c - sqrt(d)*exp(a - b*sqrt(-c)/​sqrt(d))*Ei(b*(sqrt(d)*x + sqrt(-c))/​sqrt(d))/​(S(2)*(-c)**(S(3)/​2)) + sqrt(d)*exp(a + b*sqrt(-c)/​sqrt(d))*Ei(-b*(-sqrt(d)*x + sqrt(-c))/​sqrt(d))/​(S(2)*(-c)**(S(3)/​2)) - exp(a + b*x)/​(c*x), expand=True, _diff=True, _numerical=True)564 assert rubi_test(rubi_integrate(exp(a + b*x)/​(x*(c + d*x**S(2))), x), x, exp(a)*Ei(b*x)/​c - exp(a - b*sqrt(-c)/​sqrt(d))*Ei(b*(sqrt(d)*x + sqrt(-c))/​sqrt(d))/​(S(2)*c) - exp(a + b*sqrt(-c)/​sqrt(d))*Ei(-b*(-sqrt(d)*x + sqrt(-c))/​sqrt(d))/​(S(2)*c), expand=True, _diff=True, _numerical=True)565 assert rubi_test(rubi_integrate(exp(a + b*x)/​(c + d*x**S(2)), x), x, -exp(a - b*sqrt(-c)/​sqrt(d))*Ei(b*(sqrt(d)*x + sqrt(-c))/​sqrt(d))/​(S(2)*sqrt(d)*sqrt(-c)) + exp(a + b*sqrt(-c)/​sqrt(d))*Ei(-b*(-sqrt(d)*x + sqrt(-c))/​sqrt(d))/​(S(2)*sqrt(d)*sqrt(-c)), expand=True, _diff=True, _numerical=True)566 assert rubi_test(rubi_integrate(x*exp(a + b*x)/​(c + d*x**S(2)), x), x, exp(a - b*sqrt(-c)/​sqrt(d))*Ei(b*(sqrt(d)*x + sqrt(-c))/​sqrt(d))/​(S(2)*d) + exp(a + b*sqrt(-c)/​sqrt(d))*Ei(-b*(-sqrt(d)*x + sqrt(-c))/​sqrt(d))/​(S(2)*d), expand=True, _diff=True, _numerical=True)567 assert rubi_test(rubi_integrate(x**S(2)*exp(a + b*x)/​(c + d*x**S(2)), x), x, -sqrt(-c)*exp(a - b*sqrt(-c)/​sqrt(d))*Ei(b*(sqrt(d)*x + sqrt(-c))/​sqrt(d))/​(S(2)*d**(S(3)/​2)) + sqrt(-c)*exp(a + b*sqrt(-c)/​sqrt(d))*Ei(-b*(-sqrt(d)*x + sqrt(-c))/​sqrt(d))/​(S(2)*d**(S(3)/​2)) + exp(a + b*x)/​(b*d), expand=True, _diff=True, _numerical=True)568 assert rubi_test(rubi_integrate(exp(d + e*x)/​(x**S(2)*(a + b*x + c*x**S(2))), x), x, e*exp(d)*Ei(e*x)/​a - exp(d + e*x)/​(a*x) - b*exp(d)*Ei(e*x)/​a**S(2) + (b + (-S(2)*a*c + b**S(2))/​sqrt(-S(4)*a*c + b**S(2)))*exp(d - e*(b - sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))*Ei(e*(b + S(2)*c*x - sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))/​(S(2)*a**S(2)) + (b + (S(2)*a*c - b**S(2))/​sqrt(-S(4)*a*c + b**S(2)))*exp(d - e*(b + sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))*Ei(e*(b + S(2)*c*x + sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))/​(S(2)*a**S(2)), expand=True, _diff=True, _numerical=True)569 assert rubi_test(rubi_integrate(exp(d + e*x)/​(x*(a + b*x + c*x**S(2))), x), x, -(-b/​sqrt(-S(4)*a*c + b**S(2)) + S(1))*exp(d - e*(b + sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))*Ei(e*(b + S(2)*c*x + sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))/​(S(2)*a) - (b/​sqrt(-S(4)*a*c + b**S(2)) + S(1))*exp(d - e*(b - sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))*Ei(e*(b + S(2)*c*x - sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))/​(S(2)*a) + exp(d)*Ei(e*x)/​a, expand=True, _diff=True, _numerical=True)570 assert rubi_test(rubi_integrate(exp(d + e*x)/​(a + b*x + c*x**S(2)), x), x, exp(d - e*(b - sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))*Ei(e*(b + S(2)*c*x - sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))/​sqrt(-S(4)*a*c + b**S(2)) - exp(d - e*(b + sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))*Ei(e*(b + S(2)*c*x + sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))/​sqrt(-S(4)*a*c + b**S(2)), expand=True, _diff=True, _numerical=True)571 assert rubi_test(rubi_integrate(x*exp(d + e*x)/​(a + b*x + c*x**S(2)), x), x, (-b/​sqrt(-S(4)*a*c + b**S(2)) + S(1))*exp(d - e*(b - sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))*Ei(e*(b + S(2)*c*x - sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))/​(S(2)*c) + (b/​sqrt(-S(4)*a*c + b**S(2)) + S(1))*exp(d - e*(b + sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))*Ei(e*(b + S(2)*c*x + sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))/​(S(2)*c), expand=True, _diff=True, _numerical=True)572 assert rubi_test(rubi_integrate(x**S(2)*exp(d + e*x)/​(a + b*x + c*x**S(2)), x), x, exp(d + e*x)/​(c*e) - (b + (-S(2)*a*c + b**S(2))/​sqrt(-S(4)*a*c + b**S(2)))*exp(d - e*(b + sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))*Ei(e*(b + S(2)*c*x + sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))/​(S(2)*c**S(2)) - (b + (S(2)*a*c - b**S(2))/​sqrt(-S(4)*a*c + b**S(2)))*exp(d - e*(b - sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))*Ei(e*(b + S(2)*c*x - sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))/​(S(2)*c**S(2)), expand=True, _diff=True, _numerical=True)573 assert rubi_test(rubi_integrate(x**S(3)*exp(d + e*x)/​(a + b*x + c*x**S(2)), x), x, -b*exp(d + e*x)/​(c**S(2)*e) + x*exp(d + e*x)/​(c*e) - exp(d + e*x)/​(c*e**S(2)) + (-a*c + b**S(2) - b*(-S(3)*a*c + b**S(2))/​sqrt(-S(4)*a*c + b**S(2)))*exp(d - e*(b - sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))*Ei(e*(b + S(2)*c*x - sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))/​(S(2)*c**S(3)) + (-a*c + b**S(2) + b*(-S(3)*a*c + b**S(2))/​sqrt(-S(4)*a*c + b**S(2)))*exp(d - e*(b + sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))*Ei(e*(b + S(2)*c*x + sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))/​(S(2)*c**S(3)), expand=True, _diff=True, _numerical=True)574 assert rubi_test(rubi_integrate(S(4)**x/​(S(2)**x*b + a), x), x, S(2)**x/​(b*log(S(2))) - a*log(S(2)**x*b + a)/​(b**S(2)*log(S(2))), expand=True, _diff=True, _numerical=True)575 assert rubi_test(rubi_integrate(S(2)**(S(2)*x)/​(S(2)**x*b + a), x), x, S(2)**x/​(b*log(S(2))) - a*log(S(2)**x*b + a)/​(b**S(2)*log(S(2))), expand=True, _diff=True, _numerical=True)576 assert rubi_test(rubi_integrate(S(4)**x/​(-S(2)**x*b + a), x), x, -S(2)**x/​(b*log(S(2))) - a*log(-S(2)**x*b + a)/​(b**S(2)*log(S(2))), expand=True, _diff=True, _numerical=True)577 assert rubi_test(rubi_integrate(S(2)**(S(2)*x)/​(-S(2)**x*b + a), x), x, -S(2)**x/​(b*log(S(2))) - a*log(-S(2)**x*b + a)/​(b**S(2)*log(S(2))), expand=True, _diff=True, _numerical=True)578 assert rubi_test(rubi_integrate(S(4)**x/​(a + S(2)**(-x)*b), x), x, -S(2)**x*b/​(a**S(2)*log(S(2))) + S(2)**(S(2)*x + S(-1))/​(a*log(S(2))) + b**S(2)*x/​a**S(3) + b**S(2)*log(a + S(2)**(-x)*b)/​(a**S(3)*log(S(2))), expand=True, _diff=True, _numerical=True)579 assert rubi_test(rubi_integrate(S(2)**(S(2)*x)/​(a + S(2)**(-x)*b), x), x, -S(2)**x*b/​(a**S(2)*log(S(2))) + S(2)**(S(2)*x + S(-1))/​(a*log(S(2))) + b**S(2)*x/​a**S(3) + b**S(2)*log(a + S(2)**(-x)*b)/​(a**S(3)*log(S(2))), expand=True, _diff=True, _numerical=True)580 assert rubi_test(rubi_integrate(S(4)**x/​(a - S(2)**(-x)*b), x), x, S(2)**x*b/​(a**S(2)*log(S(2))) + S(2)**(S(2)*x + S(-1))/​(a*log(S(2))) + b**S(2)*x/​a**S(3) + b**S(2)*log(a - S(2)**(-x)*b)/​(a**S(3)*log(S(2))), expand=True, _diff=True, _numerical=True)581 assert rubi_test(rubi_integrate(S(2)**(S(2)*x)/​(a - S(2)**(-x)*b), x), x, S(2)**x*b/​(a**S(2)*log(S(2))) + S(2)**(S(2)*x + S(-1))/​(a*log(S(2))) + b**S(2)*x/​a**S(3) + b**S(2)*log(a - S(2)**(-x)*b)/​(a**S(3)*log(S(2))), expand=True, _diff=True, _numerical=True)582 assert rubi_test(rubi_integrate(S(2)**x/​(S(4)**x*b + a), x), x, atan(S(2)**x*sqrt(b)/​sqrt(a))/​(sqrt(a)*sqrt(b)*log(S(2))), expand=True, _diff=True, _numerical=True)583 assert rubi_test(rubi_integrate(S(2)**x/​(S(2)**(S(2)*x)*b + a), x), x, atan(S(2)**x*sqrt(b)/​sqrt(a))/​(sqrt(a)*sqrt(b)*log(S(2))), expand=True, _diff=True, _numerical=True)584 assert rubi_test(rubi_integrate(S(2)**x/​(-S(4)**x*b + a), x), x, atanh(S(2)**x*sqrt(b)/​sqrt(a))/​(sqrt(a)*sqrt(b)*log(S(2))), expand=True, _diff=True, _numerical=True)585 assert rubi_test(rubi_integrate(S(2)**x/​(-S(2)**(S(2)*x)*b + a), x), x, atanh(S(2)**x*sqrt(b)/​sqrt(a))/​(sqrt(a)*sqrt(b)*log(S(2))), expand=True, _diff=True, _numerical=True)586 assert rubi_test(rubi_integrate(S(2)**x/​(a + S(4)**(-x)*b), x), x, S(2)**x/​(a*log(S(2))) - sqrt(b)*atan(S(2)**x*sqrt(a)/​sqrt(b))/​(a**(S(3)/​2)*log(S(2))), expand=True, _diff=True, _numerical=True)587 assert rubi_test(rubi_integrate(S(2)**x/​(a + S(2)**(-S(2)*x)*b), x), x, S(2)**x/​(a*log(S(2))) - sqrt(b)*atan(S(2)**x*sqrt(a)/​sqrt(b))/​(a**(S(3)/​2)*log(S(2))), expand=True, _diff=True, _numerical=True)588 assert rubi_test(rubi_integrate(S(2)**x/​(a - S(4)**(-x)*b), x), x, S(2)**x/​(a*log(S(2))) - sqrt(b)*atanh(S(2)**x*sqrt(a)/​sqrt(b))/​(a**(S(3)/​2)*log(S(2))), expand=True, _diff=True, _numerical=True)589 assert rubi_test(rubi_integrate(S(2)**x/​(a - S(2)**(-S(2)*x)*b), x), x, S(2)**x/​(a*log(S(2))) - sqrt(b)*atanh(S(2)**x*sqrt(a)/​sqrt(b))/​(a**(S(3)/​2)*log(S(2))), expand=True, _diff=True, _numerical=True)590 assert rubi_test(rubi_integrate(S(2)**x/​sqrt(S(4)**x*b + a), x), x, atanh(S(2)**x*sqrt(b)/​sqrt(S(2)**(S(2)*x)*b + a))/​(sqrt(b)*log(S(2))), expand=True, _diff=True, _numerical=True)591 assert rubi_test(rubi_integrate(S(2)**x/​sqrt(S(2)**(S(2)*x)*b + a), x), x, atanh(S(2)**x*sqrt(b)/​sqrt(S(2)**(S(2)*x)*b + a))/​(sqrt(b)*log(S(2))), expand=True, _diff=True, _numerical=True)592 assert rubi_test(rubi_integrate(S(2)**x/​sqrt(-S(4)**x*b + a), x), x, atan(S(2)**x*sqrt(b)/​sqrt(-S(2)**(S(2)*x)*b + a))/​(sqrt(b)*log(S(2))), expand=True, _diff=True, _numerical=True)593 assert rubi_test(rubi_integrate(S(2)**x/​sqrt(-S(2)**(S(2)*x)*b + a), x), x, atan(S(2)**x*sqrt(b)/​sqrt(-S(2)**(S(2)*x)*b + a))/​(sqrt(b)*log(S(2))), expand=True, _diff=True, _numerical=True)594 assert rubi_test(rubi_integrate(S(2)**x/​sqrt(a + S(4)**(-x)*b), x), x, S(2)**x*sqrt(a + S(2)**(-S(2)*x)*b)/​(a*log(S(2))), expand=True, _diff=True, _numerical=True)595 assert rubi_test(rubi_integrate(S(2)**x/​sqrt(a + S(2)**(-S(2)*x)*b), x), x, S(2)**x*sqrt(a + S(2)**(-S(2)*x)*b)/​(a*log(S(2))), expand=True, _diff=True, _numerical=True)596 assert rubi_test(rubi_integrate(S(2)**x/​sqrt(a - S(4)**(-x)*b), x), x, S(2)**x*sqrt(a - S(2)**(-S(2)*x)*b)/​(a*log(S(2))), expand=True, _diff=True, _numerical=True)597 assert rubi_test(rubi_integrate(S(2)**x/​sqrt(a - S(2)**(-S(2)*x)*b), x), x, S(2)**x*sqrt(a - S(2)**(-S(2)*x)*b)/​(a*log(S(2))), expand=True, _diff=True, _numerical=True)598 assert rubi_test(rubi_integrate(S(4)**x/​sqrt(S(2)**x*b + a), x), x, -S(2)*a*sqrt(S(2)**x*b + a)/​(b**S(2)*log(S(2))) + S(2)*(S(2)**x*b + a)**(S(3)/​2)/​(S(3)*b**S(2)*log(S(2))), expand=True, _diff=True, _numerical=True)599 assert rubi_test(rubi_integrate(S(2)**(S(2)*x)/​sqrt(S(2)**x*b + a), x), x, -S(2)*a*sqrt(S(2)**x*b + a)/​(b**S(2)*log(S(2))) + S(2)*(S(2)**x*b + a)**(S(3)/​2)/​(S(3)*b**S(2)*log(S(2))), expand=True, _diff=True, _numerical=True)600 assert rubi_test(rubi_integrate(S(4)**x/​sqrt(-S(2)**x*b + a), x), x, -S(2)*a*sqrt(-S(2)**x*b + a)/​(b**S(2)*log(S(2))) + S(2)*(-S(2)**x*b + a)**(S(3)/​2)/​(S(3)*b**S(2)*log(S(2))), expand=True, _diff=True, _numerical=True)601 assert rubi_test(rubi_integrate(S(2)**(S(2)*x)/​sqrt(-S(2)**x*b + a), x), x, -S(2)*a*sqrt(-S(2)**x*b + a)/​(b**S(2)*log(S(2))) + S(2)*(-S(2)**x*b + a)**(S(3)/​2)/​(S(3)*b**S(2)*log(S(2))), expand=True, _diff=True, _numerical=True)602 assert rubi_test(rubi_integrate(S(4)**x/​sqrt(a + S(2)**(-x)*b), x), x, -S(3)*S(2)**(x + S(-2))*b*sqrt(a + S(2)**(-x)*b)/​(a**S(2)*log(S(2))) + S(2)**(S(2)*x + S(-1))*sqrt(a + S(2)**(-x)*b)/​(a*log(S(2))) + S(3)*b**S(2)*atanh(sqrt(a + S(2)**(-x)*b)/​sqrt(a))/​(S(4)*a**(S(5)/​2)*log(S(2))), expand=True, _diff=True, _numerical=True)603 assert rubi_test(rubi_integrate(S(2)**(S(2)*x)/​sqrt(a + S(2)**(-x)*b), x), x, -S(3)*S(2)**(x + S(-2))*b*sqrt(a + S(2)**(-x)*b)/​(a**S(2)*log(S(2))) + S(2)**(S(2)*x + S(-1))*sqrt(a + S(2)**(-x)*b)/​(a*log(S(2))) + S(3)*b**S(2)*atanh(sqrt(a + S(2)**(-x)*b)/​sqrt(a))/​(S(4)*a**(S(5)/​2)*log(S(2))), expand=True, _diff=True, _numerical=True)604 assert rubi_test(rubi_integrate(S(4)**x/​sqrt(a - S(2)**(-x)*b), x), x, S(3)*S(2)**(x + S(-2))*b*sqrt(a - S(2)**(-x)*b)/​(a**S(2)*log(S(2))) + S(2)**(S(2)*x + S(-1))*sqrt(a - S(2)**(-x)*b)/​(a*log(S(2))) + S(3)*b**S(2)*atanh(sqrt(a - S(2)**(-x)*b)/​sqrt(a))/​(S(4)*a**(S(5)/​2)*log(S(2))), expand=True, _diff=True, _numerical=True)605 assert rubi_test(rubi_integrate(S(2)**(S(2)*x)/​sqrt(a - S(2)**(-x)*b), x), x, S(3)*S(2)**(x + S(-2))*b*sqrt(a - S(2)**(-x)*b)/​(a**S(2)*log(S(2))) + S(2)**(S(2)*x + S(-1))*sqrt(a - S(2)**(-x)*b)/​(a*log(S(2))) + S(3)*b**S(2)*atanh(sqrt(a - S(2)**(-x)*b)/​sqrt(a))/​(S(4)*a**(S(5)/​2)*log(S(2))), expand=True, _diff=True, _numerical=True)606 assert rubi_test(rubi_integrate(S(1)/​(exp(S(2)*x) + S(2)*exp(x) + S(1)), x), x, x - log(exp(x) + S(1)) + S(1)/​(exp(x) + S(1)), expand=True, _diff=True, _numerical=True)607 assert rubi_test(rubi_integrate(S(1)/​(exp(S(2)*x) + S(3)*exp(x) + S(2)), x), x, x/​S(2) - log(exp(x) + S(1)) + log(exp(x) + S(2))/​S(2), expand=True, _diff=True, _numerical=True)608 assert rubi_test(rubi_integrate(S(1)/​(exp(S(2)*x) + exp(x) + S(-1)), x), x, -x + (-sqrt(S(5)) + S(5))*log(S(2)*exp(x) + S(1) + sqrt(S(5)))/​S(10) + (sqrt(S(5)) + S(5))*log(S(2)*exp(x) - sqrt(S(5)) + S(1))/​S(10), expand=True, _diff=True, _numerical=True)609 assert rubi_test(rubi_integrate(S(1)/​(exp(S(2)*x) + S(3)*exp(x) + S(3)), x), x, x/​S(3) - log(exp(S(2)*x) + S(3)*exp(x) + S(3))/​S(6) - sqrt(S(3))*atan(sqrt(S(3))*(S(2)*exp(x) + S(3))/​S(3))/​S(3), expand=True, _diff=True, _numerical=True)610 assert rubi_test(rubi_integrate(S(1)/​(a + b*exp(x) + c*exp(S(2)*x)), x), x, b*atanh((b + S(2)*c*exp(x))/​sqrt(-S(4)*a*c + b**S(2)))/​(a*sqrt(-S(4)*a*c + b**S(2))) + x/​a - log(a + b*exp(x) + c*exp(S(2)*x))/​(S(2)*a), expand=True, _diff=True, _numerical=True)611 assert rubi_test(rubi_integrate(x/​(exp(S(2)*x) + S(2)*exp(x) + S(1)), x), x, x**S(2)/​S(2) - x*log(exp(x) + S(1)) - x + x/​(exp(x) + S(1)) + log(exp(x) + S(1)) - polylog(S(2), -exp(x)), expand=True, _diff=True, _numerical=True)612 assert rubi_test(rubi_integrate(x/​(exp(S(2)*x) + S(3)*exp(x) + S(2)), x), x, -x*log(S(1) + exp(-x)) + x*log(S(1) + S(2)*exp(-x))/​S(2) - polylog(S(2), -S(2)*exp(-x))/​S(2) + polylog(S(2), -exp(-x)), expand=True, _diff=True, _numerical=True)613 assert rubi_test(rubi_integrate(x/​(exp(S(2)*x) + exp(x) + S(-1)), x), x, S(2)*sqrt(S(5))*x*log(S(1) + (S(1)/​2 + sqrt(S(5))/​S(2))*exp(-x))/​(S(5)*(S(1) + sqrt(S(5)))) - S(2)*sqrt(S(5))*x*log(S(1) + (-sqrt(S(5))/​S(2) + S(1)/​2)*exp(-x))/​(S(5)*(-sqrt(S(5)) + S(1))) + S(2)*sqrt(S(5))*polylog(S(2), (S(-1)/​2 + sqrt(S(5))/​S(2))*exp(-x))/​(S(5)*(-sqrt(S(5)) + S(1))) - S(2)*sqrt(S(5))*polylog(S(2), (-sqrt(S(5))/​S(2) + S(-1)/​2)*exp(-x))/​(S(5)*(S(1) + sqrt(S(5)))), expand=True, _diff=True, _numerical=True)614 assert rubi_test(rubi_integrate(x/​(exp(S(2)*x) + S(3)*exp(x) + S(3)), x), x, -S(2)*sqrt(S(3))*x*log(S(1) + (S(3)/​2 - sqrt(S(3))*I/​S(2))*exp(-x))/​(S(3)*(sqrt(S(3)) + S(3)*I)) + S(2)*sqrt(S(3))*x*log(S(1) + (S(3)/​2 + sqrt(S(3))*I/​S(2))*exp(-x))/​(S(3)*(-sqrt(S(3)) + S(3)*I)) - S(2)*sqrt(S(3))*polylog(S(2), (S(-3)/​2 - sqrt(S(3))*I/​S(2))*exp(-x))/​(S(3)*(-sqrt(S(3)) + S(3)*I)) + S(2)*sqrt(S(3))*polylog(S(2), (S(-3)/​2 + sqrt(S(3))*I/​S(2))*exp(-x))/​(S(3)*(sqrt(S(3)) + S(3)*I)), expand=True, _diff=True, _numerical=True)615 assert rubi_test(rubi_integrate(x/​(a + b*exp(x) + c*exp(S(2)*x)), x), x, S(2)*c*x*log(S(1) + (b/​S(2) + sqrt(-S(4)*a*c + b**S(2))/​S(2))*exp(-x)/​c)/​(-S(4)*a*c + b**S(2) + b*sqrt(-S(4)*a*c + b**S(2))) + S(2)*c*x*log(S(1) + (b/​S(2) - sqrt(-S(4)*a*c + b**S(2))/​S(2))*exp(-x)/​c)/​(-S(4)*a*c + b**S(2) - b*sqrt(-S(4)*a*c + b**S(2))) - S(2)*c*polylog(S(2), (-b/​S(2) - sqrt(-S(4)*a*c + b**S(2))/​S(2))*exp(-x)/​c)/​(-S(4)*a*c + b**S(2) + b*sqrt(-S(4)*a*c + b**S(2))) - S(2)*c*polylog(S(2), (-b/​S(2) + sqrt(-S(4)*a*c + b**S(2))/​S(2))*exp(-x)/​c)/​(-S(4)*a*c + b**S(2) - b*sqrt(-S(4)*a*c + b**S(2))), expand=True, _diff=True, _numerical=True)616 assert rubi_test(rubi_integrate(x**S(2)/​(exp(S(2)*x) + S(2)*exp(x) + S(1)), x), x, x**S(3)/​S(3) - x**S(2)*log(exp(x) + S(1)) + x**S(2)/​(exp(x) + S(1)) + S(2)*x*log(S(1) + exp(-x)) - S(2)*x*polylog(S(2), -exp(x)) - S(2)*polylog(S(2), -exp(-x)) + S(2)*polylog(S(3), -exp(x)), expand=True, _diff=True, _numerical=True)617 assert rubi_test(rubi_integrate(x**S(2)/​(exp(S(2)*x) + S(3)*exp(x) + S(2)), x), x, -x**S(2)*log(S(1) + exp(-x)) + x**S(2)*log(S(1) + S(2)*exp(-x))/​S(2) - x*polylog(S(2), -S(2)*exp(-x)) + S(2)*x*polylog(S(2), -exp(-x)) - polylog(S(3), -S(2)*exp(-x)) + S(2)*polylog(S(3), -exp(-x)), expand=True, _diff=True, _numerical=True)618 assert rubi_test(rubi_integrate(x**S(2)/​(exp(S(2)*x) + exp(x) + S(-1)), x), x, S(2)*sqrt(S(5))*x**S(2)*log(S(1) + (S(1)/​2 + sqrt(S(5))/​S(2))*exp(-x))/​(S(5)*(S(1) + sqrt(S(5)))) - S(2)*sqrt(S(5))*x**S(2)*log(S(1) + (-sqrt(S(5))/​S(2) + S(1)/​2)*exp(-x))/​(S(5)*(-sqrt(S(5)) + S(1))) + S(4)*sqrt(S(5))*x*polylog(S(2), (S(-1)/​2 + sqrt(S(5))/​S(2))*exp(-x))/​(S(5)*(-sqrt(S(5)) + S(1))) - S(4)*sqrt(S(5))*x*polylog(S(2), (-sqrt(S(5))/​S(2) + S(-1)/​2)*exp(-x))/​(S(5)*(S(1) + sqrt(S(5)))) + S(4)*sqrt(S(5))*polylog(S(3), (S(-1)/​2 + sqrt(S(5))/​S(2))*exp(-x))/​(S(5)*(-sqrt(S(5)) + S(1))) - S(4)*sqrt(S(5))*polylog(S(3), (-sqrt(S(5))/​S(2) + S(-1)/​2)*exp(-x))/​(S(5)*(S(1) + sqrt(S(5)))), expand=True, _diff=True, _numerical=True)619 assert rubi_test(rubi_integrate(x**S(2)/​(exp(S(2)*x) + S(3)*exp(x) + S(3)), x), x, -S(2)*sqrt(S(3))*x**S(2)*log(S(1) + (S(3)/​2 - sqrt(S(3))*I/​S(2))*exp(-x))/​(S(3)*(sqrt(S(3)) + S(3)*I)) + S(2)*sqrt(S(3))*x**S(2)*log(S(1) + (S(3)/​2 + sqrt(S(3))*I/​S(2))*exp(-x))/​(S(3)*(-sqrt(S(3)) + S(3)*I)) - S(4)*sqrt(S(3))*x*polylog(S(2), (S(-3)/​2 - sqrt(S(3))*I/​S(2))*exp(-x))/​(S(3)*(-sqrt(S(3)) + S(3)*I)) + S(4)*sqrt(S(3))*x*polylog(S(2), (S(-3)/​2 + sqrt(S(3))*I/​S(2))*exp(-x))/​(S(3)*(sqrt(S(3)) + S(3)*I)) - S(4)*sqrt(S(3))*polylog(S(3), (S(-3)/​2 - sqrt(S(3))*I/​S(2))*exp(-x))/​(S(3)*(-sqrt(S(3)) + S(3)*I)) + S(4)*sqrt(S(3))*polylog(S(3), (S(-3)/​2 + sqrt(S(3))*I/​S(2))*exp(-x))/​(S(3)*(sqrt(S(3)) + S(3)*I)), expand=True, _diff=True, _numerical=True)620 assert rubi_test(rubi_integrate(x**S(2)/​(a + b*exp(x) + c*exp(S(2)*x)), x), x, S(2)*c*x**S(2)*log(S(1) + (b/​S(2) + sqrt(-S(4)*a*c + b**S(2))/​S(2))*exp(-x)/​c)/​(-S(4)*a*c + b**S(2) + b*sqrt(-S(4)*a*c + b**S(2))) + S(2)*c*x**S(2)*log(S(1) + (b/​S(2) - sqrt(-S(4)*a*c + b**S(2))/​S(2))*exp(-x)/​c)/​(-S(4)*a*c + b**S(2) - b*sqrt(-S(4)*a*c + b**S(2))) - S(4)*c*x*polylog(S(2), (-b/​S(2) - sqrt(-S(4)*a*c + b**S(2))/​S(2))*exp(-x)/​c)/​(-S(4)*a*c + b**S(2) + b*sqrt(-S(4)*a*c + b**S(2))) - S(4)*c*x*polylog(S(2), (-b/​S(2) + sqrt(-S(4)*a*c + b**S(2))/​S(2))*exp(-x)/​c)/​(-S(4)*a*c + b**S(2) - b*sqrt(-S(4)*a*c + b**S(2))) - S(4)*c*polylog(S(3), (-b/​S(2) - sqrt(-S(4)*a*c + b**S(2))/​S(2))*exp(-x)/​c)/​(-S(4)*a*c + b**S(2) + b*sqrt(-S(4)*a*c + b**S(2))) - S(4)*c*polylog(S(3), (-b/​S(2) + sqrt(-S(4)*a*c + b**S(2))/​S(2))*exp(-x)/​c)/​(-S(4)*a*c + b**S(2) - b*sqrt(-S(4)*a*c + b**S(2))), expand=True, _diff=True, _numerical=True)621 assert rubi_test(rubi_integrate(S(1)/​(S(2)*f**(c + d*x) + f**(S(2)*c + S(2)*d*x) + S(1)), x), x, x - log(f**(c + d*x) + S(1))/​(d*log(f)) + S(1)/​(d*(f**(c + d*x) + S(1))*log(f)), expand=True, _diff=True, _numerical=True)622 assert rubi_test(rubi_integrate(S(1)/​(a + b*f**(c + d*x) + c*f**(S(2)*c + S(2)*d*x)), x), x, b*atanh((b + S(2)*c*f**(c + d*x))/​sqrt(-S(4)*a*c + b**S(2)))/​(a*d*sqrt(-S(4)*a*c + b**S(2))*log(f)) + x/​a - log(a + b*f**(c + d*x) + c*f**(S(2)*c + S(2)*d*x))/​(S(2)*a*d*log(f)), expand=True, _diff=True, _numerical=True)623 assert rubi_test(rubi_integrate(S(1)/​(a + b*f**(g + h*x) + c*f**(S(2)*g + S(2)*h*x)), x), x, b*atanh((b + S(2)*c*f**(g + h*x))/​sqrt(-S(4)*a*c + b**S(2)))/​(a*h*sqrt(-S(4)*a*c + b**S(2))*log(f)) + x/​a - log(a + b*f**(g + h*x) + c*f**(S(2)*g + S(2)*h*x))/​(S(2)*a*h*log(f)), expand=True, _diff=True, _numerical=True)624 assert rubi_test(rubi_integrate(x/​(S(2)*f**(c + d*x) + f**(S(2)*c + S(2)*d*x) + S(1)), x), x, x**S(2)/​S(2) - x*log(f**(c + d*x) + S(1))/​(d*log(f)) - x/​(d*log(f)) + x/​(d*(f**(c + d*x) + S(1))*log(f)) + log(f**(c + d*x) + S(1))/​(d**S(2)*log(f)**S(2)) - polylog(S(2), -f**(c + d*x))/​(d**S(2)*log(f)**S(2)), expand=True, _diff=True, _numerical=True)625 assert rubi_test(rubi_integrate(x/​(a + b*f**(c + d*x) + c*f**(S(2)*c + S(2)*d*x)), x), x, S(2)*c*x*log(S(1) + f**(-c - d*x)*(b + sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))/​(d*(b + sqrt(-S(4)*a*c + b**S(2)))*sqrt(-S(4)*a*c + b**S(2))*log(f)) - S(2)*c*x*log(S(1) + f**(-c - d*x)*(b - sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))/​(d*(b - sqrt(-S(4)*a*c + b**S(2)))*sqrt(-S(4)*a*c + b**S(2))*log(f)) - S(2)*c*polylog(S(2), -f**(-c - d*x)*(b + sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))/​(d**S(2)*(b + sqrt(-S(4)*a*c + b**S(2)))*sqrt(-S(4)*a*c + b**S(2))*log(f)**S(2)) + S(2)*c*polylog(S(2), -f**(-c - d*x)*(b - sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))/​(d**S(2)*(b - sqrt(-S(4)*a*c + b**S(2)))*sqrt(-S(4)*a*c + b**S(2))*log(f)**S(2)), expand=True, _diff=True, _numerical=True)626 assert rubi_test(rubi_integrate(x**S(2)/​(S(2)*f**(c + d*x) + f**(S(2)*c + S(2)*d*x) + S(1)), x), x, x**S(3)/​S(3) - x**S(2)*log(f**(c + d*x) + S(1))/​(d*log(f)) + x**S(2)/​(d*(f**(c + d*x) + S(1))*log(f)) + S(2)*x*log(f**(-c - d*x) + S(1))/​(d**S(2)*log(f)**S(2)) - S(2)*x*polylog(S(2), -f**(c + d*x))/​(d**S(2)*log(f)**S(2)) - S(2)*polylog(S(2), -f**(-c - d*x))/​(d**S(3)*log(f)**S(3)) + S(2)*polylog(S(3), -f**(c + d*x))/​(d**S(3)*log(f)**S(3)), expand=True, _diff=True, _numerical=True)627 assert rubi_test(rubi_integrate(x**S(2)/​(a + b*f**(c + d*x) + c*f**(S(2)*c + S(2)*d*x)), x), x, S(2)*c*x**S(2)*log(S(1) + f**(-c - d*x)*(b + sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))/​(d*(b + sqrt(-S(4)*a*c + b**S(2)))*sqrt(-S(4)*a*c + b**S(2))*log(f)) - S(2)*c*x**S(2)*log(S(1) + f**(-c - d*x)*(b - sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))/​(d*(b - sqrt(-S(4)*a*c + b**S(2)))*sqrt(-S(4)*a*c + b**S(2))*log(f)) - S(4)*c*x*polylog(S(2), -f**(-c - d*x)*(b + sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))/​(d**S(2)*(b + sqrt(-S(4)*a*c + b**S(2)))*sqrt(-S(4)*a*c + b**S(2))*log(f)**S(2)) + S(4)*c*x*polylog(S(2), -f**(-c - d*x)*(b - sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))/​(d**S(2)*(b - sqrt(-S(4)*a*c + b**S(2)))*sqrt(-S(4)*a*c + b**S(2))*log(f)**S(2)) - S(4)*c*polylog(S(3), -f**(-c - d*x)*(b + sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))/​(d**S(3)*(b + sqrt(-S(4)*a*c + b**S(2)))*sqrt(-S(4)*a*c + b**S(2))*log(f)**S(3)) + S(4)*c*polylog(S(3), -f**(-c - d*x)*(b - sqrt(-S(4)*a*c + b**S(2)))/​(S(2)*c))/​(d**S(3)*(b - sqrt(-S(4)*a*c + b**S(2)))*sqrt(-S(4)*a*c + b**S(2))*log(f)**S(3)), expand=True, _diff=True, _numerical=True)628 assert rubi_test(rubi_integrate((d + e*f**(g + h*x))/​(a + b*f**(g + h*x) + c*f**(S(2)*g + S(2)*h*x)), x), x, d*x/​a - d*log(a + b*f**(g + h*x) + c*f**(S(2)*g + S(2)*h*x))/​(S(2)*a*h*log(f)) + (-S(2)*a*e + b*d)*atanh((b + S(2)*c*f**(g + h*x))/​sqrt(-S(4)*a*c + b**S(2)))/​(a*h*sqrt(-S(4)*a*c + b**S(2))*log(f)), expand=True, _diff=True, _numerical=True)629 assert rubi_test(rubi_integrate((d + e*f**(g + h*x))/​(a + b*f**(g + h*x) + c*f**(S(2)*g + S(2)*h*x)), x), x, d*x/​a - d*log(a + b*f**(g + h*x) + c*f**(S(2)*g + S(2)*h*x))/​(S(2)*a*h*log(f)) + (-S(2)*a*e + b*d)*atanh((b + S(2)*c*f**(g + h*x))/​sqrt(-S(4)*a*c + b**S(2)))/​(a*h*sqrt(-S(4)*a*c + b**S(2))*log(f)), expand=True, _diff=True, _numerical=True)630 assert rubi_test(rubi_integrate(S(1)/​(exp(x) + S(2) + exp(-x)), x), x, -S(1)/​(exp(x) + S(1)), expand=True, _diff=True, _numerical=True)631 assert rubi_test(rubi_integrate(x/​(exp(x) + S(2) + exp(-x)), x), x, x - x/​(exp(x) + S(1)) - log(exp(x) + S(1)), expand=True, _diff=True, _numerical=True)632 assert rubi_test(rubi_integrate(x**S(2)/​(exp(x) + S(2) + exp(-x)), x), x, -x**S(2)/​(exp(x) + S(1)) - S(2)*x*log(S(1) + exp(-x)) + S(2)*polylog(S(2), -exp(-x)), expand=True, _diff=True, _numerical=True)633 assert rubi_test(rubi_integrate(S(1)/​(f**(-c - d*x) + f**(c + d*x) + S(2)), x), x, -S(1)/​(d*(f**(c + d*x) + S(1))*log(f)), expand=True, _diff=True, _numerical=True)634 assert rubi_test(rubi_integrate(x/​(f**(-c - d*x) + f**(c + d*x) + S(2)), x), x, x/​(d*log(f)) - x/​(d*(f**(c + d*x) + S(1))*log(f)) - log(f**(c + d*x) + S(1))/​(d**S(2)*log(f)**S(2)), expand=True, _diff=True, _numerical=True)635 assert rubi_test(rubi_integrate(x**S(2)/​(f**(-c - d*x) + f**(c + d*x) + S(2)), x), x, -x**S(2)/​(d*(f**(c + d*x) + S(1))*log(f)) - S(2)*x*log(f**(-c - d*x) + S(1))/​(d**S(2)*log(f)**S(2)) + S(2)*polylog(S(2), -f**(-c - d*x))/​(d**S(3)*log(f)**S(3)), expand=True, _diff=True, _numerical=True)636 assert rubi_test(rubi_integrate(S(1)/​(S(3)**x + S(2) + S(3)**(-x)), x), x, -S(1)/​((S(3)**x + S(1))*log(S(3))), expand=True, _diff=True, _numerical=True)637 assert rubi_test(rubi_integrate(S(1)/​(S(2)*exp(x) + S(1) - exp(-x)), x), x, log(-S(2)*exp(x) + S(1))/​S(3) - log(exp(x) + S(1))/​S(3), expand=True, _diff=True, _numerical=True)638 assert rubi_test(rubi_integrate(S(1)/​(a + b*exp(-x) + c*exp(x)), x), x, -S(2)*atanh((a + S(2)*c*exp(x))/​sqrt(a**S(2) - S(4)*b*c))/​sqrt(a**S(2) - S(4)*b*c), expand=True, _diff=True, _numerical=True)639 assert rubi_test(rubi_integrate(x/​(a + b*exp(-x) + c*exp(x)), x), x, x*log(S(2)*c*exp(x)/​(a - sqrt(a**S(2) - S(4)*b*c)) + S(1))/​sqrt(a**S(2) - S(4)*b*c) - x*log(S(2)*c*exp(x)/​(a + sqrt(a**S(2) - S(4)*b*c)) + S(1))/​sqrt(a**S(2) - S(4)*b*c) + polylog(S(2), -S(2)*c*exp(x)/​(a - sqrt(a**S(2) - S(4)*b*c)))/​sqrt(a**S(2) - S(4)*b*c) - polylog(S(2), -S(2)*c*exp(x)/​(a + sqrt(a**S(2) - S(4)*b*c)))/​sqrt(a**S(2) - S(4)*b*c), expand=True, _diff=True, _numerical=True)640 assert rubi_test(rubi_integrate(x**S(2)/​(a + b*exp(-x) + c*exp(x)), x), x, x**S(2)*log(S(2)*c*exp(x)/​(a - sqrt(a**S(2) - S(4)*b*c)) + S(1))/​sqrt(a**S(2) - S(4)*b*c) - x**S(2)*log(S(2)*c*exp(x)/​(a + sqrt(a**S(2) - S(4)*b*c)) + S(1))/​sqrt(a**S(2) - S(4)*b*c) + S(2)*x*polylog(S(2), -S(2)*c*exp(x)/​(a - sqrt(a**S(2) - S(4)*b*c)))/​sqrt(a**S(2) - S(4)*b*c) - S(2)*x*polylog(S(2), -S(2)*c*exp(x)/​(a + sqrt(a**S(2) - S(4)*b*c)))/​sqrt(a**S(2) - S(4)*b*c) - S(2)*polylog(S(3), -S(2)*c*exp(x)/​(a - sqrt(a**S(2) - S(4)*b*c)))/​sqrt(a**S(2) - S(4)*b*c) + S(2)*polylog(S(3), -S(2)*c*exp(x)/​(a + sqrt(a**S(2) - S(4)*b*c)))/​sqrt(a**S(2) - S(4)*b*c), expand=True, _diff=True, _numerical=True)641 assert rubi_test(rubi_integrate(S(1)/​(a + b*f**(-c - d*x) + c*f**(c + d*x)), x), x, -S(2)*atanh((a + S(2)*c*f**(c + d*x))/​sqrt(a**S(2) - S(4)*b*c))/​(d*sqrt(a**S(2) - S(4)*b*c)*log(f)), expand=True, _diff=True, _numerical=True)642 assert rubi_test(rubi_integrate(x/​(a + b*f**(-c - d*x) + c*f**(c + d*x)), x), x, x*log(S(2)*c*f**(c + d*x)/​(a - sqrt(a**S(2) - S(4)*b*c)) + S(1))/​(d*sqrt(a**S(2) - S(4)*b*c)*log(f)) - x*log(S(2)*c*f**(c + d*x)/​(a + sqrt(a**S(2) - S(4)*b*c)) + S(1))/​(d*sqrt(a**S(2) - S(4)*b*c)*log(f)) + polylog(S(2), -S(2)*c*f**(c + d*x)/​(a - sqrt(a**S(2) - S(4)*b*c)))/​(d**S(2)*sqrt(a**S(2) - S(4)*b*c)*log(f)**S(2)) - polylog(S(2), -S(2)*c*f**(c + d*x)/​(a + sqrt(a**S(2) - S(4)*b*c)))/​(d**S(2)*sqrt(a**S(2) - S(4)*b*c)*log(f)**S(2)), expand=True, _diff=True, _numerical=True)643 assert rubi_test(rubi_integrate(x**S(2)/​(a + b*f**(-c - d*x) + c*f**(c + d*x)), x), x, x**S(2)*log(S(2)*c*f**(c + d*x)/​(a - sqrt(a**S(2) - S(4)*b*c)) + S(1))/​(d*sqrt(a**S(2) - S(4)*b*c)*log(f)) - x**S(2)*log(S(2)*c*f**(c + d*x)/​(a + sqrt(a**S(2) - S(4)*b*c)) + S(1))/​(d*sqrt(a**S(2) - S(4)*b*c)*log(f)) + S(2)*x*polylog(S(2), -S(2)*c*f**(c + d*x)/​(a - sqrt(a**S(2) - S(4)*b*c)))/​(d**S(2)*sqrt(a**S(2) - S(4)*b*c)*log(f)**S(2)) - S(2)*x*polylog(S(2), -S(2)*c*f**(c + d*x)/​(a + sqrt(a**S(2) - S(4)*b*c)))/​(d**S(2)*sqrt(a**S(2) - S(4)*b*c)*log(f)**S(2)) - S(2)*polylog(S(3), -S(2)*c*f**(c + d*x)/​(a - sqrt(a**S(2) - S(4)*b*c)))/​(d**S(3)*sqrt(a**S(2) - S(4)*b*c)*log(f)**S(3)) + S(2)*polylog(S(3), -S(2)*c*f**(c + d*x)/​(a + sqrt(a**S(2) - S(4)*b*c)))/​(d**S(3)*sqrt(a**S(2) - S(4)*b*c)*log(f)**S(3)), expand=True, _diff=True, _numerical=True)644 assert rubi_test(rubi_integrate((F**(sqrt(-a*x + S(1))/​sqrt(a*x + S(1))))**n/​(-a**S(2)*x**S(2) + S(1)), x), x, -F**(-n*sqrt(-a*x + S(1))/​sqrt(a*x + S(1)))*(F**(sqrt(-a*x + S(1))/​sqrt(a*x + S(1))))**n*Ei(n*sqrt(-a*x + S(1))*log(F)/​sqrt(a*x + S(1)))/​a, expand=True, _diff=True, _numerical=True)645 assert rubi_test(rubi_integrate(F**(S(3)*sqrt(-a*x + S(1))/​sqrt(a*x + S(1)))/​(-a**S(2)*x**S(2) + S(1)), x), x, -Ei(S(3)*sqrt(-a*x + S(1))*log(F)/​sqrt(a*x + S(1)))/​a, expand=True, _diff=True, _numerical=True)646 assert rubi_test(rubi_integrate(F**(S(2)*sqrt(-a*x + S(1))/​sqrt(a*x + S(1)))/​(-a**S(2)*x**S(2) + S(1)), x), x, -Ei(S(2)*sqrt(-a*x + S(1))*log(F)/​sqrt(a*x + S(1)))/​a, expand=True, _diff=True, _numerical=True)647 assert rubi_test(rubi_integrate(F**(sqrt(-a*x + S(1))/​sqrt(a*x + S(1)))/​(-a**S(2)*x**S(2) + S(1)), x), x, -Ei(sqrt(-a*x + S(1))*log(F)/​sqrt(a*x + S(1)))/​a, expand=True, _diff=True, _numerical=True)648 assert rubi_test(rubi_integrate(F**(-sqrt(-a*x + S(1))/​sqrt(a*x + S(1)))/​(-a**S(2)*x**S(2) + S(1)), x), x, -Ei(-sqrt(-a*x + S(1))*log(F)/​sqrt(a*x + S(1)))/​a, expand=True, _diff=True, _numerical=True)649 assert rubi_test(rubi_integrate(F**(-S(2)*sqrt(-a*x + S(1))/​sqrt(a*x + S(1)))/​(-a**S(2)*x**S(2) + S(1)), x), x, -Ei(-S(2)*sqrt(-a*x + S(1))*log(F)/​sqrt(a*x + S(1)))/​a, expand=True, _diff=True, _numerical=True)650 assert rubi_test(rubi_integrate((F**(sqrt(-c*x + S(1))/​sqrt(c*x + S(1)))*b + a)**n/​(-c**S(2)*x**S(2) + S(1)), x), x, -Integral((F**x*b + a)**n/​x, (x, sqrt(-c*x + S(1))/​sqrt(c*x + S(1))))/​c, expand=True, _diff=True, _numerical=True)651 assert rubi_test(rubi_integrate((F**(sqrt(-c*x + S(1))/​sqrt(c*x + S(1)))*b + a)**S(3)/​(-c**S(2)*x**S(2) + S(1)), x), x, -a**S(3)*log(sqrt(-c*x + S(1))/​sqrt(c*x + S(1)))/​c - S(3)*a**S(2)*b*Ei(sqrt(-c*x + S(1))*log(F)/​sqrt(c*x + S(1)))/​c - S(3)*a*b**S(2)*Ei(S(2)*sqrt(-c*x + S(1))*log(F)/​sqrt(c*x + S(1)))/​c - b**S(3)*Ei(S(3)*sqrt(-c*x + S(1))*log(F)/​sqrt(c*x + S(1)))/​c, expand=True, _diff=True, _numerical=True)652 assert rubi_test(rubi_integrate((F**(sqrt(-c*x + S(1))/​sqrt(c*x + S(1)))*b + a)**S(2)/​(-c**S(2)*x**S(2) + S(1)), x), x, -a**S(2)*log(sqrt(-c*x + S(1))/​sqrt(c*x + S(1)))/​c - S(2)*a*b*Ei(sqrt(-c*x + S(1))*log(F)/​sqrt(c*x + S(1)))/​c - b**S(2)*Ei(S(2)*sqrt(-c*x + S(1))*log(F)/​sqrt(c*x + S(1)))/​c, expand=True, _diff=True, _numerical=True)653 assert rubi_test(rubi_integrate((F**(sqrt(-c*x + S(1))/​sqrt(c*x + S(1)))*b + a)/​(-c**S(2)*x**S(2) + S(1)), x), x, -a*log(sqrt(-c*x + S(1))/​sqrt(c*x + S(1)))/​c - b*Ei(sqrt(-c*x + S(1))*log(F)/​sqrt(c*x + S(1)))/​c, expand=True, _diff=True, _numerical=True)654 assert rubi_test(rubi_integrate(S(1)/​((F**(sqrt(-c*x + S(1))/​sqrt(c*x + S(1)))*b + a)*(-c**S(2)*x**S(2) + S(1))), x), x, -Integral(S(1)/​(x*(F**x*b + a)), (x, sqrt(-c*x + S(1))/​sqrt(c*x + S(1))))/​c, expand=True, _diff=True, _numerical=True)655 assert rubi_test(rubi_integrate(S(1)/​((F**(sqrt(-c*x + S(1))/​sqrt(c*x + S(1)))*b + a)**S(2)*(-c**S(2)*x**S(2) + S(1))), x), x, -Integral(S(1)/​(x*(F**x*b + a)**S(2)), (x, sqrt(-c*x + S(1))/​sqrt(c*x + S(1))))/​c, expand=True, _diff=True, _numerical=True)656 assert rubi_test(rubi_integrate(a**x*b**x*x**S(2), x), x, a**x*b**x*x**S(2)/​(log(a) + log(b)) - S(2)*a**x*b**x*x/​(log(a) + log(b))**S(2) + S(2)*a**x*b**x/​(log(a) + log(b))**S(3), expand=True, _diff=True, _numerical=True)657 assert rubi_test(rubi_integrate(a**x*b**x*x, x), x, a**x*b**x*x/​(log(a) + log(b)) - a**x*b**x/​(log(a) + log(b))**S(2), expand=True, _diff=True, _numerical=True)658 assert rubi_test(rubi_integrate(a**x*b**x, x), x, a**x*b**x/​(log(a) + log(b)), expand=True, _diff=True, _numerical=True)659 assert rubi_test(rubi_integrate(a**x*b**x/​x, x), x, Ei(x*(log(a) + log(b))), expand=True, _diff=True, _numerical=True)660 assert rubi_test(rubi_integrate(a**x*b**x/​x**S(2), x), x, -a**x*b**x/​x + (log(a) + log(b))*Ei(x*(log(a) + log(b))), expand=True, _diff=True, _numerical=True)661 assert rubi_test(rubi_integrate(a**x*b**x/​x**S(3), x), x, -a**x*b**x*(log(a) + log(b))/​(S(2)*x) - a**x*b**x/​(S(2)*x**S(2)) + (log(a) + log(b))**S(2)*Ei(x*(log(a) + log(b)))/​S(2), expand=True, _diff=True, _numerical=True)662 assert rubi_test(rubi_integrate(a**x*b**x*c**x, x), x, a**x*b**x*c**x/​(log(a) + log(b) + log(c)), expand=True, _diff=True, _numerical=True)663 assert rubi_test(rubi_integrate(a**x*b**(-x), x), x, a**x*b**(-x)/​(log(a) - log(b)), expand=True, _diff=True, _numerical=True)664 assert rubi_test(rubi_integrate(a**x*b**(-x)*x**S(2), x), x, a**x*b**(-x)*x**S(2)/​(log(a) - log(b)) - S(2)*a**x*b**(-x)*x/​(log(a) - log(b))**S(2) + S(2)*a**x*b**(-x)/​(log(a) - log(b))**S(3), expand=True, _diff=True, _numerical=True)665 assert rubi_test(rubi_integrate(exp(S(2)*x)/​(a + b*exp(x)), x), x, -a*log(a + b*exp(x))/​b**S(2) + exp(x)/​b, expand=True, _diff=True, _numerical=True)666 assert rubi_test(rubi_integrate(exp(S(2)*x)/​(a + b*exp(x))**S(2), x), x, a/​(b**S(2)*(a + b*exp(x))) + log(a + b*exp(x))/​b**S(2), expand=True, _diff=True, _numerical=True)667 assert rubi_test(rubi_integrate(exp(S(2)*x)/​(a + b*exp(x))**S(3), x), x, exp(S(2)*x)/​(S(2)*a*(a + b*exp(x))**S(2)), expand=True, _diff=True, _numerical=True)668 assert rubi_test(rubi_integrate(exp(S(2)*x)/​(a + b*exp(x))**S(4), x), x, a/​(S(3)*b**S(2)*(a + b*exp(x))**S(3)) - S(1)/​(S(2)*b**S(2)*(a + b*exp(x))**S(2)), expand=True, _diff=True, _numerical=True)669 assert rubi_test(rubi_integrate(exp(S(4)*x)/​(a + b*exp(S(2)*x)), x), x, -a*log(a + b*exp(S(2)*x))/​(S(2)*b**S(2)) + exp(S(2)*x)/​(S(2)*b), expand=True, _diff=True, _numerical=True)670 assert rubi_test(rubi_integrate(exp(S(4)*x)/​(a + b*exp(S(2)*x))**S(2), x), x, a/​(S(2)*b**S(2)*(a + b*exp(S(2)*x))) + log(a + b*exp(S(2)*x))/​(S(2)*b**S(2)), expand=True, _diff=True, _numerical=True)671 assert rubi_test(rubi_integrate(exp(S(4)*x)/​(a + b*exp(S(2)*x))**S(3), x), x, exp(S(4)*x)/​(S(4)*a*(a + b*exp(S(2)*x))**S(2)), expand=True, _diff=True, _numerical=True)672 assert rubi_test(rubi_integrate(exp(S(4)*x)/​(a + b*exp(S(2)*x))**S(4), x), x, a/​(S(6)*b**S(2)*(a + b*exp(S(2)*x))**S(3)) - S(1)/​(S(4)*b**S(2)*(a + b*exp(S(2)*x))**S(2)), expand=True, _diff=True, _numerical=True)673 assert rubi_test(rubi_integrate(exp(S(4)*x)/​(a + b*exp(S(2)*x))**(S(2)/​3), x), x, -S(3)*a*(a + b*exp(S(2)*x))**(S(1)/​3)/​(S(2)*b**S(2)) + S(3)*(a + b*exp(S(2)*x))**(S(4)/​3)/​(S(8)*b**S(2)), expand=True, _diff=True, _numerical=True)674 assert rubi_test(rubi_integrate((a + b*exp(n*x))*exp(-n*x), x), x, -a*exp(-n*x)/​n + b*x, expand=True, _diff=True, _numerical=True)675 assert rubi_test(rubi_integrate((a + b*exp(n*x))**S(2)*exp(-n*x), x), x, -a**S(2)*exp(-n*x)/​n + S(2)*a*b*x + b**S(2)*exp(n*x)/​n, expand=True, _diff=True, _numerical=True)676 assert rubi_test(rubi_integrate((a + b*exp(n*x))**S(3)*exp(-n*x), x), x, -a**S(3)*exp(-n*x)/​n + S(3)*a**S(2)*b*x + S(3)*a*b**S(2)*exp(n*x)/​n + b**S(3)*exp(S(2)*n*x)/​(S(2)*n), expand=True, _diff=True, _numerical=True)677 assert rubi_test(rubi_integrate(exp(-n*x)/​(a + b*exp(n*x)), x), x, -exp(-n*x)/​(a*n) - b*x/​a**S(2) + b*log(a + b*exp(n*x))/​(a**S(2)*n), expand=True, _diff=True, _numerical=True)678 assert rubi_test(rubi_integrate(exp(-n*x)/​(a + b*exp(n*x))**S(2), x), x, -b/​(a**S(2)*n*(a + b*exp(n*x))) - exp(-n*x)/​(a**S(2)*n) - S(2)*b*x/​a**S(3) + S(2)*b*log(a + b*exp(n*x))/​(a**S(3)*n), expand=True, _diff=True, _numerical=True)679 assert rubi_test(rubi_integrate(exp(-n*x)/​(a + b*exp(n*x))**S(3), x), x, -b/​(S(2)*a**S(2)*n*(a + b*exp(n*x))**S(2)) - S(2)*b/​(a**S(3)*n*(a + b*exp(n*x))) - exp(-n*x)/​(a**S(3)*n) - S(3)*b*x/​a**S(4) + S(3)*b*log(a + b*exp(n*x))/​(a**S(4)*n), expand=True, _diff=True, _numerical=True)680 assert rubi_test(rubi_integrate(f**(a + b*x)/​(c + d*f**(S(2)*b*x + e)), x), x, f**(a - e/​S(2))*atan(sqrt(d)*f**(b*x + e/​S(2))/​sqrt(c))/​(b*sqrt(c)*sqrt(d)*log(f)), expand=True, _diff=True, _numerical=True)681 assert rubi_test(rubi_integrate(f**(a + S(2)*b*x)/​(c + d*f**(S(2)*b*x + e)), x), x, f**(a - e)*log(c + d*f**(S(2)*b*x + e))/​(S(2)*b*d*log(f)), expand=True, _diff=True, _numerical=True)682 assert rubi_test(rubi_integrate(f**(a + S(3)*b*x)/​(c + d*f**(S(2)*b*x + e)), x), x, -sqrt(c)*f**(a - S(3)*e/​S(2))*atan(sqrt(d)*f**(b*x + e/​S(2))/​sqrt(c))/​(b*d**(S(3)/​2)*log(f)) + f**(a + b*x - e)/​(b*d*log(f)), expand=True, _diff=True, _numerical=True)683 assert rubi_test(rubi_integrate(f**(a + S(4)*b*x)/​(c + d*f**(S(2)*b*x + e)), x), x, -c*f**(a - S(2)*e)*log(c + d*f**(S(2)*b*x + e))/​(S(2)*b*d**S(2)*log(f)) + f**(a + S(2)*b*x - e)/​(S(2)*b*d*log(f)), expand=True, _diff=True, _numerical=True)684 assert rubi_test(rubi_integrate(f**(a + S(5)*b*x)/​(c + d*f**(S(2)*b*x + e)), x), x, c**(S(3)/​2)*f**(a - S(5)*e/​S(2))*atan(sqrt(d)*f**(b*x + e/​S(2))/​sqrt(c))/​(b*d**(S(5)/​2)*log(f)) - c*f**(a + b*x - S(2)*e)/​(b*d**S(2)*log(f)) + f**(a + S(3)*b*x - e)/​(S(3)*b*d*log(f)), expand=True, _diff=True, _numerical=True)685 assert rubi_test(rubi_integrate(exp(x)/​(exp(S(2)*x) + S(1)), x), x, atan(exp(x)), expand=True, _diff=True, _numerical=True)686 assert rubi_test(rubi_integrate(exp(x)/​(-exp(S(2)*x) + S(1)), x), x, atanh(exp(x)), expand=True, _diff=True, _numerical=True)687 assert rubi_test(rubi_integrate(x*exp(x)/​(-exp(S(2)*x) + S(1)), x), x, x*atanh(exp(x)) + polylog(S(2), -exp(x))/​S(2) - polylog(S(2), exp(x))/​S(2), expand=True, _diff=True, _numerical=True)688 assert rubi_test(rubi_integrate(x**S(2)*exp(x)/​(-exp(S(2)*x) + S(1)), x), x, x**S(2)*atanh(exp(x)) + x*polylog(S(2), -exp(x)) - x*polylog(S(2), exp(x)) - polylog(S(3), -exp(x)) + polylog(S(3), exp(x)), expand=True, _diff=True, _numerical=True)689 assert rubi_test(rubi_integrate(x**S(3)*exp(x)/​(-exp(S(2)*x) + S(1)), x), x, x**S(3)*atanh(exp(x)) + S(3)*x**S(2)*polylog(S(2), -exp(x))/​S(2) - S(3)*x**S(2)*polylog(S(2), exp(x))/​S(2) - S(3)*x*polylog(S(3), -exp(x)) + S(3)*x*polylog(S(3), exp(x)) + S(3)*polylog(S(4), -exp(x)) - S(3)*polylog(S(4), exp(x)), expand=True, _diff=True, _numerical=True)690 assert rubi_test(rubi_integrate(f**x/​(a + b*f**(S(2)*x)), x), x, atan(sqrt(b)*f**x/​sqrt(a))/​(sqrt(a)*sqrt(b)*log(f)), expand=True, _diff=True, _numerical=True)691 assert rubi_test(rubi_integrate(f**x*x/​(a + b*f**(S(2)*x)), x), x, x*atan(sqrt(b)*f**x/​sqrt(a))/​(sqrt(a)*sqrt(b)*log(f)) - I*polylog(S(2), -I*sqrt(b)*f**x/​sqrt(a))/​(S(2)*sqrt(a)*sqrt(b)*log(f)**S(2)) + I*polylog(S(2), I*sqrt(b)*f**x/​sqrt(a))/​(S(2)*sqrt(a)*sqrt(b)*log(f)**S(2)), expand=True, _diff=True, _numerical=True)692 assert rubi_test(rubi_integrate(f**x*x**S(2)/​(a + b*f**(S(2)*x)), x), x, x**S(2)*atan(sqrt(b)*f**x/​sqrt(a))/​(sqrt(a)*sqrt(b)*log(f)) - I*x*polylog(S(2), -I*sqrt(b)*f**x/​sqrt(a))/​(sqrt(a)*sqrt(b)*log(f)**S(2)) + I*x*polylog(S(2), I*sqrt(b)*f**x/​sqrt(a))/​(sqrt(a)*sqrt(b)*log(f)**S(2)) + I*polylog(S(3), -I*sqrt(b)*f**x/​sqrt(a))/​(sqrt(a)*sqrt(b)*log(f)**S(3)) - I*polylog(S(3), I*sqrt(b)*f**x/​sqrt(a))/​(sqrt(a)*sqrt(b)*log(f)**S(3)), expand=True, _diff=True, _numerical=True)693 assert rubi_test(rubi_integrate(f**x*x**S(3)/​(a + b*f**(S(2)*x)), x), x, x**S(3)*atan(sqrt(b)*f**x/​sqrt(a))/​(sqrt(a)*sqrt(b)*log(f)) - S(3)*I*x**S(2)*polylog(S(2), -I*sqrt(b)*f**x/​sqrt(a))/​(S(2)*sqrt(a)*sqrt(b)*log(f)**S(2)) + S(3)*I*x**S(2)*polylog(S(2), I*sqrt(b)*f**x/​sqrt(a))/​(S(2)*sqrt(a)*sqrt(b)*log(f)**S(2)) + S(3)*I*x*polylog(S(3), -I*sqrt(b)*f**x/​sqrt(a))/​(sqrt(a)*sqrt(b)*log(f)**S(3)) - S(3)*I*x*polylog(S(3), I*sqrt(b)*f**x/​sqrt(a))/​(sqrt(a)*sqrt(b)*log(f)**S(3)) - S(3)*I*polylog(S(4), -I*sqrt(b)*f**x/​sqrt(a))/​(sqrt(a)*sqrt(b)*log(f)**S(4)) + S(3)*I*polylog(S(4), I*sqrt(b)*f**x/​sqrt(a))/​(sqrt(a)*sqrt(b)*log(f)**S(4)), expand=True, _diff=True, _numerical=True)694 assert rubi_test(rubi_integrate(f**x/​(a + b*f**(S(2)*x))**S(2), x), x, f**x/​(S(2)*a*(a + b*f**(S(2)*x))*log(f)) + atan(sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(3)/​2)*sqrt(b)*log(f)), expand=True, _diff=True, _numerical=True)695 assert rubi_test(rubi_integrate(f**x*x/​(a + b*f**(S(2)*x))**S(2), x), x, f**x*x/​(S(2)*a*(a + b*f**(S(2)*x))*log(f)) + x*atan(sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(3)/​2)*sqrt(b)*log(f)) - atan(sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(3)/​2)*sqrt(b)*log(f)**S(2)) - I*polylog(S(2), -I*sqrt(b)*f**x/​sqrt(a))/​(S(4)*a**(S(3)/​2)*sqrt(b)*log(f)**S(2)) + I*polylog(S(2), I*sqrt(b)*f**x/​sqrt(a))/​(S(4)*a**(S(3)/​2)*sqrt(b)*log(f)**S(2)), expand=True, _diff=True, _numerical=True)696 assert rubi_test(rubi_integrate(f**x*x**S(2)/​(a + b*f**(S(2)*x))**S(2), x), x, f**x*x**S(2)/​(S(2)*a*(a + b*f**(S(2)*x))*log(f)) + x**S(2)*atan(sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(3)/​2)*sqrt(b)*log(f)) - x*atan(sqrt(b)*f**x/​sqrt(a))/​(a**(S(3)/​2)*sqrt(b)*log(f)**S(2)) - I*x*polylog(S(2), -I*sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(3)/​2)*sqrt(b)*log(f)**S(2)) + I*x*polylog(S(2), I*sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(3)/​2)*sqrt(b)*log(f)**S(2)) + I*polylog(S(2), -I*sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(3)/​2)*sqrt(b)*log(f)**S(3)) - I*polylog(S(2), I*sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(3)/​2)*sqrt(b)*log(f)**S(3)) + I*polylog(S(3), -I*sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(3)/​2)*sqrt(b)*log(f)**S(3)) - I*polylog(S(3), I*sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(3)/​2)*sqrt(b)*log(f)**S(3)), expand=True, _diff=True, _numerical=True)697 assert rubi_test(rubi_integrate(f**x*x**S(3)/​(a + b*f**(S(2)*x))**S(2), x), x, f**x*x**S(3)/​(S(2)*a*(a + b*f**(S(2)*x))*log(f)) + x**S(3)*atan(sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(3)/​2)*sqrt(b)*log(f)) - S(3)*x**S(2)*atan(sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(3)/​2)*sqrt(b)*log(f)**S(2)) - S(3)*I*x**S(2)*polylog(S(2), -I*sqrt(b)*f**x/​sqrt(a))/​(S(4)*a**(S(3)/​2)*sqrt(b)*log(f)**S(2)) + S(3)*I*x**S(2)*polylog(S(2), I*sqrt(b)*f**x/​sqrt(a))/​(S(4)*a**(S(3)/​2)*sqrt(b)*log(f)**S(2)) + S(3)*I*x*polylog(S(2), -I*sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(3)/​2)*sqrt(b)*log(f)**S(3)) - S(3)*I*x*polylog(S(2), I*sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(3)/​2)*sqrt(b)*log(f)**S(3)) + S(3)*I*x*polylog(S(3), -I*sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(3)/​2)*sqrt(b)*log(f)**S(3)) - S(3)*I*x*polylog(S(3), I*sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(3)/​2)*sqrt(b)*log(f)**S(3)) - S(3)*I*polylog(S(3), -I*sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(3)/​2)*sqrt(b)*log(f)**S(4)) + S(3)*I*polylog(S(3), I*sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(3)/​2)*sqrt(b)*log(f)**S(4)) - S(3)*I*polylog(S(4), -I*sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(3)/​2)*sqrt(b)*log(f)**S(4)) + S(3)*I*polylog(S(4), I*sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(3)/​2)*sqrt(b)*log(f)**S(4)), expand=True, _diff=True, _numerical=True)698 assert rubi_test(rubi_integrate(f**x*x/​(a + b*f**(S(2)*x))**S(3), x), x, f**x*x/​(S(4)*a*(a + b*f**(S(2)*x))**S(2)*log(f)) + S(3)*f**x*x/​(S(8)*a**S(2)*(a + b*f**(S(2)*x))*log(f)) - f**x/​(S(8)*a**S(2)*(a + b*f**(S(2)*x))*log(f)**S(2)) + S(3)*x*atan(sqrt(b)*f**x/​sqrt(a))/​(S(8)*a**(S(5)/​2)*sqrt(b)*log(f)) - atan(sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(5)/​2)*sqrt(b)*log(f)**S(2)) - S(3)*I*polylog(S(2), -I*sqrt(b)*f**x/​sqrt(a))/​(S(16)*a**(S(5)/​2)*sqrt(b)*log(f)**S(2)) + S(3)*I*polylog(S(2), I*sqrt(b)*f**x/​sqrt(a))/​(S(16)*a**(S(5)/​2)*sqrt(b)*log(f)**S(2)), expand=True, _diff=True, _numerical=True)699 assert rubi_test(rubi_integrate(f**x*x**S(2)/​(a + b*f**(S(2)*x))**S(3), x), x, f**x*x**S(2)/​(S(4)*a*(a + b*f**(S(2)*x))**S(2)*log(f)) + S(3)*f**x*x**S(2)/​(S(8)*a**S(2)*(a + b*f**(S(2)*x))*log(f)) - f**x*x/​(S(4)*a**S(2)*(a + b*f**(S(2)*x))*log(f)**S(2)) + S(3)*x**S(2)*atan(sqrt(b)*f**x/​sqrt(a))/​(S(8)*a**(S(5)/​2)*sqrt(b)*log(f)) - x*atan(sqrt(b)*f**x/​sqrt(a))/​(a**(S(5)/​2)*sqrt(b)*log(f)**S(2)) - S(3)*I*x*polylog(S(2), -I*sqrt(b)*f**x/​sqrt(a))/​(S(8)*a**(S(5)/​2)*sqrt(b)*log(f)**S(2)) + S(3)*I*x*polylog(S(2), I*sqrt(b)*f**x/​sqrt(a))/​(S(8)*a**(S(5)/​2)*sqrt(b)*log(f)**S(2)) + atan(sqrt(b)*f**x/​sqrt(a))/​(S(4)*a**(S(5)/​2)*sqrt(b)*log(f)**S(3)) + I*polylog(S(2), -I*sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(5)/​2)*sqrt(b)*log(f)**S(3)) - I*polylog(S(2), I*sqrt(b)*f**x/​sqrt(a))/​(S(2)*a**(S(5)/​2)*sqrt(b)*log(f)**S(3)) + S(3)*I*polylog(S(3), -I*sqrt(b)*f**x/​sqrt(a))/​(S(8)*a**(S(5)/​2)*sqrt(b)*log(f)**S(3)) - S(3)*I*polylog(S(3), I*sqrt(b)*f**x/​sqrt(a))/​(S(8)*a**(S(5)/​2)*sqrt(b)*log(f)**S(3)), expand=True, _diff=True, _numerical=True)700 assert rubi_test(rubi_integrate(x/​(a*f**x + b*f**(-x)), x), x, x*atan(sqrt(a)*f**x/​sqrt(b))/​(sqrt(a)*sqrt(b)*log(f)) - I*polylog(S(2), -I*sqrt(a)*f**x/​sqrt(b))/​(S(2)*sqrt(a)*sqrt(b)*log(f)**S(2)) + I*polylog(S(2), I*sqrt(a)*f**x/​sqrt(b))/​(S(2)*sqrt(a)*sqrt(b)*log(f)**S(2)), expand=True, _diff=True, _numerical=True)701 assert rubi_test(rubi_integrate(x**S(2)/​(a*f**x + b*f**(-x)), x), x, x**S(2)*atan(sqrt(a)*f**x/​sqrt(b))/​(sqrt(a)*sqrt(b)*log(f)) - I*x*polylog(S(2), -I*sqrt(a)*f**x/​sqrt(b))/​(sqrt(a)*sqrt(b)*log(f)**S(2)) + I*x*polylog(S(2), I*sqrt(a)*f**x/​sqrt(b))/​(sqrt(a)*sqrt(b)*log(f)**S(2)) + I*polylog(S(3), -I*sqrt(a)*f**x/​sqrt(b))/​(sqrt(a)*sqrt(b)*log(f)**S(3)) - I*polylog(S(3), I*sqrt(a)*f**x/​sqrt(b))/​(sqrt(a)*sqrt(b)*log(f)**S(3)), expand=True, _diff=True, _numerical=True)702 assert rubi_test(rubi_integrate(x**S(3)/​(a*f**x + b*f**(-x)), x), x, x**S(3)*atan(sqrt(a)*f**x/​sqrt(b))/​(sqrt(a)*sqrt(b)*log(f)) - S(3)*I*x**S(2)*polylog(S(2), -I*sqrt(a)*f**x/​sqrt(b))/​(S(2)*sqrt(a)*sqrt(b)*log(f)**S(2)) + S(3)*I*x**S(2)*polylog(S(2), I*sqrt(a)*f**x/​sqrt(b))/​(S(2)*sqrt(a)*sqrt(b)*log(f)**S(2)) + S(3)*I*x*polylog(S(3), -I*sqrt(a)*f**x/​sqrt(b))/​(sqrt(a)*sqrt(b)*log(f)**S(3)) - S(3)*I*x*polylog(S(3), I*sqrt(a)*f**x/​sqrt(b))/​(sqrt(a)*sqrt(b)*log(f)**S(3)) - S(3)*I*polylog(S(4), -I*sqrt(a)*f**x/​sqrt(b))/​(sqrt(a)*sqrt(b)*log(f)**S(4)) + S(3)*I*polylog(S(4), I*sqrt(a)*f**x/​sqrt(b))/​(sqrt(a)*sqrt(b)*log(f)**S(4)), expand=True, _diff=True, _numerical=True)703 assert rubi_test(rubi_integrate(f**x/​(a + b*f**(S(2)*x))**S(3), x), x, f**x/​(S(4)*a*(a + b*f**(S(2)*x))**S(2)*log(f)) + S(3)*f**x/​(S(8)*a**S(2)*(a + b*f**(S(2)*x))*log(f)) + S(3)*atan(sqrt(b)*f**x/​sqrt(a))/​(S(8)*a**(S(5)/​2)*sqrt(b)*log(f)), expand=True, _diff=True, _numerical=True)704 assert rubi_test(rubi_integrate(S(1)/​(a*f**x + b*f**(-x)), x), x, atan(sqrt(a)*f**x/​sqrt(b))/​(sqrt(a)*sqrt(b)*log(f)), expand=True, _diff=True, _numerical=True)705 assert rubi_test(rubi_integrate((a*f**x + b*f**(-x))**(S(-2)), x), x, -S(1)/​(S(2)*a*(a*f**(S(2)*x) + b)*log(f)), expand=True, _diff=True, _numerical=True)706 assert rubi_test(rubi_integrate(x/​(a*f**x + b*f**(-x))**S(2), x), x, -x/​(S(2)*a*(a*f**(S(2)*x) + b)*log(f)) + x/​(S(2)*a*b*log(f)) - log(a*f**(S(2)*x) + b)/​(S(4)*a*b*log(f)**S(2)), expand=True, _diff=True, _numerical=True)707 assert rubi_test(rubi_integrate(x**S(2)/​(a*f**x + b*f**(-x))**S(2), x), x, -x**S(2)/​(S(2)*a*(a*f**(S(2)*x) + b)*log(f)) - x*log(S(1) + b*f**(-S(2)*x)/​a)/​(S(2)*a*b*log(f)**S(2)) + polylog(S(2), -b*f**(-S(2)*x)/​a)/​(S(4)*a*b*log(f)**S(3)), expand=True, _diff=True, _numerical=True)708 assert rubi_test(rubi_integrate(x**S(3)/​(a*f**x + b*f**(-x))**S(2), x), x, -x**S(3)/​(S(2)*a*(a*f**(S(2)*x) + b)*log(f)) - S(3)*x**S(2)*log(S(1) + b*f**(-S(2)*x)/​a)/​(S(4)*a*b*log(f)**S(2)) + S(3)*x*polylog(S(2), -b*f**(-S(2)*x)/​a)/​(S(4)*a*b*log(f)**S(3)) + S(3)*polylog(S(3), -b*f**(-S(2)*x)/​a)/​(S(8)*a*b*log(f)**S(4)), expand=True, _diff=True, _numerical=True)709 assert rubi_test(rubi_integrate((a*f**x + b*f**(-x))**(S(-3)), x), x, -f**x/​(S(4)*a*(a*f**(S(2)*x) + b)**S(2)*log(f)) + f**x/​(S(8)*a*b*(a*f**(S(2)*x) + b)*log(f)) + atan(sqrt(a)*f**x/​sqrt(b))/​(S(8)*a**(S(3)/​2)*b**(S(3)/​2)*log(f)), expand=True, _diff=True, _numerical=True)710 assert rubi_test(rubi_integrate(x/​(a*f**x + b*f**(-x))**S(3), x), x, -f**x*x/​(S(4)*a*(a*f**(S(2)*x) + b)**S(2)*log(f)) + f**x*x/​(S(8)*a*b*(a*f**(S(2)*x) + b)*log(f)) + f**x/​(S(8)*a*b*(a*f**(S(2)*x) + b)*log(f)**S(2)) + x*atan(sqrt(a)*f**x/​sqrt(b))/​(S(8)*a**(S(3)/​2)*b**(S(3)/​2)*log(f)) - I*polylog(S(2), -I*sqrt(a)*f**x/​sqrt(b))/​(S(16)*a**(S(3)/​2)*b**(S(3)/​2)*log(f)**S(2)) + I*polylog(S(2), I*sqrt(a)*f**x/​sqrt(b))/​(S(16)*a**(S(3)/​2)*b**(S(3)/​2)*log(f)**S(2)), expand=True, _diff=True, _numerical=True)711 assert rubi_test(rubi_integrate(x**S(2)/​(a*f**x + b*f**(-x))**S(3), x), x, -f**x*x**S(2)/​(S(4)*a*(a*f**(S(2)*x) + b)**S(2)*log(f)) + f**x*x**S(2)/​(S(8)*a*b*(a*f**(S(2)*x) + b)*log(f)) + f**x*x/​(S(4)*a*b*(a*f**(S(2)*x) + b)*log(f)**S(2)) + x**S(2)*atan(sqrt(a)*f**x/​sqrt(b))/​(S(8)*a**(S(3)/​2)*b**(S(3)/​2)*log(f)) - I*x*polylog(S(2), -I*sqrt(a)*f**x/​sqrt(b))/​(S(8)*a**(S(3)/​2)*b**(S(3)/​2)*log(f)**S(2)) + I*x*polylog(S(2), I*sqrt(a)*f**x/​sqrt(b))/​(S(8)*a**(S(3)/​2)*b**(S(3)/​2)*log(f)**S(2)) - atan(sqrt(a)*f**x/​sqrt(b))/​(S(4)*a**(S(3)/​2)*b**(S(3)/​2)*log(f)**S(3)) + I*polylog(S(3), -I*sqrt(a)*f**x/​sqrt(b))/​(S(8)*a**(S(3)/​2)*b**(S(3)/​2)*log(f)**S(3)) - I*polylog(S(3), I*sqrt(a)*f**x/​sqrt(b))/​(S(8)*a**(S(3)/​2)*b**(S(3)/​2)*log(f)**S(3)), expand=True, _diff=True, _numerical=True)712 assert rubi_test(rubi_integrate(f**(a + b*x + c*x**S(2))*g**(d + e*x + f*x**S(2)), x), x, sqrt(pi)*f**a*g**d*exp(-(b*log(f) + e*log(g))**S(2)/​(S(4)*(c*log(f) + f*log(g))))*erfi((b*log(f)/​S(2) + e*log(g)/​S(2) + x*(c*log(f) + f*log(g)))/​sqrt(c*log(f) + f*log(g)))/​(S(2)*sqrt(c*log(f) + f*log(g))), expand=True, _diff=True, _numerical=True)713 assert rubi_test(rubi_integrate(F**(e*(c + d*x))*(G**(h*(f + g*x))*b + a)**n, x), x, F**(e*(c + d*x))*(G**(h*(f + g*x))*b + a)**(n + S(1))*hyper((S(1), d*e*log(F)/​(g*h*log(G)) + n + S(1)), (d*e*log(F)/​(g*h*log(G)) + S(1),), -G**(h*(f + g*x))*b/​a)/​(a*d*e*log(F)), expand=True, _diff=True, _numerical=True)714 assert rubi_test(rubi_integrate(F**(e*(c + d*x))*H**(t*(r + s*x))/​(F**(e*(c + d*x))*b + a), x), x, H**(t*(r + s*x))*hyper((S(1), -s*t*log(H)/​(d*e*log(F))), (S(1) - s*t*log(H)/​(d*e*log(F)),), -F**(-e*(c + d*x))*a/​b)/​(b*s*t*log(H)), expand=True, _diff=True, _numerical=True)715 assert rubi_test(rubi_integrate(F**(e*(d*x + f))*H**(t*(r + s*x))/​(F**(e*(c + d*x))*b + a), x), x, F**(-e*(c - f))*H**(t*(r + s*x))*hyper((S(1), -s*t*log(H)/​(d*e*log(F))), (S(1) - s*t*log(H)/​(d*e*log(F)),), -F**(-e*(c + d*x))*a/​b)/​(b*s*t*log(H)), expand=True, _diff=True, _numerical=True)716 assert rubi_test(rubi_integrate((d + e*exp(h + i*x))*(f + g*x)**S(3)/​(a + b*exp(h + i*x) + c*exp(S(2)*h + S(2)*i*x)), x), x, S(6)*g**S(3)*(e + (b*e - S(2)*c*d)/​sqrt(-S(4)*a*c + b**S(2)))*polylog(S(4), -(b + sqrt(-S(4)*a*c + b**S(2)))*exp(-h - i*x)/​(S(2)*c))/​(i**S(4)*(b + sqrt(-S(4)*a*c + b**S(2)))) + S(6)*g**S(3)*(e + (-b*e + S(2)*c*d)/​sqrt(-S(4)*a*c + b**S(2)))*polylog(S(4), -(b - sqrt(-S(4)*a*c + b**S(2)))*exp(-h - i*x)/​(S(2)*c))/​(i**S(4)*(b - sqrt(-S(4)*a*c + b**S(2)))) + S(6)*g**S(2)*(e + (b*e - S(2)*c*d)/​sqrt(-S(4)*a*c + b**S(2)))*(f + g*x)*polylog(S(3), -(b + sqrt(-S(4)*a*c + b**S(2)))*exp(-h - i*x)/​(S(2)*c))/​(i**S(3)*(b + sqrt(-S(4)*a*c + b**S(2)))) + S(6)*g**S(2)*(e + (-b*e + S(2)*c*d)/​sqrt(-S(4)*a*c + b**S(2)))*(f + g*x)*polylog(S(3), -(b - sqrt(-S(4)*a*c + b**S(2)))*exp(-h - i*x)/​(S(2)*c))/​(i**S(3)*(b - sqrt(-S(4)*a*c + b**S(2)))) + S(3)*g*(e + (b*e - S(2)*c*d)/​sqrt(-S(4)*a*c + b**S(2)))*(f + g*x)**S(2)*polylog(S(2), -(b + sqrt(-S(4)*a*c + b**S(2)))*exp(-h - i*x)/​(S(2)*c))/​(i**S(2)*(b + sqrt(-S(4)*a*c + b**S(2)))) + S(3)*g*(e + (-b*e + S(2)*c*d)/​sqrt(-S(4)*a*c + b**S(2)))*(f + g*x)**S(2)*polylog(S(2), -(b - sqrt(-S(4)*a*c + b**S(2)))*exp(-h - i*x)/​(S(2)*c))/​(i**S(2)*(b - sqrt(-S(4)*a*c + b**S(2)))) - (e + (b*e - S(2)*c*d)/​sqrt(-S(4)*a*c + b**S(2)))*(f + g*x)**S(3)*log(S(1) + (b + sqrt(-S(4)*a*c + b**S(2)))*exp(-h - i*x)/​(S(2)*c))/​(i*(b + sqrt(-S(4)*a*c + b**S(2)))) - (e + (-b*e + S(2)*c*d)/​sqrt(-S(4)*a*c + b**S(2)))*(f + g*x)**S(3)*log(S(1) + (b - sqrt(-S(4)*a*c + b**S(2)))*exp(-h - i*x)/​(S(2)*c))/​(i*(b - sqrt(-S(4)*a*c + b**S(2)))), expand=True, _diff=True, _numerical=True)717 assert rubi_test(rubi_integrate((d + e*exp(h + i*x))*(f + g*x)**S(2)/​(a + b*exp(h + i*x) + c*exp(S(2)*h + S(2)*i*x)), x), x, S(2)*g**S(2)*(e + (b*e - S(2)*c*d)/​sqrt(-S(4)*a*c + b**S(2)))*polylog(S(3), -(b + sqrt(-S(4)*a*c + b**S(2)))*exp(-h - i*x)/​(S(2)*c))/​(i**S(3)*(b + sqrt(-S(4)*a*c + b**S(2)))) + S(2)*g**S(2)*(e + (-b*e + S(2)*c*d)/​sqrt(-S(4)*a*c + b**S(2)))*polylog(S(3), -(b - sqrt(-S(4)*a*c + b**S(2)))*exp(-h - i*x)/​(S(2)*c))/​(i**S(3)*(b - sqrt(-S(4)*a*c + b**S(2)))) + S(2)*g*(e + (b*e - S(2)*c*d)/​sqrt(-S(4)*a*c + b**S(2)))*(f + g*x)*polylog(S(2), -(b + sqrt(-S(4)*a*c + b**S(2)))*exp(-h - i*x)/​(S(2)*c))/​(i**S(2)*(b + sqrt(-S(4)*a*c + b**S(2)))) + S(2)*g*(e + (-b*e + S(2)*c*d)/​sqrt(-S(4)*a*c + b**S(2)))*(f + g*x)*polylog(S(2), -(b - sqrt(-S(4)*a*c + b**S(2)))*exp(-h - i*x)/​(S(2)*c))/​(i**S(2)*(b - sqrt(-S(4)*a*c + b**S(2)))) - (e + (b*e - S(2)*c*d)/​sqrt(-S(4)*a*c + b**S(2)))*(f + g*x)**S(2)*log(S(1) + (b + sqrt(-S(4)*a*c + b**S(2)))*exp(-h - i*x)/​(S(2)*c))/​(i*(b + sqrt(-S(4)*a*c + b**S(2)))) - (e + (-b*e + S(2)*c*d)/​sqrt(-S(4)*a*c + b**S(2)))*(f + g*x)**S(2)*log(S(1) + (b - sqrt(-S(4)*a*c + b**S(2)))*exp(-h - i*x)/​(S(2)*c))/​(i*(b - sqrt(-S(4)*a*c + b**S(2)))), expand=True, _diff=True, _numerical=True)718 assert rubi_test(rubi_integrate((d + e*exp(h + i*x))*(f + g*x)/​(a + b*exp(h + i*x) + c*exp(S(2)*h + S(2)*i*x)), x), x, g*(e + (b*e - S(2)*c*d)/​sqrt(-S(4)*a*c + b**S(2)))*polylog(S(2), -(b + sqrt(-S(4)*a*c + b**S(2)))*exp(-h - i*x)/​(S(2)*c))/​(i**S(2)*(b + sqrt(-S(4)*a*c + b**S(2)))) + g*(e + (-b*e + S(2)*c*d)/​sqrt(-S(4)*a*c + b**S(2)))*polylog(S(2), -(b - sqrt(-S(4)*a*c + b**S(2)))*exp(-h - i*x)/​(S(2)*c))/​(i**S(2)*(b - sqrt(-S(4)*a*c + b**S(2)))) - (e + (b*e - S(2)*c*d)/​sqrt(-S(4)*a*c + b**S(2)))*(f + g*x)*log(S(1) + (b + sqrt(-S(4)*a*c + b**S(2)))*exp(-h - i*x)/​(S(2)*c))/​(i*(b + sqrt(-S(4)*a*c + b**S(2)))) + (e + (-b*e + S(2)*c*d)/​sqrt(-S(4)*a*c + b**S(2)))*(-f - g*x)*log(S(1) + (b - sqrt(-S(4)*a*c + b**S(2)))*exp(-h - i*x)/​(S(2)*c))/​(i*(b - sqrt(-S(4)*a*c + b**S(2)))), expand=True, _diff=True, _numerical=True)719 assert rubi_test(rubi_integrate((d + e*exp(h + i*x))/​(a + b*exp(h + i*x) + c*exp(S(2)*h + S(2)*i*x)), x), x, d*x/​a - d*log(a + b*exp(h + i*x) + c*exp(S(2)*h + S(2)*i*x))/​(S(2)*a*i) + (-S(2)*a*e + b*d)*atanh((b + S(2)*c*exp(h + i*x))/​sqrt(-S(4)*a*c + b**S(2)))/​(a*i*sqrt(-S(4)*a*c + b**S(2))), expand=True, _diff=True, _numerical=True)720 # long time assert rubi_test(rubi_integrate((d + e*exp(h + i*x))/​((f + g*x)*(a + b*exp(h + i*x) + c*exp(S(2)*h + S(2)*i*x))), x), x, d*Integral(S(1)/​((f + g*x)*(a + b*exp(h + i*x) + c*exp(S(2)*h + S(2)*i*x))), x) + e*Integral(exp(h + i*x)/​((f + g*x)*(a + b*exp(h + i*x) + c*exp(S(2)*h + S(2)*i*x))), x), expand=True, _diff=True, _numerical=True)721 # long time assert rubi_test(rubi_integrate((d + e*exp(h + i*x))/​((f + g*x)**S(2)*(a + b*exp(h + i*x) + c*exp(S(2)*h + S(2)*i*x))), x), x, d*Integral(S(1)/​((f + g*x)**S(2)*(a + b*exp(h + i*x) + c*exp(S(2)*h + S(2)*i*x))), x) + e*Integral(exp(h + i*x)/​((f + g*x)**S(2)*(a + b*exp(h + i*x) + c*exp(S(2)*h + S(2)*i*x))), x), expand=True, _diff=True, _numerical=True)722 assert rubi_test(rubi_integrate(x*(-a*e*exp(c + d*x) + b*e)/​(-S(2)*a*e*exp(c + d*x) - b*e*exp(S(2)*c + S(2)*d*x) + b*e), x), x, -x*log(S(1) + (a - sqrt(a**S(2) + b**S(2)))*exp(-c - d*x)/​b)/​(S(2)*d) - x*log(S(1) + (a + sqrt(a**S(2) + b**S(2)))*exp(-c - d*x)/​b)/​(S(2)*d) + polylog(S(2), -(a - sqrt(a**S(2) + b**S(2)))*exp(-c - d*x)/​b)/​(S(2)*d**S(2)) + polylog(S(2), -(a + sqrt(a**S(2) + b**S(2)))*exp(-c - d*x)/​b)/​(S(2)*d**S(2)), expand=True, _diff=True, _numerical=True)723 assert rubi_test(rubi_integrate(F**(a + b*x + c*x**S(3))*(b + S(3)*c*x**S(2)), x), x, F**(a + b*x + c*x**S(3))/​log(F), expand=True, _diff=True, _numerical=True)724 assert rubi_test(rubi_integrate(F**(S(1)/​(a + b*x + c*x**S(2)))*(b + S(2)*c*x)/​(a + b*x + c*x**S(2))**S(2), x), x, -F**(S(1)/​(a + b*x + c*x**S(2)))/​log(F), expand=True, _diff=True, _numerical=True)725 assert rubi_test(rubi_integrate((b + S(2)*c*x)*(a + b*x + c*x**S(2))**m*exp(a + b*x + c*x**S(2)), x), x, (-a - b*x - c*x**S(2))**(-m)*(a + b*x + c*x**S(2))**m*Gamma(m + S(1), -a - b*x - c*x**S(2)), expand=True, _diff=True, _numerical=True)726 assert rubi_test(rubi_integrate((b + S(2)*c*x)*(a + b*x + c*x**S(2))**S(3)*exp(a + b*x + c*x**S(2)), x), x, (a + b*x + c*x**S(2))**S(3)*exp(a + b*x + c*x**S(2)) - S(3)*(a + b*x + c*x**S(2))**S(2)*exp(a + b*x + c*x**S(2)) + S(6)*(a + b*x + c*x**S(2))*exp(a + b*x + c*x**S(2)) - S(6)*exp(a + b*x + c*x**S(2)), expand=True, _diff=True, _numerical=True)727 assert rubi_test(rubi_integrate((b + S(2)*c*x)*(a + b*x + c*x**S(2))**S(2)*exp(a + b*x + c*x**S(2)), x), x, (a + b*x + c*x**S(2))**S(2)*exp(a + b*x + c*x**S(2)) - S(2)*(a + b*x + c*x**S(2))*exp(a + b*x + c*x**S(2)) + S(2)*exp(a + b*x + c*x**S(2)), expand=True, _diff=True, _numerical=True)728 assert rubi_test(rubi_integrate((b + S(2)*c*x)*(a + b*x + c*x**S(2))*exp(a + b*x + c*x**S(2)), x), x, (a + b*x + c*x**S(2))*exp(a + b*x + c*x**S(2)) - exp(a + b*x + c*x**S(2)), expand=True, _diff=True, _numerical=True)729 assert rubi_test(rubi_integrate((b + S(2)*c*x)*exp(a + b*x + c*x**S(2)), x), x, exp(a + b*x + c*x**S(2)), expand=True, _diff=True, _numerical=True)730 assert rubi_test(rubi_integrate((b + S(2)*c*x)*exp(a + b*x + c*x**S(2))/​(a + b*x + c*x**S(2)), x), x, Ei(a + b*x + c*x**S(2)), expand=True, _diff=True, _numerical=True)731 assert rubi_test(rubi_integrate((b + S(2)*c*x)*exp(a + b*x + c*x**S(2))/​(a + b*x + c*x**S(2))**S(2), x), x, Ei(a + b*x + c*x**S(2)) - exp(a + b*x + c*x**S(2))/​(a + b*x + c*x**S(2)), expand=True, _diff=True, _numerical=True)732 assert rubi_test(rubi_integrate((b + S(2)*c*x)*exp(a + b*x + c*x**S(2))/​(a + b*x + c*x**S(2))**S(3), x), x, Ei(a + b*x + c*x**S(2))/​S(2) - exp(a + b*x + c*x**S(2))/​(S(2)*(a + b*x + c*x**S(2))) - exp(a + b*x + c*x**S(2))/​(S(2)*(a + b*x + c*x**S(2))**S(2)), expand=True, _diff=True, _numerical=True)733 assert rubi_test(rubi_integrate((b + S(2)*c*x)*(a + b*x + c*x**S(2))**(S(7)/​2)*exp(a + b*x + c*x**S(2)), x), x, (a + b*x + c*x**S(2))**(S(7)/​2)*exp(a + b*x + c*x**S(2)) - S(7)*(a + b*x + c*x**S(2))**(S(5)/​2)*exp(a + b*x + c*x**S(2))/​S(2) + S(35)*(a + b*x + c*x**S(2))**(S(3)/​2)*exp(a + b*x + c*x**S(2))/​S(4) - S(105)*sqrt(a + b*x + c*x**S(2))*exp(a + b*x + c*x**S(2))/​S(8) + S(105)*sqrt(pi)*erfi(sqrt(a + b*x + c*x**S(2)))/​S(16), expand=True, _diff=True, _numerical=True)734 assert rubi_test(rubi_integrate((b + S(2)*c*x)*(a + b*x + c*x**S(2))**(S(5)/​2)*exp(a + b*x + c*x**S(2)), x), x, (a + b*x + c*x**S(2))**(S(5)/​2)*exp(a + b*x + c*x**S(2)) - S(5)*(a + b*x + c*x**S(2))**(S(3)/​2)*exp(a + b*x + c*x**S(2))/​S(2) + S(15)*sqrt(a + b*x + c*x**S(2))*exp(a + b*x + c*x**S(2))/​S(4) - S(15)*sqrt(pi)*erfi(sqrt(a + b*x + c*x**S(2)))/​S(8), expand=True, _diff=True, _numerical=True)735 assert rubi_test(rubi_integrate((b + S(2)*c*x)*(a + b*x + c*x**S(2))**(S(3)/​2)*exp(a + b*x + c*x**S(2)), x), x, (a + b*x + c*x**S(2))**(S(3)/​2)*exp(a + b*x + c*x**S(2)) - S(3)*sqrt(a + b*x + c*x**S(2))*exp(a + b*x + c*x**S(2))/​S(2) + S(3)*sqrt(pi)*erfi(sqrt(a + b*x + c*x**S(2)))/​S(4), expand=True, _diff=True, _numerical=True)736 assert rubi_test(rubi_integrate((b + S(2)*c*x)*sqrt(a + b*x + c*x**S(2))*exp(a + b*x + c*x**S(2)), x), x, sqrt(a + b*x + c*x**S(2))*exp(a + b*x + c*x**S(2)) - sqrt(pi)*erfi(sqrt(a + b*x + c*x**S(2)))/​S(2), expand=True, _diff=True, _numerical=True)737 assert rubi_test(rubi_integrate((b + S(2)*c*x)*exp(a + b*x + c*x**S(2))/​sqrt(a + b*x + c*x**S(2)), x), x, sqrt(pi)*erfi(sqrt(a + b*x + c*x**S(2))), expand=True, _diff=True, _numerical=True)738 assert rubi_test(rubi_integrate((b + S(2)*c*x)*exp(a + b*x + c*x**S(2))/​(a + b*x + c*x**S(2))**(S(3)/​2), x), x, S(2)*sqrt(pi)*erfi(sqrt(a + b*x + c*x**S(2))) - S(2)*exp(a + b*x + c*x**S(2))/​sqrt(a + b*x + c*x**S(2)), expand=True, _diff=True, _numerical=True)739 assert rubi_test(rubi_integrate((b + S(2)*c*x)*exp(a + b*x + c*x**S(2))/​(a + b*x + c*x**S(2))**(S(5)/​2), x), x, S(4)*sqrt(pi)*erfi(sqrt(a + b*x + c*x**S(2)))/​S(3) - S(4)*exp(a + b*x + c*x**S(2))/​(S(3)*sqrt(a + b*x + c*x**S(2))) - S(2)*exp(a + b*x + c*x**S(2))/​(S(3)*(a + b*x + c*x**S(2))**(S(3)/​2)), expand=True, _diff=True, _numerical=True)740 assert rubi_test(rubi_integrate((b + S(2)*c*x)*exp(a + b*x + c*x**S(2))/​(a + b*x + c*x**S(2))**(S(7)/​2), x), x, S(8)*sqrt(pi)*erfi(sqrt(a + b*x + c*x**S(2)))/​S(15) - S(8)*exp(a + b*x + c*x**S(2))/​(S(15)*sqrt(a + b*x + c*x**S(2))) - S(4)*exp(a + b*x + c*x**S(2))/​(S(15)*(a + b*x + c*x**S(2))**(S(3)/​2)) - S(2)*exp(a + b*x + c*x**S(2))/​(S(5)*(a + b*x + c*x**S(2))**(S(5)/​2)), expand=True, _diff=True, _numerical=True)741 assert rubi_test(rubi_integrate((b + S(2)*c*x)*exp(a + b*x + c*x**S(2))/​(a + b*x + c*x**S(2))**(S(9)/​2), x), x, S(16)*sqrt(pi)*erfi(sqrt(a + b*x + c*x**S(2)))/​S(105) - S(16)*exp(a + b*x + c*x**S(2))/​(S(105)*sqrt(a + b*x + c*x**S(2))) - S(8)*exp(a + b*x + c*x**S(2))/​(S(105)*(a + b*x + c*x**S(2))**(S(3)/​2)) - S(4)*exp(a + b*x + c*x**S(2))/​(S(35)*(a + b*x + c*x**S(2))**(S(5)/​2)) - S(2)*exp(a + b*x + c*x**S(2))/​(S(7)*(a + b*x + c*x**S(2))**(S(7)/​2)), expand=True, _diff=True, _numerical=True)742 assert rubi_test(rubi_integrate(exp(-x)/​sqrt(S(1) - exp(-S(2)*x)), x), x, -asin(exp(-x)), expand=True, _diff=True, _numerical=True)743 assert rubi_test(rubi_integrate(exp(x)/​(exp(S(2)*x) + S(4)), x), x, atan(exp(x)/​S(2))/​S(2), expand=True, _diff=True, _numerical=True)744 assert rubi_test(rubi_integrate(exp(x)/​(-exp(S(2)*x) + S(1)), x), x, atanh(exp(x)), expand=True, _diff=True, _numerical=True)745 assert rubi_test(rubi_integrate(exp(x)/​(-S(4)*exp(S(2)*x) + S(3)), x), x, sqrt(S(3))*atanh(S(2)*sqrt(S(3))*exp(x)/​S(3))/​S(6), expand=True, _diff=True, _numerical=True)746 assert rubi_test(rubi_integrate(sqrt(-S(4)*exp(S(2)*x) + S(3))*exp(x), x), x, sqrt(-S(4)*exp(S(2)*x) + S(3))*exp(x)/​S(2) + S(3)*asin(S(2)*sqrt(S(3))*exp(x)/​S(3))/​S(4), expand=True, _diff=True, _numerical=True)747 assert rubi_test(rubi_integrate(x**S(3)*exp(x**S(2)), x), x, x**S(2)*exp(x**S(2))/​S(2) - exp(x**S(2))/​S(2), expand=True, _diff=True, _numerical=True)748 assert rubi_test(rubi_integrate(sqrt(-exp(S(2)*x) + S(1))*exp(x), x), x, sqrt(-exp(S(2)*x) + S(1))*exp(x)/​S(2) + asin(exp(x))/​S(2), expand=True, _diff=True, _numerical=True)749 assert rubi_test(rubi_integrate(exp(x)/​sqrt(exp(S(2)*x) + exp(x) + S(1)), x), x, asinh(sqrt(S(3))*(S(2)*exp(x) + S(1))/​S(3)), expand=True, _diff=True, _numerical=True)750 assert rubi_test(rubi_integrate(exp(x)/​(exp(S(2)*x) + S(-4)), x), x, -atanh(exp(x)/​S(2))/​S(2), expand=True, _diff=True, _numerical=True)751 assert rubi_test(rubi_integrate(x*exp(-x**S(2) + S(2)), x), x, -exp(-x**S(2) + S(2))/​S(2), expand=True, _diff=True, _numerical=True)752 assert rubi_test(rubi_integrate(-x**E + exp(x), x), x, -x**(E + S(1))/​(E + S(1)) + exp(x), expand=True, _diff=True, _numerical=True)753 assert rubi_test(rubi_integrate((exp(S(2)*x) + S(-1))/​(exp(S(2)*x) + S(3)), x), x, -x/​S(3) + S(2)*log(exp(S(2)*x) + S(3))/​S(3), expand=True, _diff=True, _numerical=True)754 assert rubi_test(rubi_integrate(exp(x)/​sqrt(-exp(S(2)*x) + S(1)), x), x, asin(exp(x)), expand=True, _diff=True, _numerical=True)755 assert rubi_test(rubi_integrate(exp(S(2)*x)/​(exp(S(4)*x) + S(1)), x), x, atan(exp(S(2)*x))/​S(2), expand=True, _diff=True, _numerical=True)756 assert rubi_test(rubi_integrate(S(1)/​(exp(S(2)*x) - S(3)*exp(x)), x), x, -x/​S(9) + log(-exp(x) + S(3))/​S(9) + exp(-x)/​S(3), expand=True, _diff=True, _numerical=True)757 assert rubi_test(rubi_integrate((exp(x) + S(-2))*exp(x)/​(exp(x) + S(1)), x), x, exp(x) - S(3)*log(exp(x) + S(1)), expand=True, _diff=True, _numerical=True)758 assert rubi_test(rubi_integrate(exp(x)/​(exp(S(2)*x) + S(-1)), x), x, -atanh(exp(x)), expand=True, _diff=True, _numerical=True)759 assert rubi_test(rubi_integrate(exp(x)/​(exp(S(2)*x) + S(1)), x), x, atan(exp(x)), expand=True, _diff=True, _numerical=True)760 assert rubi_test(rubi_integrate((exp(x) + exp(-x))/​(exp(x) - exp(-x)), x), x, log(-exp(x) + exp(-x)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((exp(x) + exp(-x))/​(exp(x) - exp(-x)), x), x, -x + log(-exp(S(2)*x) + S(1)), expand=True, _diff=True, _numerical=True)761 assert rubi_test(rubi_integrate((exp(x) - exp(-x))/​(exp(x) + exp(-x)), x), x, log(exp(x) + exp(-x)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((exp(x) - exp(-x))/​(exp(x) + exp(-x)), x), x, -x + log(exp(S(2)*x) + S(1)), expand=True, _diff=True, _numerical=True)762 assert rubi_test(rubi_integrate((exp(S(2)*x) + exp(-S(2)*x))/​(exp(S(2)*x) - exp(-S(2)*x)), x), x, -x + log(-exp(S(4)*x) + S(1))/​S(2), expand=True, _diff=True, _numerical=True)763 assert rubi_test(rubi_integrate(exp(x)/​sqrt(exp(S(2)*x) + S(1)), x), x, asinh(exp(x)), expand=True, _diff=True, _numerical=True)764 assert rubi_test(rubi_integrate(exp(sqrt(x + S(4)))/​sqrt(x + S(4)), x), x, S(2)*exp(sqrt(x + S(4))), expand=True, _diff=True, _numerical=True)765 assert rubi_test(rubi_integrate(x/​sqrt(exp(S(2)*x**S(2)) + S(-1)), x), x, atan(sqrt(exp(S(2)*x**S(2)) + S(-1)))/​S(2), expand=True, _diff=True, _numerical=True)766 assert rubi_test(rubi_integrate(sqrt(exp(S(2)*x) + S(9))*exp(x), x), x, sqrt(exp(S(2)*x) + S(9))*exp(x)/​S(2) + S(9)*asinh(exp(x)/​S(3))/​S(2), expand=True, _diff=True, _numerical=True)767 assert rubi_test(rubi_integrate(sqrt(exp(S(2)*x) + S(1))*exp(x), x), x, sqrt(exp(S(2)*x) + S(1))*exp(x)/​S(2) + asinh(exp(x))/​S(2), expand=True, _diff=True, _numerical=True)768 assert rubi_test(rubi_integrate(x*exp(x**S(2))/​(exp(S(2)*x**S(2)) + S(1)), x), x, atan(exp(x**S(2)))/​S(2), expand=True, _diff=True, _numerical=True)769 assert rubi_test(rubi_integrate(x**S(2)*exp(x**(S(3)/​2)), x), x, S(2)*x**(S(3)/​2)*exp(x**(S(3)/​2))/​S(3) - S(2)*exp(x**(S(3)/​2))/​S(3), expand=True, _diff=True, _numerical=True)770 assert rubi_test(rubi_integrate(exp(x)/​sqrt(exp(S(2)*x) + S(-3)), x), x, atanh(exp(x)/​sqrt(exp(S(2)*x) + S(-3))), expand=True, _diff=True, _numerical=True)771 assert rubi_test(rubi_integrate(exp(x)/​(-exp(S(2)*x) + S(16)), x), x, atanh(exp(x)/​S(4))/​S(4), expand=True, _diff=True, _numerical=True)772 assert rubi_test(rubi_integrate(exp(S(5)*x)/​(exp(S(10)*x) + S(1)), x), x, atan(exp(S(5)*x))/​S(5), expand=True, _diff=True, _numerical=True)773 assert rubi_test(rubi_integrate(exp(S(4)*x)/​sqrt(exp(S(8)*x) + S(16)), x), x, asinh(exp(S(4)*x)/​S(4))/​S(4), expand=True, _diff=True, _numerical=True)774 assert rubi_test(rubi_integrate(x**S(2)*exp(S(4)*x**S(3))*cos(S(7)*x**S(3)), x), x, S(7)*exp(S(4)*x**S(3))*sin(S(7)*x**S(3))/​S(195) + S(4)*exp(S(4)*x**S(3))*cos(S(7)*x**S(3))/​S(195), expand=True, _diff=True, _numerical=True)775 assert rubi_test(rubi_integrate(x*exp(x**S(2) + S(1)), x), x, exp(x**S(2) + S(1))/​S(2), expand=True, _diff=True, _numerical=True)776 assert rubi_test(rubi_integrate(x**S(2)*exp(x**S(3) + S(1)), x), x, exp(x**S(3) + S(1))/​S(3), expand=True, _diff=True, _numerical=True)777 assert rubi_test(rubi_integrate(exp(sqrt(x))/​sqrt(x), x), x, S(2)*exp(sqrt(x)), expand=True, _diff=True, _numerical=True)778 assert rubi_test(rubi_integrate(exp(x**(S(1)/​3))/​x**(S(2)/​3), x), x, S(3)*exp(x**(S(1)/​3)), expand=True, _diff=True, _numerical=True)779 assert rubi_test(rubi_integrate((x**S(5) + S(2)*x**S(3) + S(-8))*exp(S(3)*x), x), x, x**S(5)*exp(S(3)*x)/​S(3) - S(5)*x**S(4)*exp(S(3)*x)/​S(9) + S(38)*x**S(3)*exp(S(3)*x)/​S(27) - S(38)*x**S(2)*exp(S(3)*x)/​S(27) + S(76)*x*exp(S(3)*x)/​S(81) - S(724)*exp(S(3)*x)/​S(243), expand=True, _diff=True, _numerical=True)780 assert rubi_test(rubi_integrate((x + exp(x))**S(2), x), x, x**S(3)/​S(3) + S(2)*x*exp(x) + exp(S(2)*x)/​S(2) - S(2)*exp(x), expand=True, _diff=True, _numerical=True)781 assert rubi_test(rubi_integrate((exp(S(3)*x) + exp(S(2)*x) + exp(x))*exp(-S(4)*x), x), x, -exp(-x) - exp(-S(2)*x)/​S(2) - exp(-S(3)*x)/​S(3), expand=True, _diff=True, _numerical=True)782 assert rubi_test(rubi_integrate(exp(x)/​(exp(S(2)*x) + S(2)*exp(x) + S(1)), x), x, -S(1)/​(exp(x) + S(1)), expand=True, _diff=True, _numerical=True)783 assert rubi_test(rubi_integrate(exp(-x)*cos(S(3)*x), x), x, S(3)*exp(-x)*sin(S(3)*x)/​S(10) - exp(-x)*cos(S(3)*x)/​S(10), expand=True, _diff=True, _numerical=True)784 assert rubi_test(rubi_integrate(exp(S(2)*x)/​(exp(S(2)*x) + S(3)*exp(x) + S(2)), x), x, -log(exp(x) + S(1)) + S(2)*log(exp(x) + S(2)), expand=True, _diff=True, _numerical=True)785 assert rubi_test(rubi_integrate(exp(S(2)*x)/​(exp(x) + S(1)), x), x, exp(x) - log(exp(x) + S(1)), expand=True, _diff=True, _numerical=True)786 assert rubi_test(rubi_integrate(exp(S(3)*x)*cos(S(5)*x), x), x, S(5)*exp(S(3)*x)*sin(S(5)*x)/​S(34) + S(3)*exp(S(3)*x)*cos(S(5)*x)/​S(34), expand=True, _diff=True, _numerical=True)787 assert rubi_test(rubi_integrate(exp(x)*sech(exp(x)), x), x, atan(sinh(exp(x))), expand=True, _diff=True, _numerical=True)788 assert rubi_test(rubi_integrate(exp(-x)/​(S(2)*exp(x) + S(1)), x), x, -S(2)*x + S(2)*log(S(2)*exp(x) + S(1)) - exp(-x), expand=True, _diff=True, _numerical=True)789 assert rubi_test(rubi_integrate(exp(x)*cos(S(3)*x + S(4)), x), x, S(3)*exp(x)*sin(S(3)*x + S(4))/​S(10) + exp(x)*cos(S(3)*x + S(4))/​S(10), expand=True, _diff=True, _numerical=True)790 assert rubi_test(rubi_integrate(x*(exp(x) + exp(-x)), x), x, x*exp(x) - x*exp(-x) - exp(x) - exp(-x), expand=True, _diff=True, _numerical=True)791 assert rubi_test(rubi_integrate(exp(x)/​(exp(S(2)*x) + S(3)*exp(x) + S(2)), x), x, -S(2)*atanh(S(2)*exp(x) + S(3)), expand=True, _diff=True, _numerical=True)792 assert rubi_test(rubi_integrate(exp(S(2)*x)/​(exp(x) + S(1))**(S(1)/​3), x), x, S(3)*(exp(x) + S(1))**(S(5)/​3)/​S(5) - S(3)*(exp(x) + S(1))**(S(2)/​3)/​S(2), expand=True, _diff=True, _numerical=True)793 assert rubi_test(rubi_integrate(exp(S(2)*x)/​(exp(x) + S(1))**(S(1)/​4), x), x, S(4)*(exp(x) + S(1))**(S(7)/​4)/​S(7) - S(4)*(exp(x) + S(1))**(S(3)/​4)/​S(3), expand=True, _diff=True, _numerical=True)794 assert rubi_test(rubi_integrate((S(2)*exp(S(2)*x) - exp(x))/​sqrt(S(3)*exp(S(2)*x) - S(6)*exp(x) + S(-1)), x), x, S(2)*sqrt(S(3)*exp(S(2)*x) - S(6)*exp(x) + S(-1))/​S(3) - sqrt(S(3))*atanh(sqrt(S(3))*(-exp(x) + S(1))/​sqrt(S(3)*exp(S(2)*x) - S(6)*exp(x) + S(-1)))/​S(3), expand=True, _diff=True, _numerical=True)795 assert rubi_test(rubi_integrate((x**S(2) - S(5)*x)*exp(x), x), x, x**S(2)*exp(x) - S(7)*x*exp(x) + S(7)*exp(x), expand=True, _diff=True, _numerical=True)796 assert rubi_test(rubi_integrate((x**S(2) - x)*exp(S(3)*x), x), x, x**S(2)*exp(S(3)*x)/​S(3) - S(5)*x*exp(S(3)*x)/​S(9) + S(5)*exp(S(3)*x)/​S(27), expand=True, _diff=True, _numerical=True)797 assert rubi_test(rubi_integrate(x**(S(2)*x)*(log(x) + S(1))*exp(x**x), x), x, (x**x + S(-1))*exp(x**x), expand=True, _diff=True, _numerical=True)798 assert rubi_test(rubi_integrate((exp(S(7)*x) + exp(S(5)*x))/​(exp(x) + exp(-x)), x), x, exp(S(6)*x)/​S(6), expand=True, _diff=True, _numerical=True)799 assert rubi_test(rubi_integrate(x**(S(-2) - S(1)/​x)*(-log(x) + S(1)), x), x, -x**(-S(1)/​x), expand=True, _diff=True, _numerical=True)800 assert rubi_test(rubi_integrate((a + b*exp(x))**S(2), x), x, a**S(2)*x + S(2)*a*b*exp(x) + b**S(2)*exp(S(2)*x)/​S(2), expand=True, _diff=True, _numerical=True)801 assert rubi_test(rubi_integrate((a + b*exp(x))**S(3), x), x, a**S(3)*x + S(3)*a**S(2)*b*exp(x) + S(3)*a*b**S(2)*exp(S(2)*x)/​S(2) + b**S(3)*exp(S(3)*x)/​S(3), expand=True, _diff=True, _numerical=True)802 assert rubi_test(rubi_integrate((a + b*exp(x))**S(4), x), x, a**S(4)*x + S(4)*a**S(3)*b*exp(x) + S(3)*a**S(2)*b**S(2)*exp(S(2)*x) + S(4)*a*b**S(3)*exp(S(3)*x)/​S(3) + b**S(4)*exp(S(4)*x)/​S(4), expand=True, _diff=True, _numerical=True)803 assert rubi_test(rubi_integrate(S(1)/​sqrt(a + b*exp(c + d*x)), x), x, -S(2)*atanh(sqrt(a + b*exp(c + d*x))/​sqrt(a))/​(sqrt(a)*d), expand=True, _diff=True, _numerical=True)804 assert rubi_test(rubi_integrate(S(1)/​sqrt(-a + b*exp(c + d*x)), x), x, S(2)*atan(sqrt(-a + b*exp(c + d*x))/​sqrt(a))/​(sqrt(a)*d), expand=True, _diff=True, _numerical=True)805 assert rubi_test(rubi_integrate(sqrt(a + b*exp(c + d*x)), x), x, -S(2)*sqrt(a)*atanh(sqrt(a + b*exp(c + d*x))/​sqrt(a))/​d + S(2)*sqrt(a + b*exp(c + d*x))/​d, expand=True, _diff=True, _numerical=True)806 assert rubi_test(rubi_integrate(sqrt(-a + b*exp(c + d*x)), x), x, -S(2)*sqrt(a)*atan(sqrt(-a + b*exp(c + d*x))/​sqrt(a))/​d + S(2)*sqrt(-a + b*exp(c + d*x))/​d, expand=True, _diff=True, _numerical=True)807 assert rubi_test(rubi_integrate(exp(S(6)*x)*sin(S(3)*x), x), x, S(2)*exp(S(6)*x)*sin(S(3)*x)/​S(15) - exp(S(6)*x)*cos(S(3)*x)/​S(15), expand=True, _diff=True, _numerical=True)808 assert rubi_test(rubi_integrate(exp(S(3)*x)/​(exp(S(2)*x) + S(1)), x), x, exp(x) - atan(exp(x)), expand=True, _diff=True, _numerical=True)809 assert rubi_test(rubi_integrate(exp(S(3)*x)/​(exp(S(2)*x) + S(-1)), x), x, exp(x) - atanh(exp(x)), expand=True, _diff=True, _numerical=True)810 assert rubi_test(rubi_integrate(exp(-x)/​sqrt(exp(S(2)*x) + S(1)), x), x, -sqrt(exp(S(2)*x) + S(1))*exp(-x), expand=True, _diff=True, _numerical=True)811 assert rubi_test(rubi_integrate(exp(x)/​(exp(S(2)*x) - S(8)*exp(x) + S(-1)), x), x, sqrt(S(17))*atanh(sqrt(S(17))*(-exp(x) + S(4))/​S(17))/​S(17), expand=True, _diff=True, _numerical=True)812 assert rubi_test(rubi_integrate(x**S(3)*exp(S(7)*x), x), x, x**S(3)*exp(S(7)*x)/​S(7) - S(3)*x**S(2)*exp(S(7)*x)/​S(49) + S(6)*x*exp(S(7)*x)/​S(343) - S(6)*exp(S(7)*x)/​S(2401), expand=True, _diff=True, _numerical=True)813 assert rubi_test(rubi_integrate(x**S(3)*exp(-S(2)*x + S(8)), x), x, -x**S(3)*exp(-S(2)*x + S(8))/​S(2) - S(3)*x**S(2)*exp(-S(2)*x + S(8))/​S(4) - S(3)*x*exp(-S(2)*x + S(8))/​S(4) - S(3)*exp(-S(2)*x + S(8))/​S(8), expand=True, _diff=True, _numerical=True)814 assert rubi_test(rubi_integrate(sqrt(-exp(S(2)*x) + S(9))*exp(x), x), x, sqrt(-exp(S(2)*x) + S(9))*exp(x)/​S(2) + S(9)*asin(exp(x)/​S(3))/​S(2), expand=True, _diff=True, _numerical=True)815 assert rubi_test(rubi_integrate(sqrt(-exp(S(2)*x) + S(9))*exp(S(6)*x), x), x, -(-exp(S(2)*x) + S(9))**(S(7)/​2)/​S(7) + S(18)*(-exp(S(2)*x) + S(9))**(S(5)/​2)/​S(5) - S(27)*(-exp(S(2)*x) + S(9))**(S(3)/​2), expand=True, _diff=True, _numerical=True)816 assert rubi_test(rubi_integrate(exp(S(6)*x)/​(-exp(x) + S(9))**(S(5)/​2), x), x, S(2)*(-exp(x) + S(9))**(S(7)/​2)/​S(7) - S(18)*(-exp(x) + S(9))**(S(5)/​2) + S(540)*(-exp(x) + S(9))**(S(3)/​2) - S(14580)*sqrt(-exp(x) + S(9)) - S(65610)/​sqrt(-exp(x) + S(9)) + S(39366)/​(-exp(x) + S(9))**(S(3)/​2), expand=True, _diff=True, _numerical=True)817 assert rubi_test(rubi_integrate(x**S(3)*(-S(7)*exp(x**S(4)) + S(2))**S(5), x), x, S(8)*x**S(4) - S(16807)*exp(S(5)*x**S(4))/​S(20) + S(12005)*exp(S(4)*x**S(4))/​S(8) - S(3430)*exp(S(3)*x**S(4))/​S(3) + S(490)*exp(S(2)*x**S(4)) - S(140)*exp(x**S(4)), expand=True, _diff=True, _numerical=True)818 assert rubi_test(rubi_integrate(x*sqrt(-exp(S(2)*x**S(2)) + S(1))*exp(x**S(2)), x), x, sqrt(-exp(S(2)*x**S(2)) + S(1))*exp(x**S(2))/​S(4) + asin(exp(x**S(2)))/​S(4), expand=True, _diff=True, _numerical=True)819 assert rubi_test(rubi_integrate(x**S(2)*(-exp(S(4)*x**S(3)) + S(1))**S(2)*exp(x**S(3)), x), x, exp(S(9)*x**S(3))/​S(27) - S(2)*exp(S(5)*x**S(3))/​S(15) + exp(x**S(3))/​S(3), expand=True, _diff=True, _numerical=True)820 assert rubi_test(rubi_integrate(exp(x + exp(x)), x), x, exp(exp(x)), expand=True, _diff=True, _numerical=True)821 assert rubi_test(rubi_integrate(exp(x + exp(x) + exp(exp(x))), x), x, exp(exp(exp(x))), expand=True, _diff=True, _numerical=True)822 assert rubi_test(rubi_integrate((exp(x) + exp(-x))**S(2), x), x, S(2)*x + exp(S(2)*x)/​S(2) - exp(-S(2)*x)/​S(2), expand=True, _diff=True, _numerical=True)823 assert rubi_test(rubi_integrate(S(1)/​(exp(x) + exp(-x)), x), x, atan(exp(x)), expand=True, _diff=True, _numerical=True)824 assert rubi_test(rubi_integrate((exp(x) + exp(-x))**(S(-2)), x), x, -S(1)/​(S(2)*(exp(S(2)*x) + S(1))), expand=True, _diff=True, _numerical=True)825 assert rubi_test(rubi_integrate(S(1)/​(exp(x) - exp(-x)), x), x, -atanh(exp(x)), expand=True, _diff=True, _numerical=True)826 assert rubi_test(rubi_integrate((exp(x) - exp(-x))**(S(-2)), x), x, S(1)/​(S(2)*(-exp(S(2)*x) + S(1))), expand=True, _diff=True, _numerical=True)827 assert rubi_test(rubi_integrate((exp(x) - exp(-x))**S(2)*exp(x), x), x, exp(S(3)*x)/​S(3) - S(2)*exp(x) - exp(-x), expand=True, _diff=True, _numerical=True)828 assert rubi_test(rubi_integrate((exp(x) - exp(-x))**S(3)*exp(x), x), x, S(3)*x + exp(S(4)*x)/​S(4) - S(3)*exp(S(2)*x)/​S(2) + exp(-S(2)*x)/​S(2), expand=True, _diff=True, _numerical=True)829 assert rubi_test(rubi_integrate((S(4)**x + S(1))/​(S(2)**x + S(1)), x), x, S(2)**x/​log(S(2)) + x - S(2)*log(S(2)**x + S(1))/​log(S(2)), expand=True, _diff=True, _numerical=True)830 assert rubi_test(rubi_integrate((S(4)**x + S(1))/​(S(1) + S(2)**(-x)), x), x, -S(2)**x/​log(S(2)) + S(2)**(S(2)*x + S(-1))/​log(S(2)) + S(2)*log(S(2)**x + S(1))/​log(S(2)), expand=True, _diff=True, _numerical=True)831 assert rubi_test(rubi_integrate(-S(2)*a*exp((a + x)**S(2))/​x + exp((a + x)**S(2))/​x**S(2), x), x, sqrt(pi)*erfi(a + x) - exp((a + x)**S(2))/​x, expand=True, _diff=True, _numerical=True)832 assert rubi_test(rubi_integrate((x**S(8) + x**S(6) + x**S(4))*exp(-x**S(2)), x), x, -x**S(7)*exp(-x**S(2))/​S(2) - S(9)*x**S(5)*exp(-x**S(2))/​S(4) - S(49)*x**S(3)*exp(-x**S(2))/​S(8) - S(147)*x*exp(-x**S(2))/​S(16) + S(147)*sqrt(pi)*erf(x)/​S(32), expand=True, _diff=True, _numerical=True)833 assert rubi_test(rubi_integrate(S(1)/​(exp(S(3)*x) - exp(x)), x), x, -atanh(exp(x)) + exp(-x), expand=True, _diff=True, _numerical=True)834 assert rubi_test(rubi_integrate((x**S(2) + x + S(-5))*exp(x)/​(x + S(-1))**S(2), x), x, exp(x) - S(3)*exp(x)/​(-x + S(1)), expand=True, _diff=True, _numerical=True)835 assert rubi_test(rubi_integrate(x**S(3)*exp(x**S(2))/​(x**S(2) + S(1))**S(2), x), x, exp(x**S(2))/​(S(2)*(x**S(2) + S(1))), expand=True, _diff=True, _numerical=True)836 assert rubi_test(rubi_integrate(exp(S(3)*x)/​sqrt(S(16)*exp(S(2)*x) + S(25)), x), x, sqrt(S(16)*exp(S(2)*x) + S(25))*exp(x)/​S(32) - S(25)*asinh(S(4)*exp(x)/​S(5))/​S(128), expand=True, _diff=True, _numerical=True)837 assert rubi_test(rubi_integrate((exp(x) + S(1))/​sqrt(x + exp(x)), x), x, S(2)*sqrt(x + exp(x)), expand=True, _diff=True, _numerical=True)838 assert rubi_test(rubi_integrate((exp(x) + S(1))/​(x + exp(x)), x), x, log(x + exp(x)), expand=True, _diff=True, _numerical=True)839 assert rubi_test(rubi_integrate(exp(x**S(2))/​x**S(2), x), x, sqrt(pi)*erfi(x) - exp(x**S(2))/​x, expand=True, _diff=True, _numerical=True)840 assert rubi_test(rubi_integrate((S(4)*x**S(4) + S(1))*exp(x**S(2))/​x**S(2), x), x, S(2)*x*exp(x**S(2)) - exp(x**S(2))/​x, expand=True, _diff=True, _numerical=True)841 assert rubi_test(rubi_integrate((a + b*x)**S(2)*sqrt(f**x), x), x, S(16)*b**S(2)*sqrt(f**x)/​log(f)**S(3) - S(8)*b*(a + b*x)*sqrt(f**x)/​log(f)**S(2) + S(2)*(a + b*x)**S(2)*sqrt(f**x)/​log(f), expand=True, _diff=True, _numerical=True)842 assert rubi_test(rubi_integrate(S(3)**(x**S(2) + S(1))*x, x), x, S(3)**(x**S(2) + S(1))/​(S(2)*log(S(3))), expand=True, _diff=True, _numerical=True)843 assert rubi_test(rubi_integrate(S(2)**(sqrt(x))/​sqrt(x), x), x, S(2)**(sqrt(x) + S(1))/​log(S(2)), expand=True, _diff=True, _numerical=True)844 assert rubi_test(rubi_integrate(S(2)**(S(1)/​x)/​x**S(2), x), x, -S(2)**(S(1)/​x)/​log(S(2)), expand=True, _diff=True, _numerical=True)845 assert rubi_test(rubi_integrate(S(2)**x + S(2)**(-x), x), x, S(2)**x/​log(S(2)) - S(2)**(-x)/​log(S(2)), expand=True, _diff=True, _numerical=True)846 assert rubi_test(rubi_integrate((x**S(2) - S(3)*x + S(2))*exp(-S(4)*x), x), x, -x**S(2)*exp(-S(4)*x)/​S(4) + S(5)*x*exp(-S(4)*x)/​S(8) - S(11)*exp(-S(4)*x)/​S(32), expand=True, _diff=True, _numerical=True)847 assert rubi_test(rubi_integrate(k**(x/​S(2)) + x**(sqrt(k)), x), x, S(2)*k**(x/​S(2))/​log(k) + x**(sqrt(k) + S(1))/​(sqrt(k) + S(1)), expand=True, _diff=True, _numerical=True)848 assert rubi_test(rubi_integrate(S(10)**(sqrt(x))/​sqrt(x), x), x, S(2)**(sqrt(x) + S(1))*S(5)**(sqrt(x))/​log(S(10)), expand=True, _diff=True, _numerical=True)849 assert rubi_test(rubi_integrate(exp(x)/​sqrt(x + exp(x)) + S(1)/​sqrt(x + exp(x)), x), x, S(2)*sqrt(x + exp(x)), expand=True, _diff=True, _numerical=True)850 assert rubi_test(rubi_integrate(x*(exp(x) + S(1))/​sqrt(x + exp(x)) + S(2)*sqrt(x + exp(x)), x), x, S(2)*x*sqrt(x + exp(x)), expand=True, _diff=True, _numerical=True)851 assert rubi_test(rubi_integrate(x*exp(x)/​sqrt(x + exp(x)) + x/​sqrt(x + exp(x)) + S(2)*sqrt(x + exp(x)), x), x, S(2)*x*sqrt(x + exp(x)), expand=True, _diff=True, _numerical=True)852 assert rubi_test(rubi_integrate(x*(exp(x) + S(1))/​sqrt(x + exp(x)), x), x, S(2)*x*sqrt(x + exp(x)) - S(2)*Integral(sqrt(x + exp(x)), x), expand=True, _diff=True, _numerical=True)853 assert rubi_test(rubi_integrate(x*exp(x)/​sqrt(x + exp(x)) + x/​sqrt(x + exp(x)), x), x, S(2)*x*sqrt(x + exp(x)) - S(2)*Integral(sqrt(x + exp(x)), x), expand=True, _diff=True, _numerical=True)854 assert rubi_test(rubi_integrate(x*exp(x)/​sqrt(x + exp(x)), x), x, S(2)*x*sqrt(x + exp(x)) + S(2)*sqrt(x + exp(x)) - Integral(S(1)/​sqrt(x + exp(x)), x) - S(3)*Integral(sqrt(x + exp(x)), x), expand=True, _diff=True, _numerical=True)855 assert rubi_test(rubi_integrate(x**S(2)*(S(3)*x**S(2) + S(5)*exp(x))/​(S(5)*sqrt(x**S(3) + S(5)*exp(x))) + S(4)*x*sqrt(x**S(3) + S(5)*exp(x))/​S(5), x), x, S(2)*x**S(2)*sqrt(x**S(3) + S(5)*exp(x))/​S(5), expand=True, _diff=True, _numerical=True)856 assert rubi_test(rubi_integrate(x**S(2)*exp(x)/​sqrt(x**S(3) + S(5)*exp(x)), x), x, S(2)*x**S(2)*sqrt(x**S(3) + S(5)*exp(x))/​S(5) - S(4)*Integral(x*sqrt(x**S(3) + S(5)*exp(x)), x)/​S(5) - S(3)*Integral(x**S(4)/​sqrt(x**S(3) + S(5)*exp(x)), x)/​S(5), expand=True, _diff=True, _numerical=True)857 assert rubi_test(rubi_integrate((-exp(x) + S(-1))/​(x + exp(x))**(S(1)/​3), x), x, -S(3)*(x + exp(x))**(S(2)/​3)/​S(2), expand=True, _diff=True, _numerical=True)858 assert rubi_test(rubi_integrate(x/​(x + exp(x))**(S(1)/​3) - (x + exp(x))**(S(2)/​3) - S(1)/​(x + exp(x))**(S(1)/​3), x), x, -S(3)*(x + exp(x))**(S(2)/​3)/​S(2), expand=True, _diff=True, _numerical=True)859 assert rubi_test(rubi_integrate(x/​(x + exp(x))**(S(1)/​3), x), x, -S(3)*(x + exp(x))**(S(2)/​3)/​S(2) + Integral((x + exp(x))**(S(-1)/​3), x) + Integral((x + exp(x))**(S(2)/​3), x), expand=True, _diff=True, _numerical=True)860 assert rubi_test(rubi_integrate((S(5)*x + (S(2)*x + S(3))*exp(x))/​(x + exp(x))**(S(1)/​3), x), x, S(3)*x*(x + exp(x))**(S(2)/​3), expand=True, _diff=True, _numerical=True)861 assert rubi_test(rubi_integrate(S(2)*x*exp(x)/​(x + exp(x))**(S(1)/​3) + S(2)*x/​(x + exp(x))**(S(1)/​3) + S(3)*(x + exp(x))**(S(2)/​3), x), x, S(3)*x*(x + exp(x))**(S(2)/​3), expand=True, _diff=True, _numerical=True)862 assert rubi_test(rubi_integrate((exp(x) - exp(-x))*(exp(x) + exp(-x))**S(2)*exp(x), x), x, -x + exp(S(4)*x)/​S(4) + exp(S(2)*x)/​S(2) + exp(-S(2)*x)/​S(2), expand=True, _diff=True, _numerical=True)863 assert rubi_test(rubi_integrate(x/​(x + exp(x)), x), x, Integral(x/​(x + exp(x)), x), expand=True, _diff=True, _numerical=True)864 assert rubi_test(rubi_integrate(x**S(2)/​sqrt(x + exp(x)), x), x, Integral(x**S(2)/​sqrt(x + exp(x)), x), expand=True, _diff=True, _numerical=True)865 assert rubi_test(rubi_integrate(exp(x)/​(x + exp(x)), x), x, Integral(exp(x)/​(x + exp(x)), x), expand=True, _diff=True, _numerical=True)866 assert rubi_test(rubi_integrate(exp(x)/​(x**S(2) + exp(x)), x), x, Integral(exp(x)/​(x**S(2) + exp(x)), x), expand=True, _diff=True, _numerical=True)867 assert rubi_test(rubi_integrate(f(x)/​(x + f(x)), x), x, x - Integral(x/​(x + f(x)), x), expand=True, _diff=True, _numerical=True)868 assert rubi_test(rubi_integrate(f(x)/​(x**S(2) + f(x)), x), x, x - Integral(x**S(2)/​(x**S(2) + f(x)), x), expand=True, _diff=True, _numerical=True)869 assert rubi_test(rubi_integrate(f(x)/​(x + f(x))**S(2), x), x, -Integral(x/​(x + f(x))**S(2), x) + Integral(S(1)/​(x + f(x)), x), expand=True, _diff=True, _numerical=True)870 assert rubi_test(rubi_integrate(f(x)/​(x**S(2) + f(x))**S(2), x), x, -Integral(x**S(2)/​(x**S(2) + f(x))**S(2), x) + Integral(S(1)/​(x**S(2) + f(x)), x), expand=True, _diff=True, _numerical=True)871 assert rubi_test(rubi_integrate((F**(c + d*x)*a)**m*(F**(e + f*x)*b)**n, x), x, (F**(c + d*x)*a)**m*(F**(e + f*x)*b)**n/​((d*m + f*n)*log(F)), expand=True, _diff=True, _numerical=True)872 assert rubi_test(rubi_integrate(exp(a + b*x**n + c + d*x**n), x), x, -x*(x**n*(-b - d))**(-S(1)/​n)*Gamma(S(1)/​n, x**n*(-b - d))*exp(a + c)/​n, expand=True, _diff=True, _numerical=True)873 # (difference in simplify `exp(a*log(f) + c*log(g))` converts to `f**a*g**c` in mathematica)874 # failing assert rubi_test(rubi_integrate(f**(a + b*x**n)*g**(c + d*x**n), x), x, -f**a*g**c*x*(-x**n*(b*log(f) + d*log(g)))**(-S(1)/​n)*Gamma(S(1)/​n, -x**n*(b*log(f) + d*log(g)))/​n, expand=True, _diff=True, _numerical=True)875 assert rubi_test(rubi_integrate(x**m*exp(x**n), x), x, -x**(m + S(1))*(-x**n)**(-(m + S(1))/​n)*Gamma((m + S(1))/​n, -x**n)/​n, expand=True, _diff=True, _numerical=True)876 assert rubi_test(rubi_integrate(f**(x**n)*x**m, x), x, -x**(m + S(1))*(-x**n*log(f))**(-(m + S(1))/​n)*Gamma((m + S(1))/​n, -x**n*log(f))/​n, expand=True, _diff=True, _numerical=True)877 assert rubi_test(rubi_integrate((a + b*x)**m*exp((a + b*x)**n), x), x, -(-(a + b*x)**n)**(-(m + S(1))/​n)*(a + b*x)**(m + S(1))*Gamma((m + S(1))/​n, -(a + b*x)**n)/​(b*n), expand=True, _diff=True, _numerical=True)878 assert rubi_test(rubi_integrate(f**((a + b*x)**n)*(a + b*x)**m, x), x, -(-(a + b*x)**n*log(f))**(-(m + S(1))/​n)*(a + b*x)**(m + S(1))*Gamma((m + S(1))/​n, -(a + b*x)**n*log(f))/​(b*n), expand=True, _diff=True, _numerical=True)...

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test_sine.py

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1import sys2from sympy.external import import_module3matchpy = import_module("matchpy")4if not matchpy:5 #bin/​test will not execute any tests now6 disabled = True7if sys.version_info[:2] < (3, 6):8 disabled = True9from sympy.integrals.rubi.utility_function import (10 sympy_op_factory, Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ,11 ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ,12 PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ,13 ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ,14 NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart,15 FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest,16 SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient,17 Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart,18 IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan,19 ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec,20 ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less,21 Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ,22 PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ,23 ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ,24 Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ,25 SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator,26 NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ,27 InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ,28 EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree,29 PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts,30 TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ,31 NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll,32 ExpandLinearProduct, GCD, ContentFactor, NumericFactor,33 NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst,34 ExpandExpression, Apart, SmartApart, MatchQ,35 PolynomialQuotientRemainder, FreeFactors, NonfreeFactors,36 RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms,37 ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup,38 AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor,39 RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon,40 MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ,41 GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList,42 PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ,43 RationalFunctionFactors, NonrationalFunctionFactors, Reverse,44 RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand,45 SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree,46 CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree,47 GeneralizedBinomialParts, GeneralizedTrinomialDegree,48 GeneralizedTrinomialParts, MonomialQ, MonomialSumQ,49 MinimumMonomialExponent, MonomialExponent, LinearMatchQ,50 PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ,51 TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ,52 QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms,53 NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial,54 PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD,55 AlgebraicFunctionFactors, NonalgebraicFunctionFactors,56 QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ,57 Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors,58 NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop,59 CombineExponents, FactorInteger, FactorAbsurdNumber,60 SubstForInverseFunction, SubstForFractionalPower,61 SubstForFractionalPowerOfQuotientOfLinears,62 FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ,63 SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ,64 FractionalPowerSubexpressionQ, Apply, FactorNumericGcd,65 MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ,66 TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest,67 OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors,68 PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn,69 PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree,70 FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify,71 FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand,72 NormalizeIntegrandAux, NormalizeIntegrandFactor,73 NormalizeIntegrandFactorBase, NormalizeTogether,74 NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors,75 SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm,76 TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum,77 UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear,78 PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ,79 IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor,80 FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ,81 FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator,82 SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand,83 SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM,84 SubstForFractionalPowerOfLinear, FractionalPowerOfLinear,85 InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig,86 FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ,87 PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ,88 KnownTangentIntegrandQ, KnownCotangentIntegrandQ,89 KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst,90 AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand,91 ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp,92 ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ,93 FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ,94 PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ,95 FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ,96 FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ,97 FunctionOfLog, PowerVariableExpn, PowerVariableDegree,98 PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic,99 SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ,100 Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ,101 SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2,102 ConstantFactor, SameQ, ReplacePart, CommonFactors,103 MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential,104 FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux,105 FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev,106 rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent,107 RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct,108 SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma,109 FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ,110 _SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify,111 _SimplifyAntiderivativeSum, SimplifyAntiderivativeSum,112 _SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux,113 TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor,114 PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral,115 CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial,116 Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ117 )118from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul, Add, Pow)119from sympy.integrals.rubi.symbol import WC120from sympy.core.symbol import symbols, Symbol121from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf, exp, log)122from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch)123from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec)124from sympy import pi as Pi125from sympy.integrals.rubi.rubi import rubi_integrate126a, b, c, d, e, f, m, n, x, u , k, p, r, s, t, i, j= symbols('a b c d e f m n x u k p r s t i j')127A, B, C, D, a, b, c, d, e, f, g, h, y, z, m, n, p, q, u, v, w, F = symbols('A B C D a b c d e f g h y z m n p q u v w F', )128def test_1():129 assert rubi_test(rubi_integrate(sin(a + b*x), x), x, -cos(a + b*x)/​b, expand=True, _diff=True, _numerical=True)130 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2), x), x, x/​S(2) - sin(a + b*x)*cos(a + b*x)/​(S(2)*b), expand=True, _diff=True, _numerical=True)131 assert rubi_test(rubi_integrate(sin(a + b*x)**S(3), x), x, cos(a + b*x)**S(3)/​(S(3)*b) - cos(a + b*x)/​b, expand=True, _diff=True, _numerical=True)132 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4), x), x, S(3)*x/​S(8) - sin(a + b*x)**S(3)*cos(a + b*x)/​(S(4)*b) - S(3)*sin(a + b*x)*cos(a + b*x)/​(S(8)*b), expand=True, _diff=True, _numerical=True)133 assert rubi_test(rubi_integrate(sin(a + b*x)**S(5), x), x, -cos(a + b*x)**S(5)/​(S(5)*b) + S(2)*cos(a + b*x)**S(3)/​(S(3)*b) - cos(a + b*x)/​b, expand=True, _diff=True, _numerical=True)134 assert rubi_test(rubi_integrate(sin(a + b*x)**S(6), x), x, S(5)*x/​S(16) - sin(a + b*x)**S(5)*cos(a + b*x)/​(S(6)*b) - S(5)*sin(a + b*x)**S(3)*cos(a + b*x)/​(S(24)*b) - S(5)*sin(a + b*x)*cos(a + b*x)/​(S(16)*b), expand=True, _diff=True, _numerical=True)135 assert rubi_test(rubi_integrate(sin(a + b*x)**S(7), x), x, cos(a + b*x)**S(7)/​(S(7)*b) - S(3)*cos(a + b*x)**S(5)/​(S(5)*b) + cos(a + b*x)**S(3)/​b - cos(a + b*x)/​b, expand=True, _diff=True, _numerical=True)136 assert rubi_test(rubi_integrate(sin(a + b*x)**S(8), x), x, S(35)*x/​S(128) - sin(a + b*x)**S(7)*cos(a + b*x)/​(S(8)*b) - S(7)*sin(a + b*x)**S(5)*cos(a + b*x)/​(S(48)*b) - S(35)*sin(a + b*x)**S(3)*cos(a + b*x)/​(S(192)*b) - S(35)*sin(a + b*x)*cos(a + b*x)/​(S(128)*b), expand=True, _diff=True, _numerical=True)137 assert rubi_test(rubi_integrate(sin(a + b*x)**(S(7)/​2), x), x, S(10)*EllipticF(-Pi/​S(4) + a/​S(2) + b*x/​S(2), S(2))/​(S(21)*b) - S(2)*sin(a + b*x)**(S(5)/​2)*cos(a + b*x)/​(S(7)*b) - S(10)*sqrt(sin(a + b*x))*cos(a + b*x)/​(S(21)*b), expand=True, _diff=True, _numerical=True)138 assert rubi_test(rubi_integrate(sin(a + b*x)**(S(5)/​2), x), x, S(6)*EllipticE(-Pi/​S(4) + a/​S(2) + b*x/​S(2), S(2))/​(S(5)*b) - S(2)*sin(a + b*x)**(S(3)/​2)*cos(a + b*x)/​(S(5)*b), expand=True, _diff=True, _numerical=True)139 assert rubi_test(rubi_integrate(sin(a + b*x)**(S(3)/​2), x), x, S(2)*EllipticF(-Pi/​S(4) + a/​S(2) + b*x/​S(2), S(2))/​(S(3)*b) - S(2)*sqrt(sin(a + b*x))*cos(a + b*x)/​(S(3)*b), expand=True, _diff=True, _numerical=True)140 assert rubi_test(rubi_integrate(sqrt(sin(a + b*x)), x), x, S(2)*EllipticE(-Pi/​S(4) + a/​S(2) + b*x/​S(2), S(2))/​b, expand=True, _diff=True, _numerical=True)141 assert rubi_test(rubi_integrate(S(1)/​sqrt(sin(a + b*x)), x), x, S(2)*EllipticF(-Pi/​S(4) + a/​S(2) + b*x/​S(2), S(2))/​b, expand=True, _diff=True, _numerical=True)142 assert rubi_test(rubi_integrate(sin(a + b*x)**(S(-3)/​2), x), x, -S(2)*EllipticE(-Pi/​S(4) + a/​S(2) + b*x/​S(2), S(2))/​b - S(2)*cos(a + b*x)/​(b*sqrt(sin(a + b*x))), expand=True, _diff=True, _numerical=True)143 assert rubi_test(rubi_integrate(sin(a + b*x)**(S(-5)/​2), x), x, S(2)*EllipticF(-Pi/​S(4) + a/​S(2) + b*x/​S(2), S(2))/​(S(3)*b) - S(2)*cos(a + b*x)/​(S(3)*b*sin(a + b*x)**(S(3)/​2)), expand=True, _diff=True, _numerical=True)144 assert rubi_test(rubi_integrate(sin(a + b*x)**(S(-7)/​2), x), x, -S(6)*EllipticE(-Pi/​S(4) + a/​S(2) + b*x/​S(2), S(2))/​(S(5)*b) - S(6)*cos(a + b*x)/​(S(5)*b*sqrt(sin(a + b*x))) - S(2)*cos(a + b*x)/​(S(5)*b*sin(a + b*x)**(S(5)/​2)), expand=True, _diff=True, _numerical=True)145 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(7)/​2), x), x, S(10)*c**S(4)*EllipticF(-Pi/​S(4) + a/​S(2) + b*x/​S(2), S(2))*sqrt(sin(a + b*x))/​(S(21)*b*sqrt(c*sin(a + b*x))) - S(10)*c**S(3)*sqrt(c*sin(a + b*x))*cos(a + b*x)/​(S(21)*b) - S(2)*c*(c*sin(a + b*x))**(S(5)/​2)*cos(a + b*x)/​(S(7)*b), expand=True, _diff=True, _numerical=True)146 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(5)/​2), x), x, S(6)*c**S(2)*sqrt(c*sin(a + b*x))*EllipticE(-Pi/​S(4) + a/​S(2) + b*x/​S(2), S(2))/​(S(5)*b*sqrt(sin(a + b*x))) - S(2)*c*(c*sin(a + b*x))**(S(3)/​2)*cos(a + b*x)/​(S(5)*b), expand=True, _diff=True, _numerical=True)147 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(3)/​2), x), x, S(2)*c**S(2)*EllipticF(-Pi/​S(4) + a/​S(2) + b*x/​S(2), S(2))*sqrt(sin(a + b*x))/​(S(3)*b*sqrt(c*sin(a + b*x))) - S(2)*c*sqrt(c*sin(a + b*x))*cos(a + b*x)/​(S(3)*b), expand=True, _diff=True, _numerical=True)148 assert rubi_test(rubi_integrate(sqrt(c*sin(a + b*x)), x), x, S(2)*sqrt(c*sin(a + b*x))*EllipticE(-Pi/​S(4) + a/​S(2) + b*x/​S(2), S(2))/​(b*sqrt(sin(a + b*x))), expand=True, _diff=True, _numerical=True)149 assert rubi_test(rubi_integrate(S(1)/​sqrt(c*sin(a + b*x)), x), x, S(2)*EllipticF(-Pi/​S(4) + a/​S(2) + b*x/​S(2), S(2))*sqrt(sin(a + b*x))/​(b*sqrt(c*sin(a + b*x))), expand=True, _diff=True, _numerical=True)150 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(-3)/​2), x), x, -S(2)*cos(a + b*x)/​(b*c*sqrt(c*sin(a + b*x))) - S(2)*sqrt(c*sin(a + b*x))*EllipticE(-Pi/​S(4) + a/​S(2) + b*x/​S(2), S(2))/​(b*c**S(2)*sqrt(sin(a + b*x))), expand=True, _diff=True, _numerical=True)151 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(-5)/​2), x), x, -S(2)*cos(a + b*x)/​(S(3)*b*c*(c*sin(a + b*x))**(S(3)/​2)) + S(2)*EllipticF(-Pi/​S(4) + a/​S(2) + b*x/​S(2), S(2))*sqrt(sin(a + b*x))/​(S(3)*b*c**S(2)*sqrt(c*sin(a + b*x))), expand=True, _diff=True, _numerical=True)152 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(-7)/​2), x), x, -S(2)*cos(a + b*x)/​(S(5)*b*c*(c*sin(a + b*x))**(S(5)/​2)) - S(6)*cos(a + b*x)/​(S(5)*b*c**S(3)*sqrt(c*sin(a + b*x))) - S(6)*sqrt(c*sin(a + b*x))*EllipticE(-Pi/​S(4) + a/​S(2) + b*x/​S(2), S(2))/​(S(5)*b*c**S(4)*sqrt(sin(a + b*x))), expand=True, _diff=True, _numerical=True)153 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(4)/​3), x), x, S(3)*(c*sin(a + b*x))**(S(7)/​3)*Hypergeometric2F1(S(1)/​2, S(7)/​6, S(13)/​6, sin(a + b*x)**S(2))*cos(a + b*x)/​(S(7)*b*c*sqrt(cos(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)154 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(2)/​3), x), x, S(3)*(c*sin(a + b*x))**(S(5)/​3)*Hypergeometric2F1(S(1)/​2, S(5)/​6, S(11)/​6, sin(a + b*x)**S(2))*cos(a + b*x)/​(S(5)*b*c*sqrt(cos(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)155 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(1)/​3), x), x, -S(3)*c**(S(1)/​3)*sqrt(S(1) - (c*sin(a + b*x))**(S(2)/​3)/​c**(S(2)/​3))*sqrt(S(9)/​2 - S(3)*sqrt(S(3))*I/​S(2))*sqrt((-sqrt(S(3)) + I)/​(-sqrt(S(3)) + S(3)*I) + S(2)*(c*sin(a + b*x))**(S(2)/​3)/​(c**(S(2)/​3)*(S(3) + sqrt(S(3))*I)))*sqrt((sqrt(S(3)) + I)/​(sqrt(S(3)) + S(3)*I) + S(2)*(c*sin(a + b*x))**(S(2)/​3)/​(c**(S(2)/​3)*(S(3) - sqrt(S(3))*I)))*EllipticE(asin(sqrt(S(2))*sqrt(S(1) - (c*sin(a + b*x))**(S(2)/​3)/​c**(S(2)/​3))/​sqrt(S(3) + sqrt(S(3))*I)), (-sqrt(S(3)) + S(3)*I)/​(sqrt(S(3)) + S(3)*I))*sec(a + b*x)/​b + S(3)*sqrt(S(2))*c**(S(1)/​3)*(S(1) - sqrt(S(3))*I)*sqrt(S(1) - (c*sin(a + b*x))**(S(2)/​3)/​c**(S(2)/​3))*sqrt(S(3) - sqrt(S(3))*I)*sqrt((-sqrt(S(3)) + I)/​(-sqrt(S(3)) + S(3)*I) + S(2)*(c*sin(a + b*x))**(S(2)/​3)/​(c**(S(2)/​3)*(S(3) + sqrt(S(3))*I)))*sqrt((sqrt(S(3)) + I)/​(sqrt(S(3)) + S(3)*I) + S(2)*(c*sin(a + b*x))**(S(2)/​3)/​(c**(S(2)/​3)*(S(3) - sqrt(S(3))*I)))*EllipticF(asin(sqrt(S(2))*sqrt(S(1) - (c*sin(a + b*x))**(S(2)/​3)/​c**(S(2)/​3))/​sqrt(S(3) - sqrt(S(3))*I)), (sqrt(S(3)) + S(3)*I)/​(-sqrt(S(3)) + S(3)*I))*sec(a + b*x)/​(S(4)*b), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((c*sin(a + b*x))**(S(1)/​3), x), x, S(3)*(c*sin(a + b*x))**(S(4)/​3)*Hypergeometric2F1(S(1)/​2, S(2)/​3, S(5)/​3, sin(a + b*x)**S(2))*cos(a + b*x)/​(S(4)*b*c*sqrt(cos(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)156 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(-1)/​3), x), x, -S(3)*sqrt(S(2))*sqrt(S(1) - (c*sin(a + b*x))**(S(2)/​3)/​c**(S(2)/​3))*sqrt(S(3) - sqrt(S(3))*I)*sqrt((-sqrt(S(3)) + I)/​(-sqrt(S(3)) + S(3)*I) + S(2)*(c*sin(a + b*x))**(S(2)/​3)/​(c**(S(2)/​3)*(S(3) + sqrt(S(3))*I)))*sqrt((sqrt(S(3)) + I)/​(sqrt(S(3)) + S(3)*I) + S(2)*(c*sin(a + b*x))**(S(2)/​3)/​(c**(S(2)/​3)*(S(3) - sqrt(S(3))*I)))*EllipticF(asin(sqrt(S(2))*sqrt(S(1) - (c*sin(a + b*x))**(S(2)/​3)/​c**(S(2)/​3))/​sqrt(S(3) - sqrt(S(3))*I)), (sqrt(S(3)) + S(3)*I)/​(-sqrt(S(3)) + S(3)*I))*sec(a + b*x)/​(S(2)*b*c**(S(1)/​3)), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((c*sin(a + b*x))**(S(-1)/​3), x), x, S(3)*(c*sin(a + b*x))**(S(2)/​3)*Hypergeometric2F1(S(1)/​3, S(1)/​2, S(4)/​3, sin(a + b*x)**S(2))*cos(a + b*x)/​(S(2)*b*c*sqrt(cos(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)157 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(-2)/​3), x), x, S(3)**(S(3)/​4)*(c*sin(a + b*x))**(S(1)/​3)*sqrt(c**(S(4)/​3)*(S(1) + (c*sin(a + b*x))**(S(2)/​3)/​c**(S(2)/​3) + (c*sin(a + b*x))**(S(4)/​3)/​c**(S(4)/​3))/​(c**(S(2)/​3) - (c*sin(a + b*x))**(S(2)/​3)*(S(1) + sqrt(S(3))))**S(2))*(c**(S(2)/​3) - (c*sin(a + b*x))**(S(2)/​3))*EllipticF(acos((c**(S(2)/​3) - (c*sin(a + b*x))**(S(2)/​3)*(-sqrt(S(3)) + S(1)))/​(c**(S(2)/​3) - (c*sin(a + b*x))**(S(2)/​3)*(S(1) + sqrt(S(3))))), sqrt(S(3))/​S(4) + S(1)/​2)*sec(a + b*x)/​(S(2)*b*c**(S(5)/​3)*sqrt(-(c*sin(a + b*x))**(S(2)/​3)*(c**(S(2)/​3) - (c*sin(a + b*x))**(S(2)/​3))/​(c**(S(2)/​3) - (c*sin(a + b*x))**(S(2)/​3)*(S(1) + sqrt(S(3))))**S(2))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((c*sin(a + b*x))**(S(-2)/​3), x), x, S(3)*(c*sin(a + b*x))**(S(1)/​3)*Hypergeometric2F1(S(1)/​6, S(1)/​2, S(7)/​6, sin(a + b*x)**S(2))*cos(a + b*x)/​(b*c*sqrt(cos(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)158 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(-4)/​3), x), x, -S(3)*Hypergeometric2F1(S(-1)/​6, S(1)/​2, S(5)/​6, sin(a + b*x)**S(2))*cos(a + b*x)/​(b*c*(c*sin(a + b*x))**(S(1)/​3)*sqrt(cos(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)159 assert rubi_test(rubi_integrate(sin(a + b*x)**n, x), x, Hypergeometric2F1(S(1)/​2, n/​S(2) + S(1)/​2, n/​S(2) + S(3)/​2, sin(a + b*x)**S(2))*sin(a + b*x)**(n + S(1))*cos(a + b*x)/​(b*(n + S(1))*sqrt(cos(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)160 assert rubi_test(rubi_integrate((c*sin(a + b*x))**n, x), x, (c*sin(a + b*x))**(n + S(1))*Hypergeometric2F1(S(1)/​2, n/​S(2) + S(1)/​2, n/​S(2) + S(3)/​2, sin(a + b*x)**S(2))*cos(a + b*x)/​(b*c*(n + S(1))*sqrt(cos(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)161 assert rubi_test(rubi_integrate((a*sin(x)**S(2))**(S(5)/​2), x), x, -S(8)*a**S(2)*sqrt(a*sin(x)**S(2))*cot(x)/​S(15) - S(4)*a*(a*sin(x)**S(2))**(S(3)/​2)*cot(x)/​S(15) - (a*sin(x)**S(2))**(S(5)/​2)*cot(x)/​S(5), expand=True, _diff=True, _numerical=True)162 assert rubi_test(rubi_integrate((a*sin(x)**S(2))**(S(3)/​2), x), x, -S(2)*a*sqrt(a*sin(x)**S(2))*cot(x)/​S(3) - (a*sin(x)**S(2))**(S(3)/​2)*cot(x)/​S(3), expand=True, _diff=True, _numerical=True)163 assert rubi_test(rubi_integrate(sqrt(a*sin(x)**S(2)), x), x, -sqrt(a*sin(x)**S(2))*cot(x), expand=True, _diff=True, _numerical=True)164 assert rubi_test(rubi_integrate(S(1)/​sqrt(a*sin(x)**S(2)), x), x, -sin(x)*atanh(cos(x))/​sqrt(a*sin(x)**S(2)), expand=True, _diff=True, _numerical=True)165 assert rubi_test(rubi_integrate((a*sin(x)**S(2))**(S(-3)/​2), x), x, -sin(x)*atanh(cos(x))/​(S(2)*a*sqrt(a*sin(x)**S(2))) - cot(x)/​(S(2)*a*sqrt(a*sin(x)**S(2))), expand=True, _diff=True, _numerical=True)166 assert rubi_test(rubi_integrate((a*sin(x)**S(2))**(S(-5)/​2), x), x, -cot(x)/​(S(4)*a*(a*sin(x)**S(2))**(S(3)/​2)) - S(3)*sin(x)*atanh(cos(x))/​(S(8)*a**S(2)*sqrt(a*sin(x)**S(2))) - S(3)*cot(x)/​(S(8)*a**S(2)*sqrt(a*sin(x)**S(2))), expand=True, _diff=True, _numerical=True)167 assert rubi_test(rubi_integrate((a*sin(x)**S(3))**(S(5)/​2), x), x, -S(26)*a**S(2)*sqrt(a*sin(x)**S(3))*EllipticF(Pi/​S(4) - x/​S(2), S(2))/​(S(77)*sin(x)**(S(3)/​2)) - S(2)*a**S(2)*sqrt(a*sin(x)**S(3))*sin(x)**S(5)*cos(x)/​S(15) - S(26)*a**S(2)*sqrt(a*sin(x)**S(3))*sin(x)**S(3)*cos(x)/​S(165) - S(78)*a**S(2)*sqrt(a*sin(x)**S(3))*sin(x)*cos(x)/​S(385) - S(26)*a**S(2)*sqrt(a*sin(x)**S(3))*cot(x)/​S(77), expand=True, _diff=True, _numerical=True)168 assert rubi_test(rubi_integrate((a*sin(x)**S(3))**(S(3)/​2), x), x, -S(14)*a*sqrt(a*sin(x)**S(3))*EllipticE(Pi/​S(4) - x/​S(2), S(2))/​(S(15)*sin(x)**(S(3)/​2)) - S(2)*a*sqrt(a*sin(x)**S(3))*sin(x)**S(2)*cos(x)/​S(9) - S(14)*a*sqrt(a*sin(x)**S(3))*cos(x)/​S(45), expand=True, _diff=True, _numerical=True)169 assert rubi_test(rubi_integrate(sqrt(a*sin(x)**S(3)), x), x, -S(2)*sqrt(a*sin(x)**S(3))*EllipticF(Pi/​S(4) - x/​S(2), S(2))/​(S(3)*sin(x)**(S(3)/​2)) - S(2)*sqrt(a*sin(x)**S(3))*cot(x)/​S(3), expand=True, _diff=True, _numerical=True)170 assert rubi_test(rubi_integrate(S(1)/​sqrt(a*sin(x)**S(3)), x), x, S(2)*EllipticE(Pi/​S(4) - x/​S(2), S(2))*sin(x)**(S(3)/​2)/​sqrt(a*sin(x)**S(3)) - S(2)*sin(x)*cos(x)/​sqrt(a*sin(x)**S(3)), expand=True, _diff=True, _numerical=True)171 assert rubi_test(rubi_integrate((a*sin(x)**S(3))**(S(-3)/​2), x), x, -S(10)*EllipticF(Pi/​S(4) - x/​S(2), S(2))*sin(x)**(S(3)/​2)/​(S(21)*a*sqrt(a*sin(x)**S(3))) - S(10)*cos(x)/​(S(21)*a*sqrt(a*sin(x)**S(3))) - S(2)*cot(x)*csc(x)/​(S(7)*a*sqrt(a*sin(x)**S(3))), expand=True, _diff=True, _numerical=True)172 assert rubi_test(rubi_integrate((a*sin(x)**S(3))**(S(-5)/​2), x), x, S(154)*EllipticE(Pi/​S(4) - x/​S(2), S(2))*sin(x)**(S(3)/​2)/​(S(195)*a**S(2)*sqrt(a*sin(x)**S(3))) - S(154)*sin(x)*cos(x)/​(S(195)*a**S(2)*sqrt(a*sin(x)**S(3))) - S(2)*cot(x)*csc(x)**S(4)/​(S(13)*a**S(2)*sqrt(a*sin(x)**S(3))) - S(22)*cot(x)*csc(x)**S(2)/​(S(117)*a**S(2)*sqrt(a*sin(x)**S(3))) - S(154)*cot(x)/​(S(585)*a**S(2)*sqrt(a*sin(x)**S(3))), expand=True, _diff=True, _numerical=True)173 assert rubi_test(rubi_integrate((a*sin(x)**S(4))**(S(5)/​2), x), x, S(63)*a**S(2)*x*sqrt(a*sin(x)**S(4))*csc(x)**S(2)/​S(256) - a**S(2)*sqrt(a*sin(x)**S(4))*sin(x)**S(7)*cos(x)/​S(10) - S(9)*a**S(2)*sqrt(a*sin(x)**S(4))*sin(x)**S(5)*cos(x)/​S(80) - S(21)*a**S(2)*sqrt(a*sin(x)**S(4))*sin(x)**S(3)*cos(x)/​S(160) - S(21)*a**S(2)*sqrt(a*sin(x)**S(4))*sin(x)*cos(x)/​S(128) - S(63)*a**S(2)*sqrt(a*sin(x)**S(4))*cot(x)/​S(256), expand=True, _diff=True, _numerical=True)174 assert rubi_test(rubi_integrate((a*sin(x)**S(4))**(S(3)/​2), x), x, S(5)*a*x*sqrt(a*sin(x)**S(4))*csc(x)**S(2)/​S(16) - a*sqrt(a*sin(x)**S(4))*sin(x)**S(3)*cos(x)/​S(6) - S(5)*a*sqrt(a*sin(x)**S(4))*sin(x)*cos(x)/​S(24) - S(5)*a*sqrt(a*sin(x)**S(4))*cot(x)/​S(16), expand=True, _diff=True, _numerical=True)175 assert rubi_test(rubi_integrate(sqrt(a*sin(x)**S(4)), x), x, x*sqrt(a*sin(x)**S(4))*csc(x)**S(2)/​S(2) - sqrt(a*sin(x)**S(4))*cot(x)/​S(2), expand=True, _diff=True, _numerical=True)176 assert rubi_test(rubi_integrate(S(1)/​sqrt(a*sin(x)**S(4)), x), x, -sin(x)*cos(x)/​sqrt(a*sin(x)**S(4)), expand=True, _diff=True, _numerical=True)177 assert rubi_test(rubi_integrate((a*sin(x)**S(4))**(S(-3)/​2), x), x, -sin(x)*cos(x)/​(a*sqrt(a*sin(x)**S(4))) - cos(x)**S(2)*cot(x)**S(3)/​(S(5)*a*sqrt(a*sin(x)**S(4))) - S(2)*cos(x)**S(2)*cot(x)/​(S(3)*a*sqrt(a*sin(x)**S(4))), expand=True, _diff=True, _numerical=True)178 assert rubi_test(rubi_integrate((a*sin(x)**S(4))**(S(-5)/​2), x), x, -sin(x)*cos(x)/​(a**S(2)*sqrt(a*sin(x)**S(4))) - cos(x)**S(2)*cot(x)**S(7)/​(S(9)*a**S(2)*sqrt(a*sin(x)**S(4))) - S(4)*cos(x)**S(2)*cot(x)**S(5)/​(S(7)*a**S(2)*sqrt(a*sin(x)**S(4))) - S(6)*cos(x)**S(2)*cot(x)**S(3)/​(S(5)*a**S(2)*sqrt(a*sin(x)**S(4))) - S(4)*cos(x)**S(2)*cot(x)/​(S(3)*a**S(2)*sqrt(a*sin(x)**S(4))), expand=True, _diff=True, _numerical=True)179 assert rubi_test(rubi_integrate((b*sin(c + d*x)**p)**n, x), x, (b*sin(c + d*x)**p)**n*Hypergeometric2F1(S(1)/​2, n*p/​S(2) + S(1)/​2, n*p/​S(2) + S(3)/​2, sin(c + d*x)**S(2))*sin(c + d*x)*cos(c + d*x)/​(d*(n*p + S(1))*sqrt(cos(c + d*x)**S(2))), expand=True, _diff=True, _numerical=True)180 assert rubi_test(rubi_integrate((c*sin(a + b*x)**S(2))**n, x), x, (c*sin(a + b*x)**S(2))**n*Hypergeometric2F1(S(1)/​2, n + S(1)/​2, n + S(3)/​2, sin(a + b*x)**S(2))*sin(a + b*x)*cos(a + b*x)/​(b*(S(2)*n + S(1))*sqrt(cos(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)181 assert rubi_test(rubi_integrate((c*sin(a + b*x)**S(3))**n, x), x, (c*sin(a + b*x)**S(3))**n*Hypergeometric2F1(S(1)/​2, S(3)*n/​S(2) + S(1)/​2, S(3)*n/​S(2) + S(3)/​2, sin(a + b*x)**S(2))*sin(a + b*x)*cos(a + b*x)/​(b*(S(3)*n + S(1))*sqrt(cos(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)182 assert rubi_test(rubi_integrate((c*sin(a + b*x)**S(4))**n, x), x, (c*sin(a + b*x)**S(4))**n*Hypergeometric2F1(S(1)/​2, S(2)*n + S(1)/​2, S(2)*n + S(3)/​2, sin(a + b*x)**S(2))*sin(a + b*x)*cos(a + b*x)/​(b*(S(4)*n + S(1))*sqrt(cos(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)183 assert rubi_test(rubi_integrate((c*sin(a + b*x)**m)**(S(5)/​2), x), x, S(2)*c**S(2)*sqrt(c*sin(a + b*x)**m)*Hypergeometric2F1(S(1)/​2, S(5)*m/​S(4) + S(1)/​2, S(5)*m/​S(4) + S(3)/​2, sin(a + b*x)**S(2))*sin(a + b*x)**(S(2)*m + S(1))*cos(a + b*x)/​(b*(S(5)*m + S(2))*sqrt(cos(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)184 assert rubi_test(rubi_integrate((c*sin(a + b*x)**m)**(S(3)/​2), x), x, S(2)*c*sqrt(c*sin(a + b*x)**m)*Hypergeometric2F1(S(1)/​2, S(3)*m/​S(4) + S(1)/​2, S(3)*m/​S(4) + S(3)/​2, sin(a + b*x)**S(2))*sin(a + b*x)**(m + S(1))*cos(a + b*x)/​(b*(S(3)*m + S(2))*sqrt(cos(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)185 assert rubi_test(rubi_integrate(sqrt(c*sin(a + b*x)**m), x), x, S(2)*sqrt(c*sin(a + b*x)**m)*Hypergeometric2F1(S(1)/​2, m/​S(4) + S(1)/​2, m/​S(4) + S(3)/​2, sin(a + b*x)**S(2))*sin(a + b*x)*cos(a + b*x)/​(b*(m + S(2))*sqrt(cos(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)186 assert rubi_test(rubi_integrate(S(1)/​sqrt(c*sin(a + b*x)**m), x), x, S(2)*Hypergeometric2F1(S(1)/​2, -m/​S(4) + S(1)/​2, -m/​S(4) + S(3)/​2, sin(a + b*x)**S(2))*sin(a + b*x)*cos(a + b*x)/​(b*sqrt(c*sin(a + b*x)**m)*(-m + S(2))*sqrt(cos(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)187 assert rubi_test(rubi_integrate((c*sin(a + b*x)**m)**(S(-3)/​2), x), x, S(2)*Hypergeometric2F1(S(1)/​2, -S(3)*m/​S(4) + S(1)/​2, -S(3)*m/​S(4) + S(3)/​2, sin(a + b*x)**S(2))*sin(a + b*x)**(-m + S(1))*cos(a + b*x)/​(b*c*sqrt(c*sin(a + b*x)**m)*(-S(3)*m + S(2))*sqrt(cos(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)188 assert rubi_test(rubi_integrate((c*sin(a + b*x)**m)**(S(-5)/​2), x), x, S(2)*Hypergeometric2F1(S(1)/​2, -S(5)*m/​S(4) + S(1)/​2, -S(5)*m/​S(4) + S(3)/​2, sin(a + b*x)**S(2))*sin(a + b*x)**(-S(2)*m + S(1))*cos(a + b*x)/​(b*c**S(2)*sqrt(c*sin(a + b*x)**m)*(-S(5)*m + S(2))*sqrt(cos(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)189 assert rubi_test(rubi_integrate((c*sin(a + b*x)**m)**(S(1)/​m), x), x, -(c*sin(a + b*x)**m)**(S(1)/​m)*cot(a + b*x)/​b, expand=True, _diff=True, _numerical=True)190 assert rubi_test(rubi_integrate((a*(b*sin(c + d*x))**p)**n, x), x, (a*(b*sin(c + d*x))**p)**n*Hypergeometric2F1(S(1)/​2, n*p/​S(2) + S(1)/​2, n*p/​S(2) + S(3)/​2, sin(c + d*x)**S(2))*sin(c + d*x)*cos(c + d*x)/​(d*(n*p + S(1))*sqrt(cos(c + d*x)**S(2))), expand=True, _diff=True, _numerical=True)191 assert rubi_test(rubi_integrate((a*sin(e + f*x))**m*(b*sin(e + f*x))**n, x), x, (a*sin(e + f*x))**(m + S(1))*(b*sin(e + f*x))**n*Hypergeometric2F1(S(1)/​2, m/​S(2) + n/​S(2) + S(1)/​2, m/​S(2) + n/​S(2) + S(3)/​2, sin(e + f*x)**S(2))*cos(e + f*x)/​(a*f*(m + n + S(1))*sqrt(cos(e + f*x)**S(2))), expand=True, _diff=True, _numerical=True)192 assert rubi_test(rubi_integrate(sin(a + b*x)*cos(a + b*x)**S(3), x), x, -cos(a + b*x)**S(4)/​(S(4)*b), expand=True, _diff=True, _numerical=True)193 assert rubi_test(rubi_integrate(sin(a + b*x)*cos(a + b*x)**S(2), x), x, -cos(a + b*x)**S(3)/​(S(3)*b), expand=True, _diff=True, _numerical=True)194 assert rubi_test(rubi_integrate(sin(a + b*x)*cos(a + b*x), x), x, sin(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)195 assert rubi_test(rubi_integrate(sin(a + b*x)*sec(a + b*x), x), x, -log(cos(a + b*x))/​b, expand=True, _diff=True, _numerical=True)196 assert rubi_test(rubi_integrate(sin(a + b*x)*sec(a + b*x)**S(2), x), x, sec(a + b*x)/​b, expand=True, _diff=True, _numerical=True)197 assert rubi_test(rubi_integrate(sin(a + b*x)*sec(a + b*x)**S(3), x), x, sec(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)198 assert rubi_test(rubi_integrate(sin(a + b*x)*sec(a + b*x)**S(4), x), x, sec(a + b*x)**S(3)/​(S(3)*b), expand=True, _diff=True, _numerical=True)199 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2)*cos(a + b*x)**S(7), x), x, -sin(a + b*x)**S(9)/​(S(9)*b) + S(3)*sin(a + b*x)**S(7)/​(S(7)*b) - S(3)*sin(a + b*x)**S(5)/​(S(5)*b) + sin(a + b*x)**S(3)/​(S(3)*b), expand=True, _diff=True, _numerical=True)200 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2)*cos(a + b*x)**S(5), x), x, sin(a + b*x)**S(7)/​(S(7)*b) - S(2)*sin(a + b*x)**S(5)/​(S(5)*b) + sin(a + b*x)**S(3)/​(S(3)*b), expand=True, _diff=True, _numerical=True)201 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2)*cos(a + b*x)**S(3), x), x, -sin(a + b*x)**S(5)/​(S(5)*b) + sin(a + b*x)**S(3)/​(S(3)*b), expand=True, _diff=True, _numerical=True)202 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2)*cos(a + b*x), x), x, sin(a + b*x)**S(3)/​(S(3)*b), expand=True, _diff=True, _numerical=True)203 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2)*sec(a + b*x)**S(2), x), x, -x + tan(a + b*x)/​b, expand=True, _diff=True, _numerical=True)204 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2)*sec(a + b*x)**S(4), x), x, tan(a + b*x)**S(3)/​(S(3)*b), expand=True, _diff=True, _numerical=True)205 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2)*sec(a + b*x)**S(6), x), x, tan(a + b*x)**S(5)/​(S(5)*b) + tan(a + b*x)**S(3)/​(S(3)*b), expand=True, _diff=True, _numerical=True)206 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2)*sec(a + b*x)**S(8), x), x, tan(a + b*x)**S(7)/​(S(7)*b) + S(2)*tan(a + b*x)**S(5)/​(S(5)*b) + tan(a + b*x)**S(3)/​(S(3)*b), expand=True, _diff=True, _numerical=True)207 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2)*sec(a + b*x)**S(10), x), x, tan(a + b*x)**S(9)/​(S(9)*b) + S(3)*tan(a + b*x)**S(7)/​(S(7)*b) + S(3)*tan(a + b*x)**S(5)/​(S(5)*b) + tan(a + b*x)**S(3)/​(S(3)*b), expand=True, _diff=True, _numerical=True)208 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2)*cos(a + b*x)**S(6), x), x, S(5)*x/​S(128) - sin(a + b*x)*cos(a + b*x)**S(7)/​(S(8)*b) + sin(a + b*x)*cos(a + b*x)**S(5)/​(S(48)*b) + S(5)*sin(a + b*x)*cos(a + b*x)**S(3)/​(S(192)*b) + S(5)*sin(a + b*x)*cos(a + b*x)/​(S(128)*b), expand=True, _diff=True, _numerical=True)209 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2)*cos(a + b*x)**S(4), x), x, x/​S(16) - sin(a + b*x)*cos(a + b*x)**S(5)/​(S(6)*b) + sin(a + b*x)*cos(a + b*x)**S(3)/​(S(24)*b) + sin(a + b*x)*cos(a + b*x)/​(S(16)*b), expand=True, _diff=True, _numerical=True)210 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2)*cos(a + b*x)**S(2), x), x, x/​S(8) - sin(a + b*x)*cos(a + b*x)**S(3)/​(S(4)*b) + sin(a + b*x)*cos(a + b*x)/​(S(8)*b), expand=True, _diff=True, _numerical=True)211 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2), x), x, x/​S(2) - sin(a + b*x)*cos(a + b*x)/​(S(2)*b), expand=True, _diff=True, _numerical=True)212 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2)*sec(a + b*x), x), x, -sin(a + b*x)/​b + atanh(sin(a + b*x))/​b, expand=True, _diff=True, _numerical=True)213 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2)*sec(a + b*x)**S(3), x), x, tan(a + b*x)*sec(a + b*x)/​(S(2)*b) - atanh(sin(a + b*x))/​(S(2)*b), expand=True, _diff=True, _numerical=True)214 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2)*sec(a + b*x)**S(5), x), x, tan(a + b*x)*sec(a + b*x)**S(3)/​(S(4)*b) - tan(a + b*x)*sec(a + b*x)/​(S(8)*b) - atanh(sin(a + b*x))/​(S(8)*b), expand=True, _diff=True, _numerical=True)215 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2)*sec(a + b*x)**S(7), x), x, tan(a + b*x)*sec(a + b*x)**S(5)/​(S(6)*b) - tan(a + b*x)*sec(a + b*x)**S(3)/​(S(24)*b) - tan(a + b*x)*sec(a + b*x)/​(S(16)*b) - atanh(sin(a + b*x))/​(S(16)*b), expand=True, _diff=True, _numerical=True)216 assert rubi_test(rubi_integrate(sin(a + b*x)**S(3)*cos(a + b*x)**S(5), x), x, cos(a + b*x)**S(8)/​(S(8)*b) - cos(a + b*x)**S(6)/​(S(6)*b), expand=True, _diff=True, _numerical=True)217 assert rubi_test(rubi_integrate(sin(a + b*x)**S(3)*cos(a + b*x)**S(4), x), x, cos(a + b*x)**S(7)/​(S(7)*b) - cos(a + b*x)**S(5)/​(S(5)*b), expand=True, _diff=True, _numerical=True)218 assert rubi_test(rubi_integrate(sin(a + b*x)**S(3)*cos(a + b*x)**S(3), x), x, -sin(a + b*x)**S(6)/​(S(6)*b) + sin(a + b*x)**S(4)/​(S(4)*b), expand=True, _diff=True, _numerical=True)219 assert rubi_test(rubi_integrate(sin(a + b*x)**S(3)*cos(a + b*x)**S(2), x), x, cos(a + b*x)**S(5)/​(S(5)*b) - cos(a + b*x)**S(3)/​(S(3)*b), expand=True, _diff=True, _numerical=True)220 assert rubi_test(rubi_integrate(sin(a + b*x)**S(3)*cos(a + b*x), x), x, sin(a + b*x)**S(4)/​(S(4)*b), expand=True, _diff=True, _numerical=True)221 assert rubi_test(rubi_integrate(sin(a + b*x)**S(3)*sec(a + b*x), x), x, -log(cos(a + b*x))/​b + cos(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)222 assert rubi_test(rubi_integrate(sin(a + b*x)**S(3)*sec(a + b*x)**S(2), x), x, cos(a + b*x)/​b + sec(a + b*x)/​b, expand=True, _diff=True, _numerical=True)223 assert rubi_test(rubi_integrate(sin(a + b*x)**S(3)*sec(a + b*x)**S(3), x), x, log(cos(a + b*x))/​b + tan(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)224 assert rubi_test(rubi_integrate(sin(a + b*x)**S(3)*sec(a + b*x)**S(4), x), x, sec(a + b*x)**S(3)/​(S(3)*b) - sec(a + b*x)/​b, expand=True, _diff=True, _numerical=True)225 assert rubi_test(rubi_integrate(sin(a + b*x)**S(3)*sec(a + b*x)**S(5), x), x, tan(a + b*x)**S(4)/​(S(4)*b), expand=True, _diff=True, _numerical=True)226 assert rubi_test(rubi_integrate(sin(a + b*x)**S(3)*sec(a + b*x)**S(6), x), x, sec(a + b*x)**S(5)/​(S(5)*b) - sec(a + b*x)**S(3)/​(S(3)*b), expand=True, _diff=True, _numerical=True)227 assert rubi_test(rubi_integrate(sin(a + b*x)**S(3)*sec(a + b*x)**S(7), x), x, sec(a + b*x)**S(6)/​(S(6)*b) - sec(a + b*x)**S(4)/​(S(4)*b), expand=True, _diff=True, _numerical=True)228 assert rubi_test(rubi_integrate(sin(a + b*x)**S(3)*sec(a + b*x)**S(8), x), x, sec(a + b*x)**S(7)/​(S(7)*b) - sec(a + b*x)**S(5)/​(S(5)*b), expand=True, _diff=True, _numerical=True)229 assert rubi_test(rubi_integrate(sin(a + b*x)**S(3)*sec(a + b*x)**S(9), x), x, sec(a + b*x)**S(8)/​(S(8)*b) - sec(a + b*x)**S(6)/​(S(6)*b), expand=True, _diff=True, _numerical=True)230 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4)*cos(a + b*x)**S(7), x), x, -sin(a + b*x)**S(11)/​(S(11)*b) + sin(a + b*x)**S(9)/​(S(3)*b) - S(3)*sin(a + b*x)**S(7)/​(S(7)*b) + sin(a + b*x)**S(5)/​(S(5)*b), expand=True, _diff=True, _numerical=True)231 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4)*cos(a + b*x)**S(5), x), x, sin(a + b*x)**S(9)/​(S(9)*b) - S(2)*sin(a + b*x)**S(7)/​(S(7)*b) + sin(a + b*x)**S(5)/​(S(5)*b), expand=True, _diff=True, _numerical=True)232 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4)*cos(a + b*x)**S(3), x), x, -sin(a + b*x)**S(7)/​(S(7)*b) + sin(a + b*x)**S(5)/​(S(5)*b), expand=True, _diff=True, _numerical=True)233 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4)*cos(a + b*x), x), x, sin(a + b*x)**S(5)/​(S(5)*b), expand=True, _diff=True, _numerical=True)234 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4)*sec(a + b*x)**S(2), x), x, -S(3)*x/​S(2) - sin(a + b*x)**S(2)*tan(a + b*x)/​(S(2)*b) + S(3)*tan(a + b*x)/​(S(2)*b), expand=True, _diff=True, _numerical=True)235 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4)*sec(a + b*x)**S(4), x), x, x + tan(a + b*x)**S(3)/​(S(3)*b) - tan(a + b*x)/​b, expand=True, _diff=True, _numerical=True)236 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4)*sec(a + b*x)**S(6), x), x, tan(a + b*x)**S(5)/​(S(5)*b), expand=True, _diff=True, _numerical=True)237 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4)*sec(a + b*x)**S(8), x), x, tan(a + b*x)**S(7)/​(S(7)*b) + tan(a + b*x)**S(5)/​(S(5)*b), expand=True, _diff=True, _numerical=True)238 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4)*sec(a + b*x)**S(10), x), x, tan(a + b*x)**S(9)/​(S(9)*b) + S(2)*tan(a + b*x)**S(7)/​(S(7)*b) + tan(a + b*x)**S(5)/​(S(5)*b), expand=True, _diff=True, _numerical=True)239 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4)*cos(a + b*x)**S(6), x), x, S(3)*x/​S(256) - sin(a + b*x)**S(3)*cos(a + b*x)**S(7)/​(S(10)*b) - S(3)*sin(a + b*x)*cos(a + b*x)**S(7)/​(S(80)*b) + sin(a + b*x)*cos(a + b*x)**S(5)/​(S(160)*b) + sin(a + b*x)*cos(a + b*x)**S(3)/​(S(128)*b) + S(3)*sin(a + b*x)*cos(a + b*x)/​(S(256)*b), expand=True, _diff=True, _numerical=True)240 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4)*cos(a + b*x)**S(4), x), x, S(3)*x/​S(128) - sin(a + b*x)**S(3)*cos(a + b*x)**S(5)/​(S(8)*b) - sin(a + b*x)*cos(a + b*x)**S(5)/​(S(16)*b) + sin(a + b*x)*cos(a + b*x)**S(3)/​(S(64)*b) + S(3)*sin(a + b*x)*cos(a + b*x)/​(S(128)*b), expand=True, _diff=True, _numerical=True)241 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4)*cos(a + b*x)**S(2), x), x, x/​S(16) - sin(a + b*x)**S(3)*cos(a + b*x)**S(3)/​(S(6)*b) - sin(a + b*x)*cos(a + b*x)**S(3)/​(S(8)*b) + sin(a + b*x)*cos(a + b*x)/​(S(16)*b), expand=True, _diff=True, _numerical=True)242 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4), x), x, S(3)*x/​S(8) - sin(a + b*x)**S(3)*cos(a + b*x)/​(S(4)*b) - S(3)*sin(a + b*x)*cos(a + b*x)/​(S(8)*b), expand=True, _diff=True, _numerical=True)243 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4)*sec(a + b*x), x), x, -sin(a + b*x)**S(3)/​(S(3)*b) - sin(a + b*x)/​b + atanh(sin(a + b*x))/​b, expand=True, _diff=True, _numerical=True)244 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4)*sec(a + b*x)**S(3), x), x, sin(a + b*x)*tan(a + b*x)**S(2)/​(S(2)*b) + S(3)*sin(a + b*x)/​(S(2)*b) - S(3)*atanh(sin(a + b*x))/​(S(2)*b), expand=True, _diff=True, _numerical=True)245 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4)*sec(a + b*x)**S(5), x), x, tan(a + b*x)**S(3)*sec(a + b*x)/​(S(4)*b) - S(3)*tan(a + b*x)*sec(a + b*x)/​(S(8)*b) + S(3)*atanh(sin(a + b*x))/​(S(8)*b), expand=True, _diff=True, _numerical=True)246 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4)*sec(a + b*x)**S(7), x), x, tan(a + b*x)**S(3)*sec(a + b*x)**S(3)/​(S(6)*b) - tan(a + b*x)*sec(a + b*x)**S(3)/​(S(8)*b) + tan(a + b*x)*sec(a + b*x)/​(S(16)*b) + atanh(sin(a + b*x))/​(S(16)*b), expand=True, _diff=True, _numerical=True)247 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4)*sec(a + b*x)**S(9), x), x, tan(a + b*x)**S(3)*sec(a + b*x)**S(5)/​(S(8)*b) - tan(a + b*x)*sec(a + b*x)**S(5)/​(S(16)*b) + tan(a + b*x)*sec(a + b*x)**S(3)/​(S(64)*b) + S(3)*tan(a + b*x)*sec(a + b*x)/​(S(128)*b) + S(3)*atanh(sin(a + b*x))/​(S(128)*b), expand=True, _diff=True, _numerical=True)248 assert rubi_test(rubi_integrate(sin(a + b*x)**S(5)*cos(a + b*x)**S(7), x), x, -cos(a + b*x)**S(12)/​(S(12)*b) + cos(a + b*x)**S(10)/​(S(5)*b) - cos(a + b*x)**S(8)/​(S(8)*b), expand=True, _diff=True, _numerical=True)249 assert rubi_test(rubi_integrate(sin(a + b*x)**S(5)*cos(a + b*x)**S(6), x), x, -cos(a + b*x)**S(11)/​(S(11)*b) + S(2)*cos(a + b*x)**S(9)/​(S(9)*b) - cos(a + b*x)**S(7)/​(S(7)*b), expand=True, _diff=True, _numerical=True)250 assert rubi_test(rubi_integrate(sin(a + b*x)**S(5)*cos(a + b*x)**S(5), x), x, sin(a + b*x)**S(10)/​(S(10)*b) - sin(a + b*x)**S(8)/​(S(4)*b) + sin(a + b*x)**S(6)/​(S(6)*b), expand=True, _diff=True, _numerical=True)251 assert rubi_test(rubi_integrate(sin(a + b*x)**S(5)*cos(a + b*x)**S(4), x), x, -cos(a + b*x)**S(9)/​(S(9)*b) + S(2)*cos(a + b*x)**S(7)/​(S(7)*b) - cos(a + b*x)**S(5)/​(S(5)*b), expand=True, _diff=True, _numerical=True)252 assert rubi_test(rubi_integrate(sin(a + b*x)**S(5)*cos(a + b*x)**S(3), x), x, -sin(a + b*x)**S(8)/​(S(8)*b) + sin(a + b*x)**S(6)/​(S(6)*b), expand=True, _diff=True, _numerical=True)253 assert rubi_test(rubi_integrate(sin(a + b*x)**S(5)*cos(a + b*x)**S(2), x), x, -cos(a + b*x)**S(7)/​(S(7)*b) + S(2)*cos(a + b*x)**S(5)/​(S(5)*b) - cos(a + b*x)**S(3)/​(S(3)*b), expand=True, _diff=True, _numerical=True)254 assert rubi_test(rubi_integrate(sin(a + b*x)**S(5)*cos(a + b*x), x), x, sin(a + b*x)**S(6)/​(S(6)*b), expand=True, _diff=True, _numerical=True)255 assert rubi_test(rubi_integrate(sin(a + b*x)**S(5)*sec(a + b*x), x), x, -log(cos(a + b*x))/​b - cos(a + b*x)**S(4)/​(S(4)*b) + cos(a + b*x)**S(2)/​b, expand=True, _diff=True, _numerical=True)256 assert rubi_test(rubi_integrate(sin(a + b*x)**S(5)*sec(a + b*x)**S(2), x), x, -cos(a + b*x)**S(3)/​(S(3)*b) + S(2)*cos(a + b*x)/​b + sec(a + b*x)/​b, expand=True, _diff=True, _numerical=True)257 assert rubi_test(rubi_integrate(sin(a + b*x)**S(5)*sec(a + b*x)**S(3), x), x, S(2)*log(cos(a + b*x))/​b - cos(a + b*x)**S(2)/​(S(2)*b) + sec(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)258 assert rubi_test(rubi_integrate(sin(a + b*x)**S(5)*sec(a + b*x)**S(4), x), x, -cos(a + b*x)/​b + sec(a + b*x)**S(3)/​(S(3)*b) - S(2)*sec(a + b*x)/​b, expand=True, _diff=True, _numerical=True)259 assert rubi_test(rubi_integrate(sin(a + b*x)**S(5)*sec(a + b*x)**S(5), x), x, -log(cos(a + b*x))/​b + tan(a + b*x)**S(4)/​(S(4)*b) - tan(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)260 assert rubi_test(rubi_integrate(sin(a + b*x)**S(5)*sec(a + b*x)**S(6), x), x, sec(a + b*x)**S(5)/​(S(5)*b) - S(2)*sec(a + b*x)**S(3)/​(S(3)*b) + sec(a + b*x)/​b, expand=True, _diff=True, _numerical=True)261 assert rubi_test(rubi_integrate(sin(a + b*x)**S(5)*sec(a + b*x)**S(7), x), x, tan(a + b*x)**S(6)/​(S(6)*b), expand=True, _diff=True, _numerical=True)262 assert rubi_test(rubi_integrate(sin(a + b*x)**S(5)*sec(a + b*x)**S(8), x), x, sec(a + b*x)**S(7)/​(S(7)*b) - S(2)*sec(a + b*x)**S(5)/​(S(5)*b) + sec(a + b*x)**S(3)/​(S(3)*b), expand=True, _diff=True, _numerical=True)263 assert rubi_test(rubi_integrate(sin(a + b*x)**S(5)*sec(a + b*x)**S(9), x), x, tan(a + b*x)**S(8)/​(S(8)*b) + tan(a + b*x)**S(6)/​(S(6)*b), expand=True, _diff=True, _numerical=True)264 assert rubi_test(rubi_integrate(sin(a + b*x)**S(5)*sec(a + b*x)**S(10), x), x, sec(a + b*x)**S(9)/​(S(9)*b) - S(2)*sec(a + b*x)**S(7)/​(S(7)*b) + sec(a + b*x)**S(5)/​(S(5)*b), expand=True, _diff=True, _numerical=True)265 assert rubi_test(rubi_integrate(sin(a + b*x)**S(5)*sec(a + b*x)**S(11), x), x, sec(a + b*x)**S(10)/​(S(10)*b) - sec(a + b*x)**S(8)/​(S(4)*b) + sec(a + b*x)**S(6)/​(S(6)*b), expand=True, _diff=True, _numerical=True)266 assert rubi_test(rubi_integrate(sin(a + b*x)**S(5)*sec(a + b*x)**S(12), x), x, sec(a + b*x)**S(11)/​(S(11)*b) - S(2)*sec(a + b*x)**S(9)/​(S(9)*b) + sec(a + b*x)**S(7)/​(S(7)*b), expand=True, _diff=True, _numerical=True)267 assert rubi_test(rubi_integrate(sin(a + b*x)**S(5)*sec(a + b*x)**S(13), x), x, sec(a + b*x)**S(12)/​(S(12)*b) - sec(a + b*x)**S(10)/​(S(5)*b) + sec(a + b*x)**S(8)/​(S(8)*b), expand=True, _diff=True, _numerical=True)268 assert rubi_test(rubi_integrate(sin(a + b*x)**S(6)*sec(a + b*x)**S(3), x), x, sin(a + b*x)**S(3)*tan(a + b*x)**S(2)/​(S(2)*b) + S(5)*sin(a + b*x)**S(3)/​(S(6)*b) + S(5)*sin(a + b*x)/​(S(2)*b) - S(5)*atanh(sin(a + b*x))/​(S(2)*b), expand=True, _diff=True, _numerical=True)269 assert rubi_test(rubi_integrate(sin(a + b*x)**S(7)*sec(a + b*x)**S(6), x), x, cos(a + b*x)/​b + sec(a + b*x)**S(5)/​(S(5)*b) - sec(a + b*x)**S(3)/​b + S(3)*sec(a + b*x)/​b, expand=True, _diff=True, _numerical=True)270 assert rubi_test(rubi_integrate(cos(a + b*x)**S(6)/​sin(a + b*x), x), x, cos(a + b*x)**S(5)/​(S(5)*b) + cos(a + b*x)**S(3)/​(S(3)*b) + cos(a + b*x)/​b - atanh(cos(a + b*x))/​b, expand=True, _diff=True, _numerical=True)271 assert rubi_test(rubi_integrate(cos(a + b*x)**S(5)/​sin(a + b*x), x), x, log(sin(a + b*x))/​b + sin(a + b*x)**S(4)/​(S(4)*b) - sin(a + b*x)**S(2)/​b, expand=True, _diff=True, _numerical=True)272 assert rubi_test(rubi_integrate(cos(a + b*x)**S(4)/​sin(a + b*x), x), x, cos(a + b*x)**S(3)/​(S(3)*b) + cos(a + b*x)/​b - atanh(cos(a + b*x))/​b, expand=True, _diff=True, _numerical=True)273 assert rubi_test(rubi_integrate(cos(a + b*x)**S(3)/​sin(a + b*x), x), x, log(sin(a + b*x))/​b - sin(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)274 assert rubi_test(rubi_integrate(cos(a + b*x)**S(2)/​sin(a + b*x), x), x, cos(a + b*x)/​b - atanh(cos(a + b*x))/​b, expand=True, _diff=True, _numerical=True)275 assert rubi_test(rubi_integrate(cos(a + b*x)/​sin(a + b*x), x), x, log(sin(a + b*x))/​b, expand=True, _diff=True, _numerical=True)276 assert rubi_test(rubi_integrate(sec(a + b*x)/​sin(a + b*x), x), x, log(tan(a + b*x))/​b, expand=True, _diff=True, _numerical=True)277 assert rubi_test(rubi_integrate(sec(a + b*x)**S(2)/​sin(a + b*x), x), x, -atanh(cos(a + b*x))/​b + sec(a + b*x)/​b, expand=True, _diff=True, _numerical=True)278 assert rubi_test(rubi_integrate(sec(a + b*x)**S(3)/​sin(a + b*x), x), x, log(tan(a + b*x))/​b + tan(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)279 assert rubi_test(rubi_integrate(sec(a + b*x)**S(4)/​sin(a + b*x), x), x, -atanh(cos(a + b*x))/​b + sec(a + b*x)**S(3)/​(S(3)*b) + sec(a + b*x)/​b, expand=True, _diff=True, _numerical=True)280 assert rubi_test(rubi_integrate(sec(a + b*x)**S(5)/​sin(a + b*x), x), x, log(tan(a + b*x))/​b + tan(a + b*x)**S(4)/​(S(4)*b) + tan(a + b*x)**S(2)/​b, expand=True, _diff=True, _numerical=True)281 assert rubi_test(rubi_integrate(sec(a + b*x)**S(6)/​sin(a + b*x), x), x, -atanh(cos(a + b*x))/​b + sec(a + b*x)**S(5)/​(S(5)*b) + sec(a + b*x)**S(3)/​(S(3)*b) + sec(a + b*x)/​b, expand=True, _diff=True, _numerical=True)282 assert rubi_test(rubi_integrate(sec(a + b*x)**S(7)/​sin(a + b*x), x), x, log(tan(a + b*x))/​b + tan(a + b*x)**S(6)/​(S(6)*b) + S(3)*tan(a + b*x)**S(4)/​(S(4)*b) + S(3)*tan(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)283 assert rubi_test(rubi_integrate(cos(a + b*x)**S(7)/​sin(a + b*x)**S(2), x), x, -sin(a + b*x)**S(5)/​(S(5)*b) + sin(a + b*x)**S(3)/​b - S(3)*sin(a + b*x)/​b - csc(a + b*x)/​b, expand=True, _diff=True, _numerical=True)284 assert rubi_test(rubi_integrate(cos(a + b*x)**S(6)/​sin(a + b*x)**S(2), x), x, -S(15)*x/​S(8) + cos(a + b*x)**S(4)*cot(a + b*x)/​(S(4)*b) + S(5)*cos(a + b*x)**S(2)*cot(a + b*x)/​(S(8)*b) - S(15)*cot(a + b*x)/​(S(8)*b), expand=True, _diff=True, _numerical=True)285 assert rubi_test(rubi_integrate(cos(a + b*x)**S(5)/​sin(a + b*x)**S(2), x), x, sin(a + b*x)**S(3)/​(S(3)*b) - S(2)*sin(a + b*x)/​b - csc(a + b*x)/​b, expand=True, _diff=True, _numerical=True)286 assert rubi_test(rubi_integrate(cos(a + b*x)**S(4)/​sin(a + b*x)**S(2), x), x, -S(3)*x/​S(2) + cos(a + b*x)**S(2)*cot(a + b*x)/​(S(2)*b) - S(3)*cot(a + b*x)/​(S(2)*b), expand=True, _diff=True, _numerical=True)287 assert rubi_test(rubi_integrate(cos(a + b*x)**S(3)/​sin(a + b*x)**S(2), x), x, -sin(a + b*x)/​b - csc(a + b*x)/​b, expand=True, _diff=True, _numerical=True)288 assert rubi_test(rubi_integrate(cos(a + b*x)**S(2)/​sin(a + b*x)**S(2), x), x, -x - cot(a + b*x)/​b, expand=True, _diff=True, _numerical=True)289 assert rubi_test(rubi_integrate(cos(a + b*x)/​sin(a + b*x)**S(2), x), x, -csc(a + b*x)/​b, expand=True, _diff=True, _numerical=True)290 assert rubi_test(rubi_integrate(sec(a + b*x)/​sin(a + b*x)**S(2), x), x, atanh(sin(a + b*x))/​b - csc(a + b*x)/​b, expand=True, _diff=True, _numerical=True)291 assert rubi_test(rubi_integrate(sec(a + b*x)**S(2)/​sin(a + b*x)**S(2), x), x, tan(a + b*x)/​b - cot(a + b*x)/​b, expand=True, _diff=True, _numerical=True)292 assert rubi_test(rubi_integrate(sec(a + b*x)**S(3)/​sin(a + b*x)**S(2), x), x, S(3)*atanh(sin(a + b*x))/​(S(2)*b) + csc(a + b*x)*sec(a + b*x)**S(2)/​(S(2)*b) - S(3)*csc(a + b*x)/​(S(2)*b), expand=True, _diff=True, _numerical=True)293 assert rubi_test(rubi_integrate(sec(a + b*x)**S(4)/​sin(a + b*x)**S(2), x), x, tan(a + b*x)**S(3)/​(S(3)*b) + S(2)*tan(a + b*x)/​b - cot(a + b*x)/​b, expand=True, _diff=True, _numerical=True)294 assert rubi_test(rubi_integrate(sec(a + b*x)**S(5)/​sin(a + b*x)**S(2), x), x, S(15)*atanh(sin(a + b*x))/​(S(8)*b) + csc(a + b*x)*sec(a + b*x)**S(4)/​(S(4)*b) + S(5)*csc(a + b*x)*sec(a + b*x)**S(2)/​(S(8)*b) - S(15)*csc(a + b*x)/​(S(8)*b), expand=True, _diff=True, _numerical=True)295 assert rubi_test(rubi_integrate(cos(a + b*x)**S(7)/​sin(a + b*x)**S(3), x), x, -S(3)*log(sin(a + b*x))/​b - sin(a + b*x)**S(4)/​(S(4)*b) + S(3)*sin(a + b*x)**S(2)/​(S(2)*b) - csc(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)296 assert rubi_test(rubi_integrate(cos(a + b*x)**S(6)/​sin(a + b*x)**S(3), x), x, -cos(a + b*x)**S(3)*cot(a + b*x)**S(2)/​(S(2)*b) - S(5)*cos(a + b*x)**S(3)/​(S(6)*b) - S(5)*cos(a + b*x)/​(S(2)*b) + S(5)*atanh(cos(a + b*x))/​(S(2)*b), expand=True, _diff=True, _numerical=True)297 assert rubi_test(rubi_integrate(cos(a + b*x)**S(5)/​sin(a + b*x)**S(3), x), x, -S(2)*log(sin(a + b*x))/​b + sin(a + b*x)**S(2)/​(S(2)*b) - csc(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)298 assert rubi_test(rubi_integrate(cos(a + b*x)**S(4)/​sin(a + b*x)**S(3), x), x, -cos(a + b*x)*cot(a + b*x)**S(2)/​(S(2)*b) - S(3)*cos(a + b*x)/​(S(2)*b) + S(3)*atanh(cos(a + b*x))/​(S(2)*b), expand=True, _diff=True, _numerical=True)299 assert rubi_test(rubi_integrate(cos(a + b*x)**S(3)/​sin(a + b*x)**S(3), x), x, -log(sin(a + b*x))/​b - cot(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)300 assert rubi_test(rubi_integrate(cos(a + b*x)**S(2)/​sin(a + b*x)**S(3), x), x, -cot(a + b*x)*csc(a + b*x)/​(S(2)*b) + atanh(cos(a + b*x))/​(S(2)*b), expand=True, _diff=True, _numerical=True)301 assert rubi_test(rubi_integrate(cos(a + b*x)/​sin(a + b*x)**S(3), x), x, -csc(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)302 assert rubi_test(rubi_integrate(sec(a + b*x)/​sin(a + b*x)**S(3), x), x, log(tan(a + b*x))/​b - cot(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)303 assert rubi_test(rubi_integrate(sec(a + b*x)**S(2)/​sin(a + b*x)**S(3), x), x, -S(3)*atanh(cos(a + b*x))/​(S(2)*b) - csc(a + b*x)**S(2)*sec(a + b*x)/​(S(2)*b) + S(3)*sec(a + b*x)/​(S(2)*b), expand=True, _diff=True, _numerical=True)304 assert rubi_test(rubi_integrate(sec(a + b*x)**S(3)/​sin(a + b*x)**S(3), x), x, S(2)*log(tan(a + b*x))/​b + tan(a + b*x)**S(2)/​(S(2)*b) - cot(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)305 assert rubi_test(rubi_integrate(sec(a + b*x)**S(4)/​sin(a + b*x)**S(3), x), x, -S(5)*atanh(cos(a + b*x))/​(S(2)*b) - csc(a + b*x)**S(2)*sec(a + b*x)**S(3)/​(S(2)*b) + S(5)*sec(a + b*x)**S(3)/​(S(6)*b) + S(5)*sec(a + b*x)/​(S(2)*b), expand=True, _diff=True, _numerical=True)306 assert rubi_test(rubi_integrate(sec(a + b*x)**S(5)/​sin(a + b*x)**S(3), x), x, S(3)*log(tan(a + b*x))/​b + tan(a + b*x)**S(4)/​(S(4)*b) + S(3)*tan(a + b*x)**S(2)/​(S(2)*b) - cot(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)307 assert rubi_test(rubi_integrate(cos(a + b*x)**S(9)/​sin(a + b*x)**S(4), x), x, sin(a + b*x)**S(5)/​(S(5)*b) - S(4)*sin(a + b*x)**S(3)/​(S(3)*b) + S(6)*sin(a + b*x)/​b - csc(a + b*x)**S(3)/​(S(3)*b) + S(4)*csc(a + b*x)/​b, expand=True, _diff=True, _numerical=True)308 assert rubi_test(rubi_integrate(cos(a + b*x)**S(8)/​sin(a + b*x)**S(4), x), x, S(35)*x/​S(8) + cos(a + b*x)**S(4)*cot(a + b*x)**S(3)/​(S(4)*b) + S(7)*cos(a + b*x)**S(2)*cot(a + b*x)**S(3)/​(S(8)*b) - S(35)*cot(a + b*x)**S(3)/​(S(24)*b) + S(35)*cot(a + b*x)/​(S(8)*b), expand=True, _diff=True, _numerical=True)309 assert rubi_test(rubi_integrate(cos(a + b*x)**S(7)/​sin(a + b*x)**S(4), x), x, -sin(a + b*x)**S(3)/​(S(3)*b) + S(3)*sin(a + b*x)/​b - csc(a + b*x)**S(3)/​(S(3)*b) + S(3)*csc(a + b*x)/​b, expand=True, _diff=True, _numerical=True)310 assert rubi_test(rubi_integrate(cos(a + b*x)**S(6)/​sin(a + b*x)**S(4), x), x, S(5)*x/​S(2) + cos(a + b*x)**S(2)*cot(a + b*x)**S(3)/​(S(2)*b) - S(5)*cot(a + b*x)**S(3)/​(S(6)*b) + S(5)*cot(a + b*x)/​(S(2)*b), expand=True, _diff=True, _numerical=True)311 assert rubi_test(rubi_integrate(cos(a + b*x)**S(5)/​sin(a + b*x)**S(4), x), x, sin(a + b*x)/​b - csc(a + b*x)**S(3)/​(S(3)*b) + S(2)*csc(a + b*x)/​b, expand=True, _diff=True, _numerical=True)312 assert rubi_test(rubi_integrate(cos(a + b*x)**S(4)/​sin(a + b*x)**S(4), x), x, x - cot(a + b*x)**S(3)/​(S(3)*b) + cot(a + b*x)/​b, expand=True, _diff=True, _numerical=True)313 assert rubi_test(rubi_integrate(cos(a + b*x)**S(3)/​sin(a + b*x)**S(4), x), x, -csc(a + b*x)**S(3)/​(S(3)*b) + csc(a + b*x)/​b, expand=True, _diff=True, _numerical=True)314 assert rubi_test(rubi_integrate(cos(a + b*x)**S(2)/​sin(a + b*x)**S(4), x), x, -cot(a + b*x)**S(3)/​(S(3)*b), expand=True, _diff=True, _numerical=True)315 assert rubi_test(rubi_integrate(cos(a + b*x)/​sin(a + b*x)**S(4), x), x, -csc(a + b*x)**S(3)/​(S(3)*b), expand=True, _diff=True, _numerical=True)316 assert rubi_test(rubi_integrate(sec(a + b*x)/​sin(a + b*x)**S(4), x), x, atanh(sin(a + b*x))/​b - csc(a + b*x)**S(3)/​(S(3)*b) - csc(a + b*x)/​b, expand=True, _diff=True, _numerical=True)317 assert rubi_test(rubi_integrate(sec(a + b*x)**S(2)/​sin(a + b*x)**S(4), x), x, tan(a + b*x)/​b - cot(a + b*x)**S(3)/​(S(3)*b) - S(2)*cot(a + b*x)/​b, expand=True, _diff=True, _numerical=True)318 assert rubi_test(rubi_integrate(sec(a + b*x)**S(3)/​sin(a + b*x)**S(4), x), x, S(5)*atanh(sin(a + b*x))/​(S(2)*b) + csc(a + b*x)**S(3)*sec(a + b*x)**S(2)/​(S(2)*b) - S(5)*csc(a + b*x)**S(3)/​(S(6)*b) - S(5)*csc(a + b*x)/​(S(2)*b), expand=True, _diff=True, _numerical=True)319 assert rubi_test(rubi_integrate(sec(a + b*x)**S(4)/​sin(a + b*x)**S(4), x), x, tan(a + b*x)**S(3)/​(S(3)*b) + S(3)*tan(a + b*x)/​b - cot(a + b*x)**S(3)/​(S(3)*b) - S(3)*cot(a + b*x)/​b, expand=True, _diff=True, _numerical=True)320 assert rubi_test(rubi_integrate(sec(a + b*x)**S(5)/​sin(a + b*x)**S(4), x), x, S(35)*atanh(sin(a + b*x))/​(S(8)*b) + csc(a + b*x)**S(3)*sec(a + b*x)**S(4)/​(S(4)*b) + S(7)*csc(a + b*x)**S(3)*sec(a + b*x)**S(2)/​(S(8)*b) - S(35)*csc(a + b*x)**S(3)/​(S(24)*b) - S(35)*csc(a + b*x)/​(S(8)*b), expand=True, _diff=True, _numerical=True)321 assert rubi_test(rubi_integrate(cos(a + b*x)**S(9)/​sin(a + b*x)**S(5), x), x, S(6)*log(sin(a + b*x))/​b + sin(a + b*x)**S(4)/​(S(4)*b) - S(2)*sin(a + b*x)**S(2)/​b - csc(a + b*x)**S(4)/​(S(4)*b) + S(2)*csc(a + b*x)**S(2)/​b, expand=True, _diff=True, _numerical=True)322 assert rubi_test(rubi_integrate(cos(a + b*x)**S(8)/​sin(a + b*x)**S(5), x), x, -cos(a + b*x)**S(3)*cot(a + b*x)**S(4)/​(S(4)*b) + S(7)*cos(a + b*x)**S(3)*cot(a + b*x)**S(2)/​(S(8)*b) + S(35)*cos(a + b*x)**S(3)/​(S(24)*b) + S(35)*cos(a + b*x)/​(S(8)*b) - S(35)*atanh(cos(a + b*x))/​(S(8)*b), expand=True, _diff=True, _numerical=True)323 assert rubi_test(rubi_integrate(cos(a + b*x)**S(7)/​sin(a + b*x)**S(5), x), x, S(3)*log(sin(a + b*x))/​b - sin(a + b*x)**S(2)/​(S(2)*b) - csc(a + b*x)**S(4)/​(S(4)*b) + S(3)*csc(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)324 assert rubi_test(rubi_integrate(cos(a + b*x)**S(6)/​sin(a + b*x)**S(5), x), x, -cos(a + b*x)*cot(a + b*x)**S(4)/​(S(4)*b) + S(5)*cos(a + b*x)*cot(a + b*x)**S(2)/​(S(8)*b) + S(15)*cos(a + b*x)/​(S(8)*b) - S(15)*atanh(cos(a + b*x))/​(S(8)*b), expand=True, _diff=True, _numerical=True)325 assert rubi_test(rubi_integrate(cos(a + b*x)**S(5)/​sin(a + b*x)**S(5), x), x, log(sin(a + b*x))/​b - cot(a + b*x)**S(4)/​(S(4)*b) + cot(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)326 assert rubi_test(rubi_integrate(cos(a + b*x)**S(4)/​sin(a + b*x)**S(5), x), x, -cot(a + b*x)**S(3)*csc(a + b*x)/​(S(4)*b) + S(3)*cot(a + b*x)*csc(a + b*x)/​(S(8)*b) - S(3)*atanh(cos(a + b*x))/​(S(8)*b), expand=True, _diff=True, _numerical=True)327 assert rubi_test(rubi_integrate(cos(a + b*x)**S(3)/​sin(a + b*x)**S(5), x), x, -cot(a + b*x)**S(4)/​(S(4)*b), expand=True, _diff=True, _numerical=True)328 assert rubi_test(rubi_integrate(cos(a + b*x)**S(2)/​sin(a + b*x)**S(5), x), x, -cot(a + b*x)*csc(a + b*x)**S(3)/​(S(4)*b) + cot(a + b*x)*csc(a + b*x)/​(S(8)*b) + atanh(cos(a + b*x))/​(S(8)*b), expand=True, _diff=True, _numerical=True)329 assert rubi_test(rubi_integrate(cos(a + b*x)/​sin(a + b*x)**S(5), x), x, -csc(a + b*x)**S(4)/​(S(4)*b), expand=True, _diff=True, _numerical=True)330 assert rubi_test(rubi_integrate(sec(a + b*x)/​sin(a + b*x)**S(5), x), x, log(tan(a + b*x))/​b - cot(a + b*x)**S(4)/​(S(4)*b) - cot(a + b*x)**S(2)/​b, expand=True, _diff=True, _numerical=True)331 assert rubi_test(rubi_integrate(sec(a + b*x)**S(2)/​sin(a + b*x)**S(5), x), x, -S(15)*atanh(cos(a + b*x))/​(S(8)*b) - csc(a + b*x)**S(4)*sec(a + b*x)/​(S(4)*b) - S(5)*csc(a + b*x)**S(2)*sec(a + b*x)/​(S(8)*b) + S(15)*sec(a + b*x)/​(S(8)*b), expand=True, _diff=True, _numerical=True)332 assert rubi_test(rubi_integrate(sec(a + b*x)**S(3)/​sin(a + b*x)**S(5), x), x, S(3)*log(tan(a + b*x))/​b + tan(a + b*x)**S(2)/​(S(2)*b) - cot(a + b*x)**S(4)/​(S(4)*b) - S(3)*cot(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)333 assert rubi_test(rubi_integrate(sec(a + b*x)**S(4)/​sin(a + b*x)**S(5), x), x, -S(35)*atanh(cos(a + b*x))/​(S(8)*b) - csc(a + b*x)**S(4)*sec(a + b*x)**S(3)/​(S(4)*b) - S(7)*csc(a + b*x)**S(2)*sec(a + b*x)**S(3)/​(S(8)*b) + S(35)*sec(a + b*x)**S(3)/​(S(24)*b) + S(35)*sec(a + b*x)/​(S(8)*b), expand=True, _diff=True, _numerical=True)334 assert rubi_test(rubi_integrate(sec(a + b*x)**S(5)/​sin(a + b*x)**S(5), x), x, S(6)*log(tan(a + b*x))/​b + tan(a + b*x)**S(4)/​(S(4)*b) + S(2)*tan(a + b*x)**S(2)/​b - cot(a + b*x)**S(4)/​(S(4)*b) - S(2)*cot(a + b*x)**S(2)/​b, expand=True, _diff=True, _numerical=True)335 assert rubi_test(rubi_integrate(cos(x)**S(2)/​sin(x)**S(6), x), x, -cot(x)**S(5)/​S(5) - cot(x)**S(3)/​S(3), expand=True, _diff=True, _numerical=True)336 assert rubi_test(rubi_integrate(cos(x)**S(3)/​sin(x)**S(7), x), x, -csc(x)**S(6)/​S(6) + csc(x)**S(4)/​S(4), expand=True, _diff=True, _numerical=True)337 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(3)/​2)*sin(a + b*x), x), x, -S(2)*(d*cos(a + b*x))**(S(5)/​2)/​(S(5)*b*d), expand=True, _diff=True, _numerical=True)338 assert rubi_test(rubi_integrate(sqrt(d*cos(a + b*x))*sin(a + b*x), x), x, -S(2)*(d*cos(a + b*x))**(S(3)/​2)/​(S(3)*b*d), expand=True, _diff=True, _numerical=True)339 assert rubi_test(rubi_integrate(sin(a + b*x)/​sqrt(d*cos(a + b*x)), x), x, -S(2)*sqrt(d*cos(a + b*x))/​(b*d), expand=True, _diff=True, _numerical=True)340 assert rubi_test(rubi_integrate(sin(a + b*x)/​(d*cos(a + b*x))**(S(3)/​2), x), x, S(2)/​(b*d*sqrt(d*cos(a + b*x))), expand=True, _diff=True, _numerical=True)341 assert rubi_test(rubi_integrate(sin(a + b*x)/​(d*cos(a + b*x))**(S(5)/​2), x), x, S(2)/​(S(3)*b*d*(d*cos(a + b*x))**(S(3)/​2)), expand=True, _diff=True, _numerical=True)342 assert rubi_test(rubi_integrate(sin(a + b*x)/​(d*cos(a + b*x))**(S(7)/​2), x), x, S(2)/​(S(5)*b*d*(d*cos(a + b*x))**(S(5)/​2)), expand=True, _diff=True, _numerical=True)343 assert rubi_test(rubi_integrate(sin(a + b*x)/​(d*cos(a + b*x))**(S(9)/​2), x), x, S(2)/​(S(7)*b*d*(d*cos(a + b*x))**(S(7)/​2)), expand=True, _diff=True, _numerical=True)344 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(9)/​2)*sin(a + b*x)**S(2), x), x, S(28)*d**S(4)*sqrt(d*cos(a + b*x))*EllipticE(a/​S(2) + b*x/​S(2), S(2))/​(S(195)*b*sqrt(cos(a + b*x))) + S(28)*d**S(3)*(d*cos(a + b*x))**(S(3)/​2)*sin(a + b*x)/​(S(585)*b) + S(4)*d*(d*cos(a + b*x))**(S(7)/​2)*sin(a + b*x)/​(S(117)*b) - S(2)*(d*cos(a + b*x))**(S(11)/​2)*sin(a + b*x)/​(S(13)*b*d), expand=True, _diff=True, _numerical=True)345 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(7)/​2)*sin(a + b*x)**S(2), x), x, S(20)*d**S(4)*EllipticF(a/​S(2) + b*x/​S(2), S(2))*sqrt(cos(a + b*x))/​(S(231)*b*sqrt(d*cos(a + b*x))) + S(20)*d**S(3)*sqrt(d*cos(a + b*x))*sin(a + b*x)/​(S(231)*b) + S(4)*d*(d*cos(a + b*x))**(S(5)/​2)*sin(a + b*x)/​(S(77)*b) - S(2)*(d*cos(a + b*x))**(S(9)/​2)*sin(a + b*x)/​(S(11)*b*d), expand=True, _diff=True, _numerical=True)346 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(5)/​2)*sin(a + b*x)**S(2), x), x, S(4)*d**S(2)*sqrt(d*cos(a + b*x))*EllipticE(a/​S(2) + b*x/​S(2), S(2))/​(S(15)*b*sqrt(cos(a + b*x))) + S(4)*d*(d*cos(a + b*x))**(S(3)/​2)*sin(a + b*x)/​(S(45)*b) - S(2)*(d*cos(a + b*x))**(S(7)/​2)*sin(a + b*x)/​(S(9)*b*d), expand=True, _diff=True, _numerical=True)347 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(3)/​2)*sin(a + b*x)**S(2), x), x, S(4)*d**S(2)*EllipticF(a/​S(2) + b*x/​S(2), S(2))*sqrt(cos(a + b*x))/​(S(21)*b*sqrt(d*cos(a + b*x))) + S(4)*d*sqrt(d*cos(a + b*x))*sin(a + b*x)/​(S(21)*b) - S(2)*(d*cos(a + b*x))**(S(5)/​2)*sin(a + b*x)/​(S(7)*b*d), expand=True, _diff=True, _numerical=True)348 assert rubi_test(rubi_integrate(sqrt(d*cos(a + b*x))*sin(a + b*x)**S(2), x), x, S(4)*sqrt(d*cos(a + b*x))*EllipticE(a/​S(2) + b*x/​S(2), S(2))/​(S(5)*b*sqrt(cos(a + b*x))) - S(2)*(d*cos(a + b*x))**(S(3)/​2)*sin(a + b*x)/​(S(5)*b*d), expand=True, _diff=True, _numerical=True)349 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2)/​sqrt(d*cos(a + b*x)), x), x, S(4)*EllipticF(a/​S(2) + b*x/​S(2), S(2))*sqrt(cos(a + b*x))/​(S(3)*b*sqrt(d*cos(a + b*x))) - S(2)*sqrt(d*cos(a + b*x))*sin(a + b*x)/​(S(3)*b*d), expand=True, _diff=True, _numerical=True)350 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2)/​(d*cos(a + b*x))**(S(3)/​2), x), x, S(2)*sin(a + b*x)/​(b*d*sqrt(d*cos(a + b*x))) - S(4)*sqrt(d*cos(a + b*x))*EllipticE(a/​S(2) + b*x/​S(2), S(2))/​(b*d**S(2)*sqrt(cos(a + b*x))), expand=True, _diff=True, _numerical=True)351 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2)/​(d*cos(a + b*x))**(S(5)/​2), x), x, S(2)*sin(a + b*x)/​(S(3)*b*d*(d*cos(a + b*x))**(S(3)/​2)) - S(4)*EllipticF(a/​S(2) + b*x/​S(2), S(2))*sqrt(cos(a + b*x))/​(S(3)*b*d**S(2)*sqrt(d*cos(a + b*x))), expand=True, _diff=True, _numerical=True)352 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2)/​(d*cos(a + b*x))**(S(7)/​2), x), x, S(2)*sin(a + b*x)/​(S(5)*b*d*(d*cos(a + b*x))**(S(5)/​2)) - S(4)*sin(a + b*x)/​(S(5)*b*d**S(3)*sqrt(d*cos(a + b*x))) + S(4)*sqrt(d*cos(a + b*x))*EllipticE(a/​S(2) + b*x/​S(2), S(2))/​(S(5)*b*d**S(4)*sqrt(cos(a + b*x))), expand=True, _diff=True, _numerical=True)353 assert rubi_test(rubi_integrate(sin(a + b*x)**S(2)/​(d*cos(a + b*x))**(S(9)/​2), x), x, S(2)*sin(a + b*x)/​(S(7)*b*d*(d*cos(a + b*x))**(S(7)/​2)) - S(4)*sin(a + b*x)/​(S(21)*b*d**S(3)*(d*cos(a + b*x))**(S(3)/​2)) - S(4)*EllipticF(a/​S(2) + b*x/​S(2), S(2))*sqrt(cos(a + b*x))/​(S(21)*b*d**S(4)*sqrt(d*cos(a + b*x))), expand=True, _diff=True, _numerical=True)354 assert rubi_test(rubi_integrate(sqrt(d*cos(a + b*x))*sin(a + b*x)**S(3), x), x, -S(2)*(d*cos(a + b*x))**(S(3)/​2)/​(S(3)*b*d) + S(2)*(d*cos(a + b*x))**(S(7)/​2)/​(S(7)*b*d**S(3)), expand=True, _diff=True, _numerical=True)355 assert rubi_test(rubi_integrate(sin(a + b*x)**S(3)/​sqrt(d*cos(a + b*x)), x), x, -S(2)*sqrt(d*cos(a + b*x))/​(b*d) + S(2)*(d*cos(a + b*x))**(S(5)/​2)/​(S(5)*b*d**S(3)), expand=True, _diff=True, _numerical=True)356 assert rubi_test(rubi_integrate(sin(a + b*x)**S(3)/​(d*cos(a + b*x))**(S(3)/​2), x), x, S(2)/​(b*d*sqrt(d*cos(a + b*x))) + S(2)*(d*cos(a + b*x))**(S(3)/​2)/​(S(3)*b*d**S(3)), expand=True, _diff=True, _numerical=True)357 assert rubi_test(rubi_integrate(sin(a + b*x)**S(3)/​(d*cos(a + b*x))**(S(5)/​2), x), x, S(2)/​(S(3)*b*d*(d*cos(a + b*x))**(S(3)/​2)) + S(2)*sqrt(d*cos(a + b*x))/​(b*d**S(3)), expand=True, _diff=True, _numerical=True)358 assert rubi_test(rubi_integrate(sin(a + b*x)**S(3)/​(d*cos(a + b*x))**(S(7)/​2), x), x, S(2)/​(S(5)*b*d*(d*cos(a + b*x))**(S(5)/​2)) - S(2)/​(b*d**S(3)*sqrt(d*cos(a + b*x))), expand=True, _diff=True, _numerical=True)359 assert rubi_test(rubi_integrate(sin(a + b*x)**S(3)/​(d*cos(a + b*x))**(S(9)/​2), x), x, S(2)/​(S(7)*b*d*(d*cos(a + b*x))**(S(7)/​2)) - S(2)/​(S(3)*b*d**S(3)*(d*cos(a + b*x))**(S(3)/​2)), expand=True, _diff=True, _numerical=True)360 assert rubi_test(rubi_integrate(sin(a + b*x)**S(3)/​(d*cos(a + b*x))**(S(11)/​2), x), x, S(2)/​(S(9)*b*d*(d*cos(a + b*x))**(S(9)/​2)) - S(2)/​(S(5)*b*d**S(3)*(d*cos(a + b*x))**(S(5)/​2)), expand=True, _diff=True, _numerical=True)361 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(9)/​2)*sin(a + b*x)**S(4), x), x, S(56)*d**S(4)*sqrt(d*cos(a + b*x))*EllipticE(a/​S(2) + b*x/​S(2), S(2))/​(S(1105)*b*sqrt(cos(a + b*x))) + S(56)*d**S(3)*(d*cos(a + b*x))**(S(3)/​2)*sin(a + b*x)/​(S(3315)*b) + S(8)*d*(d*cos(a + b*x))**(S(7)/​2)*sin(a + b*x)/​(S(663)*b) - S(2)*(d*cos(a + b*x))**(S(11)/​2)*sin(a + b*x)**S(3)/​(S(17)*b*d) - S(12)*(d*cos(a + b*x))**(S(11)/​2)*sin(a + b*x)/​(S(221)*b*d), expand=True, _diff=True, _numerical=True)362 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(7)/​2)*sin(a + b*x)**S(4), x), x, S(8)*d**S(4)*EllipticF(a/​S(2) + b*x/​S(2), S(2))*sqrt(cos(a + b*x))/​(S(231)*b*sqrt(d*cos(a + b*x))) + S(8)*d**S(3)*sqrt(d*cos(a + b*x))*sin(a + b*x)/​(S(231)*b) + S(8)*d*(d*cos(a + b*x))**(S(5)/​2)*sin(a + b*x)/​(S(385)*b) - S(2)*(d*cos(a + b*x))**(S(9)/​2)*sin(a + b*x)**S(3)/​(S(15)*b*d) - S(4)*(d*cos(a + b*x))**(S(9)/​2)*sin(a + b*x)/​(S(55)*b*d), expand=True, _diff=True, _numerical=True)363 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(5)/​2)*sin(a + b*x)**S(4), x), x, S(8)*d**S(2)*sqrt(d*cos(a + b*x))*EllipticE(a/​S(2) + b*x/​S(2), S(2))/​(S(65)*b*sqrt(cos(a + b*x))) + S(8)*d*(d*cos(a + b*x))**(S(3)/​2)*sin(a + b*x)/​(S(195)*b) - S(2)*(d*cos(a + b*x))**(S(7)/​2)*sin(a + b*x)**S(3)/​(S(13)*b*d) - S(4)*(d*cos(a + b*x))**(S(7)/​2)*sin(a + b*x)/​(S(39)*b*d), expand=True, _diff=True, _numerical=True)364 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(3)/​2)*sin(a + b*x)**S(4), x), x, S(8)*d**S(2)*EllipticF(a/​S(2) + b*x/​S(2), S(2))*sqrt(cos(a + b*x))/​(S(77)*b*sqrt(d*cos(a + b*x))) + S(8)*d*sqrt(d*cos(a + b*x))*sin(a + b*x)/​(S(77)*b) - S(2)*(d*cos(a + b*x))**(S(5)/​2)*sin(a + b*x)**S(3)/​(S(11)*b*d) - S(12)*(d*cos(a + b*x))**(S(5)/​2)*sin(a + b*x)/​(S(77)*b*d), expand=True, _diff=True, _numerical=True)365 assert rubi_test(rubi_integrate(sqrt(d*cos(a + b*x))*sin(a + b*x)**S(4), x), x, S(8)*sqrt(d*cos(a + b*x))*EllipticE(a/​S(2) + b*x/​S(2), S(2))/​(S(15)*b*sqrt(cos(a + b*x))) - S(2)*(d*cos(a + b*x))**(S(3)/​2)*sin(a + b*x)**S(3)/​(S(9)*b*d) - S(4)*(d*cos(a + b*x))**(S(3)/​2)*sin(a + b*x)/​(S(15)*b*d), expand=True, _diff=True, _numerical=True)366 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4)/​sqrt(d*cos(a + b*x)), x), x, S(8)*EllipticF(a/​S(2) + b*x/​S(2), S(2))*sqrt(cos(a + b*x))/​(S(7)*b*sqrt(d*cos(a + b*x))) - S(2)*sqrt(d*cos(a + b*x))*sin(a + b*x)**S(3)/​(S(7)*b*d) - S(4)*sqrt(d*cos(a + b*x))*sin(a + b*x)/​(S(7)*b*d), expand=True, _diff=True, _numerical=True)367 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4)/​(d*cos(a + b*x))**(S(3)/​2), x), x, S(2)*sin(a + b*x)**S(3)/​(b*d*sqrt(d*cos(a + b*x))) - S(24)*sqrt(d*cos(a + b*x))*EllipticE(a/​S(2) + b*x/​S(2), S(2))/​(S(5)*b*d**S(2)*sqrt(cos(a + b*x))) + S(12)*(d*cos(a + b*x))**(S(3)/​2)*sin(a + b*x)/​(S(5)*b*d**S(3)), expand=True, _diff=True, _numerical=True)368 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4)/​(d*cos(a + b*x))**(S(5)/​2), x), x, S(2)*sin(a + b*x)**S(3)/​(S(3)*b*d*(d*cos(a + b*x))**(S(3)/​2)) - S(8)*EllipticF(a/​S(2) + b*x/​S(2), S(2))*sqrt(cos(a + b*x))/​(S(3)*b*d**S(2)*sqrt(d*cos(a + b*x))) + S(4)*sqrt(d*cos(a + b*x))*sin(a + b*x)/​(S(3)*b*d**S(3)), expand=True, _diff=True, _numerical=True)369 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4)/​(d*cos(a + b*x))**(S(7)/​2), x), x, S(2)*sin(a + b*x)**S(3)/​(S(5)*b*d*(d*cos(a + b*x))**(S(5)/​2)) - S(12)*sin(a + b*x)/​(S(5)*b*d**S(3)*sqrt(d*cos(a + b*x))) + S(24)*sqrt(d*cos(a + b*x))*EllipticE(a/​S(2) + b*x/​S(2), S(2))/​(S(5)*b*d**S(4)*sqrt(cos(a + b*x))), expand=True, _diff=True, _numerical=True)370 assert rubi_test(rubi_integrate(sin(a + b*x)**S(4)/​(d*cos(a + b*x))**(S(9)/​2), x), x, S(2)*sin(a + b*x)**S(3)/​(S(7)*b*d*(d*cos(a + b*x))**(S(7)/​2)) - S(4)*sin(a + b*x)/​(S(7)*b*d**S(3)*(d*cos(a + b*x))**(S(3)/​2)) + S(8)*EllipticF(a/​S(2) + b*x/​S(2), S(2))*sqrt(cos(a + b*x))/​(S(7)*b*d**S(4)*sqrt(d*cos(a + b*x))), expand=True, _diff=True, _numerical=True)371 assert rubi_test(rubi_integrate(sin(a + b*x)**S(5)*cos(a + b*x)**(S(3)/​2), x), x, -S(2)*cos(a + b*x)**(S(13)/​2)/​(S(13)*b) + S(4)*cos(a + b*x)**(S(9)/​2)/​(S(9)*b) - S(2)*cos(a + b*x)**(S(5)/​2)/​(S(5)*b), expand=True, _diff=True, _numerical=True)372 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(9)/​2)*csc(a + b*x), x), x, d**(S(9)/​2)*ArcTan(sqrt(d*cos(a + b*x))/​sqrt(d))/​b - d**(S(9)/​2)*atanh(sqrt(d*cos(a + b*x))/​sqrt(d))/​b + S(2)*d**S(3)*(d*cos(a + b*x))**(S(3)/​2)/​(S(3)*b) + S(2)*d*(d*cos(a + b*x))**(S(7)/​2)/​(S(7)*b), expand=True, _diff=True, _numerical=True)373 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(7)/​2)*csc(a + b*x), x), x, -d**(S(7)/​2)*ArcTan(sqrt(d*cos(a + b*x))/​sqrt(d))/​b - d**(S(7)/​2)*atanh(sqrt(d*cos(a + b*x))/​sqrt(d))/​b + S(2)*d**S(3)*sqrt(d*cos(a + b*x))/​b + S(2)*d*(d*cos(a + b*x))**(S(5)/​2)/​(S(5)*b), expand=True, _diff=True, _numerical=True)374 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(5)/​2)*csc(a + b*x), x), x, d**(S(5)/​2)*ArcTan(sqrt(d*cos(a + b*x))/​sqrt(d))/​b - d**(S(5)/​2)*atanh(sqrt(d*cos(a + b*x))/​sqrt(d))/​b + S(2)*d*(d*cos(a + b*x))**(S(3)/​2)/​(S(3)*b), expand=True, _diff=True, _numerical=True)375 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(3)/​2)*csc(a + b*x), x), x, -d**(S(3)/​2)*ArcTan(sqrt(d*cos(a + b*x))/​sqrt(d))/​b - d**(S(3)/​2)*atanh(sqrt(d*cos(a + b*x))/​sqrt(d))/​b + S(2)*d*sqrt(d*cos(a + b*x))/​b, expand=True, _diff=True, _numerical=True)376 assert rubi_test(rubi_integrate(sqrt(d*cos(a + b*x))*csc(a + b*x), x), x, sqrt(d)*ArcTan(sqrt(d*cos(a + b*x))/​sqrt(d))/​b - sqrt(d)*atanh(sqrt(d*cos(a + b*x))/​sqrt(d))/​b, expand=True, _diff=True, _numerical=True)377 assert rubi_test(rubi_integrate(csc(a + b*x)/​sqrt(d*cos(a + b*x)), x), x, -ArcTan(sqrt(d*cos(a + b*x))/​sqrt(d))/​(b*sqrt(d)) - atanh(sqrt(d*cos(a + b*x))/​sqrt(d))/​(b*sqrt(d)), expand=True, _diff=True, _numerical=True)378 assert rubi_test(rubi_integrate(csc(a + b*x)/​(d*cos(a + b*x))**(S(3)/​2), x), x, S(2)/​(b*d*sqrt(d*cos(a + b*x))) + ArcTan(sqrt(d*cos(a + b*x))/​sqrt(d))/​(b*d**(S(3)/​2)) - atanh(sqrt(d*cos(a + b*x))/​sqrt(d))/​(b*d**(S(3)/​2)), expand=True, _diff=True, _numerical=True)379 assert rubi_test(rubi_integrate(csc(a + b*x)/​(d*cos(a + b*x))**(S(5)/​2), x), x, S(2)/​(S(3)*b*d*(d*cos(a + b*x))**(S(3)/​2)) - ArcTan(sqrt(d*cos(a + b*x))/​sqrt(d))/​(b*d**(S(5)/​2)) - atanh(sqrt(d*cos(a + b*x))/​sqrt(d))/​(b*d**(S(5)/​2)), expand=True, _diff=True, _numerical=True)380 assert rubi_test(rubi_integrate(csc(a + b*x)/​(d*cos(a + b*x))**(S(7)/​2), x), x, S(2)/​(S(5)*b*d*(d*cos(a + b*x))**(S(5)/​2)) + S(2)/​(b*d**S(3)*sqrt(d*cos(a + b*x))) + ArcTan(sqrt(d*cos(a + b*x))/​sqrt(d))/​(b*d**(S(7)/​2)) - atanh(sqrt(d*cos(a + b*x))/​sqrt(d))/​(b*d**(S(7)/​2)), expand=True, _diff=True, _numerical=True)381 assert rubi_test(rubi_integrate(csc(a + b*x)/​(d*cos(a + b*x))**(S(9)/​2), x), x, S(2)/​(S(7)*b*d*(d*cos(a + b*x))**(S(7)/​2)) + S(2)/​(S(3)*b*d**S(3)*(d*cos(a + b*x))**(S(3)/​2)) - ArcTan(sqrt(d*cos(a + b*x))/​sqrt(d))/​(b*d**(S(9)/​2)) - atanh(sqrt(d*cos(a + b*x))/​sqrt(d))/​(b*d**(S(9)/​2)), expand=True, _diff=True, _numerical=True)382 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(11)/​2)*csc(a + b*x)**S(2), x), x, -S(15)*d**S(6)*EllipticF(a/​S(2) + b*x/​S(2), S(2))*sqrt(cos(a + b*x))/​(S(7)*b*sqrt(d*cos(a + b*x))) - S(15)*d**S(5)*sqrt(d*cos(a + b*x))*sin(a + b*x)/​(S(7)*b) - S(9)*d**S(3)*(d*cos(a + b*x))**(S(5)/​2)*sin(a + b*x)/​(S(7)*b) - d*(d*cos(a + b*x))**(S(9)/​2)*csc(a + b*x)/​b, expand=True, _diff=True, _numerical=True)383 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(9)/​2)*csc(a + b*x)**S(2), x), x, -S(21)*d**S(4)*sqrt(d*cos(a + b*x))*EllipticE(a/​S(2) + b*x/​S(2), S(2))/​(S(5)*b*sqrt(cos(a + b*x))) - S(7)*d**S(3)*(d*cos(a + b*x))**(S(3)/​2)*sin(a + b*x)/​(S(5)*b) - d*(d*cos(a + b*x))**(S(7)/​2)*csc(a + b*x)/​b, expand=True, _diff=True, _numerical=True)384 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(7)/​2)*csc(a + b*x)**S(2), x), x, -S(5)*d**S(4)*EllipticF(a/​S(2) + b*x/​S(2), S(2))*sqrt(cos(a + b*x))/​(S(3)*b*sqrt(d*cos(a + b*x))) - S(5)*d**S(3)*sqrt(d*cos(a + b*x))*sin(a + b*x)/​(S(3)*b) - d*(d*cos(a + b*x))**(S(5)/​2)*csc(a + b*x)/​b, expand=True, _diff=True, _numerical=True)385 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(5)/​2)*csc(a + b*x)**S(2), x), x, -S(3)*d**S(2)*sqrt(d*cos(a + b*x))*EllipticE(a/​S(2) + b*x/​S(2), S(2))/​(b*sqrt(cos(a + b*x))) - d*(d*cos(a + b*x))**(S(3)/​2)*csc(a + b*x)/​b, expand=True, _diff=True, _numerical=True)386 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(3)/​2)*csc(a + b*x)**S(2), x), x, -d**S(2)*EllipticF(a/​S(2) + b*x/​S(2), S(2))*sqrt(cos(a + b*x))/​(b*sqrt(d*cos(a + b*x))) - d*sqrt(d*cos(a + b*x))*csc(a + b*x)/​b, expand=True, _diff=True, _numerical=True)387 assert rubi_test(rubi_integrate(sqrt(d*cos(a + b*x))*csc(a + b*x)**S(2), x), x, -sqrt(d*cos(a + b*x))*EllipticE(a/​S(2) + b*x/​S(2), S(2))/​(b*sqrt(cos(a + b*x))) - (d*cos(a + b*x))**(S(3)/​2)*csc(a + b*x)/​(b*d), expand=True, _diff=True, _numerical=True)388 assert rubi_test(rubi_integrate(csc(a + b*x)**S(2)/​sqrt(d*cos(a + b*x)), x), x, EllipticF(a/​S(2) + b*x/​S(2), S(2))*sqrt(cos(a + b*x))/​(b*sqrt(d*cos(a + b*x))) - sqrt(d*cos(a + b*x))*csc(a + b*x)/​(b*d), expand=True, _diff=True, _numerical=True)389 assert rubi_test(rubi_integrate(csc(a + b*x)**S(2)/​(d*cos(a + b*x))**(S(3)/​2), x), x, S(3)*sin(a + b*x)/​(b*d*sqrt(d*cos(a + b*x))) - csc(a + b*x)/​(b*d*sqrt(d*cos(a + b*x))) - S(3)*sqrt(d*cos(a + b*x))*EllipticE(a/​S(2) + b*x/​S(2), S(2))/​(b*d**S(2)*sqrt(cos(a + b*x))), expand=True, _diff=True, _numerical=True)390 assert rubi_test(rubi_integrate(csc(a + b*x)**S(2)/​(d*cos(a + b*x))**(S(5)/​2), x), x, S(5)*sin(a + b*x)/​(S(3)*b*d*(d*cos(a + b*x))**(S(3)/​2)) - csc(a + b*x)/​(b*d*(d*cos(a + b*x))**(S(3)/​2)) + S(5)*EllipticF(a/​S(2) + b*x/​S(2), S(2))*sqrt(cos(a + b*x))/​(S(3)*b*d**S(2)*sqrt(d*cos(a + b*x))), expand=True, _diff=True, _numerical=True)391 assert rubi_test(rubi_integrate(csc(a + b*x)**S(2)/​(d*cos(a + b*x))**(S(7)/​2), x), x, S(7)*sin(a + b*x)/​(S(5)*b*d*(d*cos(a + b*x))**(S(5)/​2)) - csc(a + b*x)/​(b*d*(d*cos(a + b*x))**(S(5)/​2)) + S(21)*sin(a + b*x)/​(S(5)*b*d**S(3)*sqrt(d*cos(a + b*x))) - S(21)*sqrt(d*cos(a + b*x))*EllipticE(a/​S(2) + b*x/​S(2), S(2))/​(S(5)*b*d**S(4)*sqrt(cos(a + b*x))), expand=True, _diff=True, _numerical=True)392 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(11)/​2)*csc(a + b*x)**S(3), x), x, S(9)*d**(S(11)/​2)*ArcTan(sqrt(d*cos(a + b*x))/​sqrt(d))/​(S(4)*b) + S(9)*d**(S(11)/​2)*atanh(sqrt(d*cos(a + b*x))/​sqrt(d))/​(S(4)*b) - S(9)*d**S(5)*sqrt(d*cos(a + b*x))/​(S(2)*b) - S(9)*d**S(3)*(d*cos(a + b*x))**(S(5)/​2)/​(S(10)*b) - d*(d*cos(a + b*x))**(S(9)/​2)*csc(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)393 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(9)/​2)*csc(a + b*x)**S(3), x), x, -S(7)*d**(S(9)/​2)*ArcTan(sqrt(d*cos(a + b*x))/​sqrt(d))/​(S(4)*b) + S(7)*d**(S(9)/​2)*atanh(sqrt(d*cos(a + b*x))/​sqrt(d))/​(S(4)*b) - S(7)*d**S(3)*(d*cos(a + b*x))**(S(3)/​2)/​(S(6)*b) - d*(d*cos(a + b*x))**(S(7)/​2)*csc(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)394 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(7)/​2)*csc(a + b*x)**S(3), x), x, S(5)*d**(S(7)/​2)*ArcTan(sqrt(d*cos(a + b*x))/​sqrt(d))/​(S(4)*b) + S(5)*d**(S(7)/​2)*atanh(sqrt(d*cos(a + b*x))/​sqrt(d))/​(S(4)*b) - S(5)*d**S(3)*sqrt(d*cos(a + b*x))/​(S(2)*b) - d*(d*cos(a + b*x))**(S(5)/​2)*csc(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)395 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(5)/​2)*csc(a + b*x)**S(3), x), x, -S(3)*d**(S(5)/​2)*ArcTan(sqrt(d*cos(a + b*x))/​sqrt(d))/​(S(4)*b) + S(3)*d**(S(5)/​2)*atanh(sqrt(d*cos(a + b*x))/​sqrt(d))/​(S(4)*b) - d*(d*cos(a + b*x))**(S(3)/​2)*csc(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)396 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(3)/​2)*csc(a + b*x)**S(3), x), x, d**(S(3)/​2)*ArcTan(sqrt(d*cos(a + b*x))/​sqrt(d))/​(S(4)*b) + d**(S(3)/​2)*atanh(sqrt(d*cos(a + b*x))/​sqrt(d))/​(S(4)*b) - d*sqrt(d*cos(a + b*x))*csc(a + b*x)**S(2)/​(S(2)*b), expand=True, _diff=True, _numerical=True)397 assert rubi_test(rubi_integrate(sqrt(d*cos(a + b*x))*csc(a + b*x)**S(3), x), x, sqrt(d)*ArcTan(sqrt(d*cos(a + b*x))/​sqrt(d))/​(S(4)*b) - sqrt(d)*atanh(sqrt(d*cos(a + b*x))/​sqrt(d))/​(S(4)*b) - (d*cos(a + b*x))**(S(3)/​2)*csc(a + b*x)**S(2)/​(S(2)*b*d), expand=True, _diff=True, _numerical=True)398 assert rubi_test(rubi_integrate(csc(a + b*x)**S(3)/​sqrt(d*cos(a + b*x)), x), x, -sqrt(d*cos(a + b*x))*csc(a + b*x)**S(2)/​(S(2)*b*d) - S(3)*ArcTan(sqrt(d*cos(a + b*x))/​sqrt(d))/​(S(4)*b*sqrt(d)) - S(3)*atanh(sqrt(d*cos(a + b*x))/​sqrt(d))/​(S(4)*b*sqrt(d)), expand=True, _diff=True, _numerical=True)399 assert rubi_test(rubi_integrate(csc(a + b*x)**S(3)/​(d*cos(a + b*x))**(S(3)/​2), x), x, -csc(a + b*x)**S(2)/​(S(2)*b*d*sqrt(d*cos(a + b*x))) + S(5)/​(S(2)*b*d*sqrt(d*cos(a + b*x))) + S(5)*ArcTan(sqrt(d*cos(a + b*x))/​sqrt(d))/​(S(4)*b*d**(S(3)/​2)) - S(5)*atanh(sqrt(d*cos(a + b*x))/​sqrt(d))/​(S(4)*b*d**(S(3)/​2)), expand=True, _diff=True, _numerical=True)400 assert rubi_test(rubi_integrate(csc(a + b*x)**S(3)/​(d*cos(a + b*x))**(S(5)/​2), x), x, -csc(a + b*x)**S(2)/​(S(2)*b*d*(d*cos(a + b*x))**(S(3)/​2)) + S(7)/​(S(6)*b*d*(d*cos(a + b*x))**(S(3)/​2)) - S(7)*ArcTan(sqrt(d*cos(a + b*x))/​sqrt(d))/​(S(4)*b*d**(S(5)/​2)) - S(7)*atanh(sqrt(d*cos(a + b*x))/​sqrt(d))/​(S(4)*b*d**(S(5)/​2)), expand=True, _diff=True, _numerical=True)401 assert rubi_test(rubi_integrate(csc(a + b*x)**S(3)/​(d*cos(a + b*x))**(S(7)/​2), x), x, -csc(a + b*x)**S(2)/​(S(2)*b*d*(d*cos(a + b*x))**(S(5)/​2)) + S(9)/​(S(10)*b*d*(d*cos(a + b*x))**(S(5)/​2)) + S(9)/​(S(2)*b*d**S(3)*sqrt(d*cos(a + b*x))) + S(9)*ArcTan(sqrt(d*cos(a + b*x))/​sqrt(d))/​(S(4)*b*d**(S(7)/​2)) - S(9)*atanh(sqrt(d*cos(a + b*x))/​sqrt(d))/​(S(4)*b*d**(S(7)/​2)), expand=True, _diff=True, _numerical=True)402 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(1)/​5)*sin(a + b*x), x), x, -S(5)*(d*cos(a + b*x))**(S(6)/​5)/​(S(6)*b*d), expand=True, _diff=True, _numerical=True)403 assert rubi_test(rubi_integrate(sqrt(sin(x))*cos(x)**S(3), x), x, -S(2)*sin(x)**(S(7)/​2)/​S(7) + S(2)*sin(x)**(S(3)/​2)/​S(3), expand=True, _diff=True, _numerical=True)404 assert rubi_test(rubi_integrate(sin(x)**(S(3)/​2)*cos(x)**S(3), x), x, -S(2)*sin(x)**(S(9)/​2)/​S(9) + S(2)*sin(x)**(S(5)/​2)/​S(5), expand=True, _diff=True, _numerical=True)405 assert rubi_test(rubi_integrate(sin(x)**(S(5)/​2)*cos(x)**S(3), x), x, -S(2)*sin(x)**(S(11)/​2)/​S(11) + S(2)*sin(x)**(S(7)/​2)/​S(7), expand=True, _diff=True, _numerical=True)406 assert rubi_test(rubi_integrate(cos(x)**S(3)/​sqrt(sin(x)), x), x, -S(2)*sin(x)**(S(5)/​2)/​S(5) + S(2)*sqrt(sin(x)), expand=True, _diff=True, _numerical=True)407 assert rubi_test(rubi_integrate(sqrt(c*sin(a + b*x))*(d*cos(a + b*x))**(S(9)/​2), x), x, S(7)*d**S(4)*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))*EllipticE(-Pi/​S(4) + a + b*x, S(2))/​(S(20)*b*sqrt(sin(S(2)*a + S(2)*b*x))) + S(7)*d**S(3)*(c*sin(a + b*x))**(S(3)/​2)*(d*cos(a + b*x))**(S(3)/​2)/​(S(30)*b*c) + d*(c*sin(a + b*x))**(S(3)/​2)*(d*cos(a + b*x))**(S(7)/​2)/​(S(5)*b*c), expand=True, _diff=True, _numerical=True)408 assert rubi_test(rubi_integrate(sqrt(c*sin(a + b*x))*(d*cos(a + b*x))**(S(5)/​2), x), x, d**S(2)*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))*EllipticE(-Pi/​S(4) + a + b*x, S(2))/​(S(2)*b*sqrt(sin(S(2)*a + S(2)*b*x))) + d*(c*sin(a + b*x))**(S(3)/​2)*(d*cos(a + b*x))**(S(3)/​2)/​(S(3)*b*c), expand=True, _diff=True, _numerical=True)409 assert rubi_test(rubi_integrate(sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x)), x), x, sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))*EllipticE(-Pi/​S(4) + a + b*x, S(2))/​(b*sqrt(sin(S(2)*a + S(2)*b*x))), expand=True, _diff=True, _numerical=True)410 assert rubi_test(rubi_integrate(sqrt(c*sin(a + b*x))/​(d*cos(a + b*x))**(S(3)/​2), x), x, -S(2)*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))*EllipticE(-Pi/​S(4) + a + b*x, S(2))/​(b*d**S(2)*sqrt(sin(S(2)*a + S(2)*b*x))) + S(2)*(c*sin(a + b*x))**(S(3)/​2)/​(b*c*d*sqrt(d*cos(a + b*x))), expand=True, _diff=True, _numerical=True)411 assert rubi_test(rubi_integrate(sqrt(c*sin(a + b*x))/​(d*cos(a + b*x))**(S(7)/​2), x), x, -S(4)*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))*EllipticE(-Pi/​S(4) + a + b*x, S(2))/​(S(5)*b*d**S(4)*sqrt(sin(S(2)*a + S(2)*b*x))) + S(2)*(c*sin(a + b*x))**(S(3)/​2)/​(S(5)*b*c*d*(d*cos(a + b*x))**(S(5)/​2)) + S(4)*(c*sin(a + b*x))**(S(3)/​2)/​(S(5)*b*c*d**S(3)*sqrt(d*cos(a + b*x))), expand=True, _diff=True, _numerical=True)412 assert rubi_test(rubi_integrate(sqrt(c*sin(a + b*x))*(d*cos(a + b*x))**(S(3)/​2), x), x, -sqrt(S(2))*sqrt(c)*d**(S(3)/​2)*ArcTan(S(1) - sqrt(S(2))*sqrt(d)*sqrt(c*sin(a + b*x))/​(sqrt(c)*sqrt(d*cos(a + b*x))))/​(S(8)*b) + sqrt(S(2))*sqrt(c)*d**(S(3)/​2)*ArcTan(S(1) + sqrt(S(2))*sqrt(d)*sqrt(c*sin(a + b*x))/​(sqrt(c)*sqrt(d*cos(a + b*x))))/​(S(8)*b) + sqrt(S(2))*sqrt(c)*d**(S(3)/​2)*log(sqrt(c)*tan(a + b*x) + sqrt(c) - sqrt(S(2))*sqrt(d)*sqrt(c*sin(a + b*x))/​sqrt(d*cos(a + b*x)))/​(S(16)*b) - sqrt(S(2))*sqrt(c)*d**(S(3)/​2)*log(sqrt(c)*tan(a + b*x) + sqrt(c) + sqrt(S(2))*sqrt(d)*sqrt(c*sin(a + b*x))/​sqrt(d*cos(a + b*x)))/​(S(16)*b) + d*(c*sin(a + b*x))**(S(3)/​2)*sqrt(d*cos(a + b*x))/​(S(2)*b*c), expand=True, _diff=True, _numerical=True)413 assert rubi_test(rubi_integrate(sqrt(c*sin(a + b*x))/​sqrt(d*cos(a + b*x)), x), x, -sqrt(S(2))*sqrt(c)*ArcTan(S(1) - sqrt(S(2))*sqrt(d)*sqrt(c*sin(a + b*x))/​(sqrt(c)*sqrt(d*cos(a + b*x))))/​(S(2)*b*sqrt(d)) + sqrt(S(2))*sqrt(c)*ArcTan(S(1) + sqrt(S(2))*sqrt(d)*sqrt(c*sin(a + b*x))/​(sqrt(c)*sqrt(d*cos(a + b*x))))/​(S(2)*b*sqrt(d)) + sqrt(S(2))*sqrt(c)*log(sqrt(c)*tan(a + b*x) + sqrt(c) - sqrt(S(2))*sqrt(d)*sqrt(c*sin(a + b*x))/​sqrt(d*cos(a + b*x)))/​(S(4)*b*sqrt(d)) - sqrt(S(2))*sqrt(c)*log(sqrt(c)*tan(a + b*x) + sqrt(c) + sqrt(S(2))*sqrt(d)*sqrt(c*sin(a + b*x))/​sqrt(d*cos(a + b*x)))/​(S(4)*b*sqrt(d)), expand=True, _diff=True, _numerical=True)414 assert rubi_test(rubi_integrate(sqrt(c*sin(a + b*x))/​(d*cos(a + b*x))**(S(5)/​2), x), x, S(2)*(c*sin(a + b*x))**(S(3)/​2)/​(S(3)*b*c*d*(d*cos(a + b*x))**(S(3)/​2)), expand=True, _diff=True, _numerical=True)415 assert rubi_test(rubi_integrate(sqrt(c*sin(a + b*x))/​(d*cos(a + b*x))**(S(9)/​2), x), x, S(2)*(c*sin(a + b*x))**(S(3)/​2)/​(S(7)*b*c*d*(d*cos(a + b*x))**(S(7)/​2)) + S(8)*(c*sin(a + b*x))**(S(3)/​2)/​(S(21)*b*c*d**S(3)*(d*cos(a + b*x))**(S(3)/​2)), expand=True, _diff=True, _numerical=True)416 assert rubi_test(rubi_integrate(sqrt(c*sin(a + b*x))/​(d*cos(a + b*x))**(S(13)/​2), x), x, S(2)*(c*sin(a + b*x))**(S(3)/​2)/​(S(11)*b*c*d*(d*cos(a + b*x))**(S(11)/​2)) + S(16)*(c*sin(a + b*x))**(S(3)/​2)/​(S(77)*b*c*d**S(3)*(d*cos(a + b*x))**(S(7)/​2)) + S(64)*(c*sin(a + b*x))**(S(3)/​2)/​(S(231)*b*c*d**S(5)*(d*cos(a + b*x))**(S(3)/​2)), expand=True, _diff=True, _numerical=True)417 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(3)/​2)*(d*cos(a + b*x))**(S(3)/​2), x), x, c**S(2)*d**S(2)*EllipticF(-Pi/​S(4) + a + b*x, S(2))*sqrt(sin(S(2)*a + S(2)*b*x))/​(S(12)*b*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))) + c*d*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))/​(S(6)*b) - c*sqrt(c*sin(a + b*x))*(d*cos(a + b*x))**(S(5)/​2)/​(S(3)*b*d), expand=True, _diff=True, _numerical=True)418 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(3)/​2)/​sqrt(d*cos(a + b*x)), x), x, c**S(2)*EllipticF(-Pi/​S(4) + a + b*x, S(2))*sqrt(sin(S(2)*a + S(2)*b*x))/​(S(2)*b*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))) - c*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))/​(b*d), expand=True, _diff=True, _numerical=True)419 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(3)/​2)/​(d*cos(a + b*x))**(S(5)/​2), x), x, -c**S(2)*EllipticF(-Pi/​S(4) + a + b*x, S(2))*sqrt(sin(S(2)*a + S(2)*b*x))/​(S(3)*b*d**S(2)*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))) + S(2)*c*sqrt(c*sin(a + b*x))/​(S(3)*b*d*(d*cos(a + b*x))**(S(3)/​2)), expand=True, _diff=True, _numerical=True)420 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(3)/​2)/​(d*cos(a + b*x))**(S(9)/​2), x), x, -S(2)*c**S(2)*EllipticF(-Pi/​S(4) + a + b*x, S(2))*sqrt(sin(S(2)*a + S(2)*b*x))/​(S(21)*b*d**S(4)*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))) + S(2)*c*sqrt(c*sin(a + b*x))/​(S(7)*b*d*(d*cos(a + b*x))**(S(7)/​2)) - S(2)*c*sqrt(c*sin(a + b*x))/​(S(21)*b*d**S(3)*(d*cos(a + b*x))**(S(3)/​2)), expand=True, _diff=True, _numerical=True)421 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(3)/​2)*sqrt(d*cos(a + b*x)), x), x, sqrt(S(2))*c**(S(3)/​2)*sqrt(d)*ArcTan(-sqrt(S(2))*sqrt(c)*sqrt(d*cos(a + b*x))/​(sqrt(d)*sqrt(c*sin(a + b*x))) + S(1))/​(S(8)*b) - sqrt(S(2))*c**(S(3)/​2)*sqrt(d)*ArcTan(sqrt(S(2))*sqrt(c)*sqrt(d*cos(a + b*x))/​(sqrt(d)*sqrt(c*sin(a + b*x))) + S(1))/​(S(8)*b) - sqrt(S(2))*c**(S(3)/​2)*sqrt(d)*log(-sqrt(S(2))*sqrt(c)*sqrt(d*cos(a + b*x))/​sqrt(c*sin(a + b*x)) + sqrt(d)*cot(a + b*x) + sqrt(d))/​(S(16)*b) + sqrt(S(2))*c**(S(3)/​2)*sqrt(d)*log(sqrt(S(2))*sqrt(c)*sqrt(d*cos(a + b*x))/​sqrt(c*sin(a + b*x)) + sqrt(d)*cot(a + b*x) + sqrt(d))/​(S(16)*b) - c*sqrt(c*sin(a + b*x))*(d*cos(a + b*x))**(S(3)/​2)/​(S(2)*b*d), expand=True, _diff=True, _numerical=True)422 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(3)/​2)/​(d*cos(a + b*x))**(S(3)/​2), x), x, -sqrt(S(2))*c**(S(3)/​2)*ArcTan(-sqrt(S(2))*sqrt(c)*sqrt(d*cos(a + b*x))/​(sqrt(d)*sqrt(c*sin(a + b*x))) + S(1))/​(S(2)*b*d**(S(3)/​2)) + sqrt(S(2))*c**(S(3)/​2)*ArcTan(sqrt(S(2))*sqrt(c)*sqrt(d*cos(a + b*x))/​(sqrt(d)*sqrt(c*sin(a + b*x))) + S(1))/​(S(2)*b*d**(S(3)/​2)) + sqrt(S(2))*c**(S(3)/​2)*log(-sqrt(S(2))*sqrt(c)*sqrt(d*cos(a + b*x))/​sqrt(c*sin(a + b*x)) + sqrt(d)*cot(a + b*x) + sqrt(d))/​(S(4)*b*d**(S(3)/​2)) - sqrt(S(2))*c**(S(3)/​2)*log(sqrt(S(2))*sqrt(c)*sqrt(d*cos(a + b*x))/​sqrt(c*sin(a + b*x)) + sqrt(d)*cot(a + b*x) + sqrt(d))/​(S(4)*b*d**(S(3)/​2)) + S(2)*c*sqrt(c*sin(a + b*x))/​(b*d*sqrt(d*cos(a + b*x))), expand=True, _diff=True, _numerical=True)423 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(3)/​2)/​(d*cos(a + b*x))**(S(7)/​2), x), x, S(2)*(c*sin(a + b*x))**(S(5)/​2)/​(S(5)*b*c*d*(d*cos(a + b*x))**(S(5)/​2)), expand=True, _diff=True, _numerical=True)424 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(3)/​2)/​(d*cos(a + b*x))**(S(11)/​2), x), x, S(2)*c*sqrt(c*sin(a + b*x))/​(S(9)*b*d*(d*cos(a + b*x))**(S(9)/​2)) - S(2)*c*sqrt(c*sin(a + b*x))/​(S(45)*b*d**S(3)*(d*cos(a + b*x))**(S(5)/​2)) - S(8)*c*sqrt(c*sin(a + b*x))/​(S(45)*b*d**S(5)*sqrt(d*cos(a + b*x))), expand=True, _diff=True, _numerical=True)425 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(3)/​2)/​(d*cos(a + b*x))**(S(15)/​2), x), x, S(2)*c*sqrt(c*sin(a + b*x))/​(S(13)*b*d*(d*cos(a + b*x))**(S(13)/​2)) - S(2)*c*sqrt(c*sin(a + b*x))/​(S(117)*b*d**S(3)*(d*cos(a + b*x))**(S(9)/​2)) - S(16)*c*sqrt(c*sin(a + b*x))/​(S(585)*b*d**S(5)*(d*cos(a + b*x))**(S(5)/​2)) - S(64)*c*sqrt(c*sin(a + b*x))/​(S(585)*b*d**S(7)*sqrt(d*cos(a + b*x))), expand=True, _diff=True, _numerical=True)426 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(5)/​2)*(d*cos(a + b*x))**(S(9)/​2), x), x, S(3)*c**S(2)*d**S(4)*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))*EllipticE(-Pi/​S(4) + a + b*x, S(2))/​(S(40)*b*sqrt(sin(S(2)*a + S(2)*b*x))) + c*d**S(3)*(c*sin(a + b*x))**(S(3)/​2)*(d*cos(a + b*x))**(S(3)/​2)/​(S(20)*b) + S(3)*c*d*(c*sin(a + b*x))**(S(3)/​2)*(d*cos(a + b*x))**(S(7)/​2)/​(S(70)*b) - c*(c*sin(a + b*x))**(S(3)/​2)*(d*cos(a + b*x))**(S(11)/​2)/​(S(7)*b*d), expand=True, _diff=True, _numerical=True)427 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(5)/​2)*(d*cos(a + b*x))**(S(5)/​2), x), x, S(3)*c**S(2)*d**S(2)*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))*EllipticE(-Pi/​S(4) + a + b*x, S(2))/​(S(20)*b*sqrt(sin(S(2)*a + S(2)*b*x))) + c*d*(c*sin(a + b*x))**(S(3)/​2)*(d*cos(a + b*x))**(S(3)/​2)/​(S(10)*b) - c*(c*sin(a + b*x))**(S(3)/​2)*(d*cos(a + b*x))**(S(7)/​2)/​(S(5)*b*d), expand=True, _diff=True, _numerical=True)428 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(5)/​2)*sqrt(d*cos(a + b*x)), x), x, c**S(2)*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))*EllipticE(-Pi/​S(4) + a + b*x, S(2))/​(S(2)*b*sqrt(sin(S(2)*a + S(2)*b*x))) - c*(c*sin(a + b*x))**(S(3)/​2)*(d*cos(a + b*x))**(S(3)/​2)/​(S(3)*b*d), expand=True, _diff=True, _numerical=True)429 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(5)/​2)/​(d*cos(a + b*x))**(S(3)/​2), x), x, -S(3)*c**S(2)*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))*EllipticE(-Pi/​S(4) + a + b*x, S(2))/​(b*d**S(2)*sqrt(sin(S(2)*a + S(2)*b*x))) + S(2)*c*(c*sin(a + b*x))**(S(3)/​2)/​(b*d*sqrt(d*cos(a + b*x))), expand=True, _diff=True, _numerical=True)430 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(5)/​2)/​(d*cos(a + b*x))**(S(7)/​2), x), x, S(6)*c**S(2)*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))*EllipticE(-Pi/​S(4) + a + b*x, S(2))/​(S(5)*b*d**S(4)*sqrt(sin(S(2)*a + S(2)*b*x))) + S(2)*c*(c*sin(a + b*x))**(S(3)/​2)/​(S(5)*b*d*(d*cos(a + b*x))**(S(5)/​2)) - S(6)*c*(c*sin(a + b*x))**(S(3)/​2)/​(S(5)*b*d**S(3)*sqrt(d*cos(a + b*x))), expand=True, _diff=True, _numerical=True)431 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(5)/​2)/​(d*cos(a + b*x))**(S(11)/​2), x), x, S(4)*c**S(2)*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))*EllipticE(-Pi/​S(4) + a + b*x, S(2))/​(S(15)*b*d**S(6)*sqrt(sin(S(2)*a + S(2)*b*x))) + S(2)*c*(c*sin(a + b*x))**(S(3)/​2)/​(S(9)*b*d*(d*cos(a + b*x))**(S(9)/​2)) - S(2)*c*(c*sin(a + b*x))**(S(3)/​2)/​(S(15)*b*d**S(3)*(d*cos(a + b*x))**(S(5)/​2)) - S(4)*c*(c*sin(a + b*x))**(S(3)/​2)/​(S(15)*b*d**S(5)*sqrt(d*cos(a + b*x))), expand=True, _diff=True, _numerical=True)432 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(5)/​2)/​sqrt(d*cos(a + b*x)), x), x, -S(3)*sqrt(S(2))*c**(S(5)/​2)*ArcTan(S(1) - sqrt(S(2))*sqrt(d)*sqrt(c*sin(a + b*x))/​(sqrt(c)*sqrt(d*cos(a + b*x))))/​(S(8)*b*sqrt(d)) + S(3)*sqrt(S(2))*c**(S(5)/​2)*ArcTan(S(1) + sqrt(S(2))*sqrt(d)*sqrt(c*sin(a + b*x))/​(sqrt(c)*sqrt(d*cos(a + b*x))))/​(S(8)*b*sqrt(d)) + S(3)*sqrt(S(2))*c**(S(5)/​2)*log(sqrt(c)*tan(a + b*x) + sqrt(c) - sqrt(S(2))*sqrt(d)*sqrt(c*sin(a + b*x))/​sqrt(d*cos(a + b*x)))/​(S(16)*b*sqrt(d)) - S(3)*sqrt(S(2))*c**(S(5)/​2)*log(sqrt(c)*tan(a + b*x) + sqrt(c) + sqrt(S(2))*sqrt(d)*sqrt(c*sin(a + b*x))/​sqrt(d*cos(a + b*x)))/​(S(16)*b*sqrt(d)) - c*(c*sin(a + b*x))**(S(3)/​2)*sqrt(d*cos(a + b*x))/​(S(2)*b*d), expand=True, _diff=True, _numerical=True)433 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(5)/​2)/​(d*cos(a + b*x))**(S(5)/​2), x), x, sqrt(S(2))*c**(S(5)/​2)*ArcTan(S(1) - sqrt(S(2))*sqrt(d)*sqrt(c*sin(a + b*x))/​(sqrt(c)*sqrt(d*cos(a + b*x))))/​(S(2)*b*d**(S(5)/​2)) - sqrt(S(2))*c**(S(5)/​2)*ArcTan(S(1) + sqrt(S(2))*sqrt(d)*sqrt(c*sin(a + b*x))/​(sqrt(c)*sqrt(d*cos(a + b*x))))/​(S(2)*b*d**(S(5)/​2)) - sqrt(S(2))*c**(S(5)/​2)*log(sqrt(c)*tan(a + b*x) + sqrt(c) - sqrt(S(2))*sqrt(d)*sqrt(c*sin(a + b*x))/​sqrt(d*cos(a + b*x)))/​(S(4)*b*d**(S(5)/​2)) + sqrt(S(2))*c**(S(5)/​2)*log(sqrt(c)*tan(a + b*x) + sqrt(c) + sqrt(S(2))*sqrt(d)*sqrt(c*sin(a + b*x))/​sqrt(d*cos(a + b*x)))/​(S(4)*b*d**(S(5)/​2)) + S(2)*c*(c*sin(a + b*x))**(S(3)/​2)/​(S(3)*b*d*(d*cos(a + b*x))**(S(3)/​2)), expand=True, _diff=True, _numerical=True)434 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(5)/​2)/​(d*cos(a + b*x))**(S(9)/​2), x), x, S(2)*(c*sin(a + b*x))**(S(7)/​2)/​(S(7)*b*c*d*(d*cos(a + b*x))**(S(7)/​2)), expand=True, _diff=True, _numerical=True)435 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(5)/​2)/​(d*cos(a + b*x))**(S(13)/​2), x), x, S(2)*c*(c*sin(a + b*x))**(S(3)/​2)/​(S(11)*b*d*(d*cos(a + b*x))**(S(11)/​2)) - S(6)*c*(c*sin(a + b*x))**(S(3)/​2)/​(S(77)*b*d**S(3)*(d*cos(a + b*x))**(S(7)/​2)) - S(8)*c*(c*sin(a + b*x))**(S(3)/​2)/​(S(77)*b*d**S(5)*(d*cos(a + b*x))**(S(3)/​2)), expand=True, _diff=True, _numerical=True)436 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(5)/​2)/​(d*cos(a + b*x))**(S(17)/​2), x), x, S(2)*c*(c*sin(a + b*x))**(S(3)/​2)/​(S(15)*b*d*(d*cos(a + b*x))**(S(15)/​2)) - S(2)*c*(c*sin(a + b*x))**(S(3)/​2)/​(S(55)*b*d**S(3)*(d*cos(a + b*x))**(S(11)/​2)) - S(16)*c*(c*sin(a + b*x))**(S(3)/​2)/​(S(385)*b*d**S(5)*(d*cos(a + b*x))**(S(7)/​2)) - S(64)*c*(c*sin(a + b*x))**(S(3)/​2)/​(S(1155)*b*d**S(7)*(d*cos(a + b*x))**(S(3)/​2)), expand=True, _diff=True, _numerical=True)437 assert rubi_test(rubi_integrate(sin(a + b*x)**(S(7)/​2)/​cos(a + b*x)**(S(7)/​2), x), x, sqrt(S(2))*ArcTan(S(1) - sqrt(S(2))*sqrt(cos(a + b*x))/​sqrt(sin(a + b*x)))/​(S(2)*b) - sqrt(S(2))*ArcTan(S(1) + sqrt(S(2))*sqrt(cos(a + b*x))/​sqrt(sin(a + b*x)))/​(S(2)*b) - sqrt(S(2))*log(cot(a + b*x) + S(1) - sqrt(S(2))*sqrt(cos(a + b*x))/​sqrt(sin(a + b*x)))/​(S(4)*b) + sqrt(S(2))*log(cot(a + b*x) + S(1) + sqrt(S(2))*sqrt(cos(a + b*x))/​sqrt(sin(a + b*x)))/​(S(4)*b) + S(2)*sin(a + b*x)**(S(5)/​2)/​(S(5)*b*cos(a + b*x)**(S(5)/​2)) - S(2)*sqrt(sin(a + b*x))/​(b*sqrt(cos(a + b*x))), expand=True, _diff=True, _numerical=True)438 assert rubi_test(rubi_integrate(sin(x)**(S(3)/​2)/​cos(x)**(S(7)/​2), x), x, S(2)*sin(x)**(S(5)/​2)/​(S(5)*cos(x)**(S(5)/​2)), expand=True, _diff=True, _numerical=True)439 assert rubi_test(rubi_integrate(sqrt(sin(x))/​sqrt(cos(x)), x), x, -sqrt(S(2))*ArcTan(-sqrt(S(2))*sqrt(sin(x))/​sqrt(cos(x)) + S(1))/​S(2) + sqrt(S(2))*ArcTan(sqrt(S(2))*sqrt(sin(x))/​sqrt(cos(x)) + S(1))/​S(2) + sqrt(S(2))*log(-sqrt(S(2))*sqrt(sin(x))/​sqrt(cos(x)) + tan(x) + S(1))/​S(4) - sqrt(S(2))*log(sqrt(S(2))*sqrt(sin(x))/​sqrt(cos(x)) + tan(x) + S(1))/​S(4), expand=True, _diff=True, _numerical=True)440 assert rubi_test(rubi_integrate(sin(x)**(S(5)/​2)/​sqrt(cos(x)), x), x, -S(3)*sqrt(S(2))*ArcTan(-sqrt(S(2))*sqrt(sin(x))/​sqrt(cos(x)) + S(1))/​S(8) + S(3)*sqrt(S(2))*ArcTan(sqrt(S(2))*sqrt(sin(x))/​sqrt(cos(x)) + S(1))/​S(8) + S(3)*sqrt(S(2))*log(-sqrt(S(2))*sqrt(sin(x))/​sqrt(cos(x)) + tan(x) + S(1))/​S(16) - S(3)*sqrt(S(2))*log(sqrt(S(2))*sqrt(sin(x))/​sqrt(cos(x)) + tan(x) + S(1))/​S(16) - sin(x)**(S(3)/​2)*sqrt(cos(x))/​S(2), expand=True, _diff=True, _numerical=True)441 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(7)/​2)/​sqrt(c*sin(a + b*x)), x), x, S(5)*d**S(4)*EllipticF(-Pi/​S(4) + a + b*x, S(2))*sqrt(sin(S(2)*a + S(2)*b*x))/​(S(12)*b*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))) + S(5)*d**S(3)*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))/​(S(6)*b*c) + d*sqrt(c*sin(a + b*x))*(d*cos(a + b*x))**(S(5)/​2)/​(S(3)*b*c), expand=True, _diff=True, _numerical=True)442 assert rubi_test(rubi_integrate((d*cos(a + b*x))**(S(3)/​2)/​sqrt(c*sin(a + b*x)), x), x, d**S(2)*EllipticF(-Pi/​S(4) + a + b*x, S(2))*sqrt(sin(S(2)*a + S(2)*b*x))/​(S(2)*b*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))) + d*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))/​(b*c), expand=True, _diff=True, _numerical=True)443 assert rubi_test(rubi_integrate(S(1)/​(sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))), x), x, EllipticF(-Pi/​S(4) + a + b*x, S(2))*sqrt(sin(S(2)*a + S(2)*b*x))/​(b*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))), expand=True, _diff=True, _numerical=True)444 assert rubi_test(rubi_integrate(S(1)/​(sqrt(c*sin(a + b*x))*(d*cos(a + b*x))**(S(5)/​2)), x), x, S(2)*EllipticF(-Pi/​S(4) + a + b*x, S(2))*sqrt(sin(S(2)*a + S(2)*b*x))/​(S(3)*b*d**S(2)*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))) + S(2)*sqrt(c*sin(a + b*x))/​(S(3)*b*c*d*(d*cos(a + b*x))**(S(3)/​2)), expand=True, _diff=True, _numerical=True)445 assert rubi_test(rubi_integrate(S(1)/​(sqrt(c*sin(a + b*x))*(d*cos(a + b*x))**(S(9)/​2)), x), x, S(4)*EllipticF(-Pi/​S(4) + a + b*x, S(2))*sqrt(sin(S(2)*a + S(2)*b*x))/​(S(7)*b*d**S(4)*sqrt(c*sin(a + b*x))*sqrt(d*cos(a + b*x))) + S(2)*sqrt(c*sin(a + b*x))/​(S(7)*b*c*d*(d*cos(a + b*x))**(S(7)/​2)) + S(4)*sqrt(c*sin(a + b*x))/​(S(7)*b*c*d**S(3)*(d*cos(a + b*x))**(S(3)/​2)), expand=True, _diff=True, _numerical=True)446 assert rubi_test(rubi_integrate(sqrt(d*cos(a + b*x))/​sqrt(c*sin(a + b*x)), x), x, sqrt(S(2))*sqrt(d)*ArcTan(-sqrt(S(2))*sqrt(c)*sqrt(d*cos(a + b*x))/​(sqrt(d)*sqrt(c*sin(a + b*x))) + S(1))/​(S(2)*b*sqrt(c)) - sqrt(S(2))*sqrt(d)*ArcTan(sqrt(S(2))*sqrt(c)*sqrt(d*cos(a + b*x))/​(sqrt(d)*sqrt(c*sin(a + b*x))) + S(1))/​(S(2)*b*sqrt(c)) - sqrt(S(2))*sqrt(d)*log(-sqrt(S(2))*sqrt(c)*sqrt(d*cos(a + b*x))/​sqrt(c*sin(a + b*x)) + sqrt(d)*cot(a + b*x) + sqrt(d))/​(S(4)*b*sqrt(c)) + sqrt(S(2))*sqrt(d)*log(sqrt(S(2))*sqrt(c)*sqrt(d*cos(a + b*x))/​sqrt(c*sin(a + b*x)) + sqrt(d)*cot(a + b*x) + sqrt(d))/​(S(4)*b*sqrt(c)), expand=True, _diff=True, _numerical=True)447 assert rubi_test(rubi_integrate(S(1)/​(sqrt(c*sin(a + b*x))*(d*cos(a + b*x))**(S(3)/​2)), x), x, S(2)*sqrt(c*sin(a + b*x))/​(b*c*d*sqrt(d*cos(a + b*x))), expand=True, _diff=True, _numerical=True)448 assert rubi_test(rubi_integrate(S(1)/​(sqrt(c*sin(a + b*x))*(d*cos(a + b*x))**(S(7)/​2)), x), x, S(2)*sqrt(c*sin(a + b*x))/​(S(5)*b*c*d*(d*cos(a + b*x))**(S(5)/​2)) + S(8)*sqrt(c*sin(a + b*x))/​(S(5)*b*c*d**S(3)*sqrt(d*cos(a + b*x))), expand=True, _diff=True, _numerical=True)449 assert rubi_test(rubi_integrate(S(1)/​(sqrt(c*sin(a + b*x))*(d*cos(a + b*x))**(S(11)/​2)), x), x, S(2)*sqrt(c*sin(a + b*x))/​(S(9)*b*c*d*(d*cos(a + b*x))**(S(9)/​2)) + S(16)*sqrt(c*sin(a + b*x))/​(S(45)*b*c*d**S(3)*(d*cos(a + b*x))**(S(5)/​2)) + S(64)*sqrt(c*sin(a + b*x))/​(S(45)*b*c*d**S(5)*sqrt(d*cos(a + b*x))), expand=True, _diff=True, _numerical=True)450 assert rubi_test(rubi_integrate(sqrt(cos(a + b*x))/​sqrt(sin(a + b*x)), x), x, sqrt(S(2))*ArcTan(S(1) - sqrt(S(2))*sqrt(cos(a + b*x))/​sqrt(sin(a + b*x)))/​(S(2)*b) - sqrt(S(2))*ArcTan(S(1) + sqrt(S(2))*sqrt(cos(a + b*x))/​sqrt(sin(a + b*x)))/​(S(2)*b) - sqrt(S(2))*log(cot(a + b*x) + S(1) - sqrt(S(2))*sqrt(cos(a + b*x))/​sqrt(sin(a + b*x)))/​(S(4)*b) + sqrt(S(2))*log(cot(a + b*x) + S(1) + sqrt(S(2))*sqrt(cos(a + b*x))/​sqrt(sin(a + b*x)))/​(S(4)*b), expand=True, _diff=True, _numerical=True)451 assert rubi_test(rubi_integrate(cos(a + b*x)**(S(3)/​2)/​sin(a + b*x)**(S(3)/​2), x), x, sqrt(S(2))*ArcTan(-sqrt(S(2))*sqrt(sin(a + b*x))/​sqrt(cos(a + b*x)) + S(1))/​(S(2)*b) - sqrt(S(2))*ArcTan(sqrt(S(2))*sqrt(sin(a + b*x))/​sqrt(cos(a + b*x)) + S(1))/​(S(2)*b) - sqrt(S(2))*log(-sqrt(S(2))*sqrt(sin(a + b*x))/​sqrt(cos(a + b*x)) + tan(a + b*x) + S(1))/​(S(4)*b) + sqrt(S(2))*log(sqrt(S(2))*sqrt(sin(a + b*x))/​sqrt(cos(a + b*x)) + tan(a + b*x) + S(1))/​(S(4)*b) - S(2)*sqrt(cos(a + b*x))/​(b*sqrt(sin(a + b*x))), expand=True, _diff=True, _numerical=True)452 assert rubi_test(rubi_integrate(cos(a + b*x)**(S(5)/​2)/​sin(a + b*x)**(S(5)/​2), x), x, -sqrt(S(2))*ArcTan(S(1) - sqrt(S(2))*sqrt(cos(a + b*x))/​sqrt(sin(a + b*x)))/​(S(2)*b) + sqrt(S(2))*ArcTan(S(1) + sqrt(S(2))*sqrt(cos(a + b*x))/​sqrt(sin(a + b*x)))/​(S(2)*b) + sqrt(S(2))*log(cot(a + b*x) + S(1) - sqrt(S(2))*sqrt(cos(a + b*x))/​sqrt(sin(a + b*x)))/​(S(4)*b) - sqrt(S(2))*log(cot(a + b*x) + S(1) + sqrt(S(2))*sqrt(cos(a + b*x))/​sqrt(sin(a + b*x)))/​(S(4)*b) - S(2)*cos(a + b*x)**(S(3)/​2)/​(S(3)*b*sin(a + b*x)**(S(3)/​2)), expand=True, _diff=True, _numerical=True)453 assert rubi_test(rubi_integrate(cos(a + b*x)**(S(7)/​2)/​sin(a + b*x)**(S(7)/​2), x), x, -sqrt(S(2))*ArcTan(-sqrt(S(2))*sqrt(sin(a + b*x))/​sqrt(cos(a + b*x)) + S(1))/​(S(2)*b) + sqrt(S(2))*ArcTan(sqrt(S(2))*sqrt(sin(a + b*x))/​sqrt(cos(a + b*x)) + S(1))/​(S(2)*b) + sqrt(S(2))*log(-sqrt(S(2))*sqrt(sin(a + b*x))/​sqrt(cos(a + b*x)) + tan(a + b*x) + S(1))/​(S(4)*b) - sqrt(S(2))*log(sqrt(S(2))*sqrt(sin(a + b*x))/​sqrt(cos(a + b*x)) + tan(a + b*x) + S(1))/​(S(4)*b) + S(2)*sqrt(cos(a + b*x))/​(b*sqrt(sin(a + b*x))) - S(2)*cos(a + b*x)**(S(5)/​2)/​(S(5)*b*sin(a + b*x)**(S(5)/​2)), expand=True, _diff=True, _numerical=True)454 assert rubi_test(rubi_integrate((b*sin(e + f*x))**(S(1)/​3)*cos(e + f*x)**S(4), x), x, S(3)*(b*sin(e + f*x))**(S(4)/​3)*Hypergeometric2F1(S(-3)/​2, S(2)/​3, S(5)/​3, sin(e + f*x)**S(2))*cos(e + f*x)/​(S(4)*b*f*sqrt(cos(e + f*x)**S(2))), expand=True, _diff=True, _numerical=True)455 assert rubi_test(rubi_integrate((b*sin(e + f*x))**(S(1)/​3)*cos(e + f*x)**S(2), x), x, S(3)*(b*sin(e + f*x))**(S(4)/​3)*Hypergeometric2F1(S(-1)/​2, S(2)/​3, S(5)/​3, sin(e + f*x)**S(2))*cos(e + f*x)/​(S(4)*b*f*sqrt(cos(e + f*x)**S(2))), expand=True, _diff=True, _numerical=True)456 assert rubi_test(rubi_integrate((b*sin(e + f*x))**(S(1)/​3), x), x, S(3)*(b*sin(e + f*x))**(S(4)/​3)*Hypergeometric2F1(S(1)/​2, S(2)/​3, S(5)/​3, sin(e + f*x)**S(2))*cos(e + f*x)/​(S(4)*b*f*sqrt(cos(e + f*x)**S(2))), expand=True, _diff=True, _numerical=True)457 assert rubi_test(rubi_integrate((b*sin(e + f*x))**(S(1)/​3)*sec(e + f*x)**S(2), x), x, S(3)*(b*sin(e + f*x))**(S(4)/​3)*sqrt(cos(e + f*x)**S(2))*Hypergeometric2F1(S(2)/​3, S(3)/​2, S(5)/​3, sin(e + f*x)**S(2))*sec(e + f*x)/​(S(4)*b*f), expand=True, _diff=True, _numerical=True)458 assert rubi_test(rubi_integrate((b*sin(e + f*x))**(S(1)/​3)*sec(e + f*x)**S(4), x), x, S(3)*(b*sin(e + f*x))**(S(4)/​3)*sqrt(cos(e + f*x)**S(2))*Hypergeometric2F1(S(2)/​3, S(5)/​2, S(5)/​3, sin(e + f*x)**S(2))*sec(e + f*x)/​(S(4)*b*f), expand=True, _diff=True, _numerical=True)459 assert rubi_test(rubi_integrate((b*sin(e + f*x))**(S(5)/​3)*cos(e + f*x)**S(4), x), x, S(3)*(b*sin(e + f*x))**(S(8)/​3)*Hypergeometric2F1(S(-3)/​2, S(4)/​3, S(7)/​3, sin(e + f*x)**S(2))*cos(e + f*x)/​(S(8)*b*f*sqrt(cos(e + f*x)**S(2))), expand=True, _diff=True, _numerical=True)460 assert rubi_test(rubi_integrate((b*sin(e + f*x))**(S(5)/​3)*cos(e + f*x)**S(2), x), x, S(3)*(b*sin(e + f*x))**(S(8)/​3)*Hypergeometric2F1(S(-1)/​2, S(4)/​3, S(7)/​3, sin(e + f*x)**S(2))*cos(e + f*x)/​(S(8)*b*f*sqrt(cos(e + f*x)**S(2))), expand=True, _diff=True, _numerical=True)461 assert rubi_test(rubi_integrate((b*sin(e + f*x))**(S(5)/​3), x), x, S(3)*(b*sin(e + f*x))**(S(8)/​3)*Hypergeometric2F1(S(1)/​2, S(4)/​3, S(7)/​3, sin(e + f*x)**S(2))*cos(e + f*x)/​(S(8)*b*f*sqrt(cos(e + f*x)**S(2))), expand=True, _diff=True, _numerical=True)462 assert rubi_test(rubi_integrate((b*sin(e + f*x))**(S(5)/​3)*sec(e + f*x)**S(2), x), x, S(3)*(b*sin(e + f*x))**(S(8)/​3)*sqrt(cos(e + f*x)**S(2))*Hypergeometric2F1(S(4)/​3, S(3)/​2, S(7)/​3, sin(e + f*x)**S(2))*sec(e + f*x)/​(S(8)*b*f), expand=True, _diff=True, _numerical=True)463 assert rubi_test(rubi_integrate((b*sin(e + f*x))**(S(5)/​3)*sec(e + f*x)**S(4), x), x, S(3)*(b*sin(e + f*x))**(S(8)/​3)*sqrt(cos(e + f*x)**S(2))*Hypergeometric2F1(S(4)/​3, S(5)/​2, S(7)/​3, sin(e + f*x)**S(2))*sec(e + f*x)/​(S(8)*b*f), expand=True, _diff=True, _numerical=True)464 assert rubi_test(rubi_integrate(cos(e + f*x)**S(4)/​(b*sin(e + f*x))**(S(1)/​3), x), x, S(3)*(b*sin(e + f*x))**(S(2)/​3)*Hypergeometric2F1(S(-3)/​2, S(1)/​3, S(4)/​3, sin(e + f*x)**S(2))*cos(e + f*x)/​(S(2)*b*f*sqrt(cos(e + f*x)**S(2))), expand=True, _diff=True, _numerical=True)465 assert rubi_test(rubi_integrate(cos(e + f*x)**S(2)/​(b*sin(e + f*x))**(S(1)/​3), x), x, S(3)*(b*sin(e + f*x))**(S(2)/​3)*Hypergeometric2F1(S(-1)/​2, S(1)/​3, S(4)/​3, sin(e + f*x)**S(2))*cos(e + f*x)/​(S(2)*b*f*sqrt(cos(e + f*x)**S(2))), expand=True, _diff=True, _numerical=True)466 assert rubi_test(rubi_integrate((b*sin(e + f*x))**(S(-1)/​3), x), x, S(3)*(b*sin(e + f*x))**(S(2)/​3)*Hypergeometric2F1(S(1)/​3, S(1)/​2, S(4)/​3, sin(e + f*x)**S(2))*cos(e + f*x)/​(S(2)*b*f*sqrt(cos(e + f*x)**S(2))), expand=True, _diff=True, _numerical=True)467 assert rubi_test(rubi_integrate(sec(e + f*x)**S(2)/​(b*sin(e + f*x))**(S(1)/​3), x), x, S(3)*(b*sin(e + f*x))**(S(2)/​3)*sqrt(cos(e + f*x)**S(2))*Hypergeometric2F1(S(1)/​3, S(3)/​2, S(4)/​3, sin(e + f*x)**S(2))*sec(e + f*x)/​(S(2)*b*f), expand=True, _diff=True, _numerical=True)468 assert rubi_test(rubi_integrate(sec(e + f*x)**S(4)/​(b*sin(e + f*x))**(S(1)/​3), x), x, S(3)*(b*sin(e + f*x))**(S(2)/​3)*sqrt(cos(e + f*x)**S(2))*Hypergeometric2F1(S(1)/​3, S(5)/​2, S(4)/​3, sin(e + f*x)**S(2))*sec(e + f*x)/​(S(2)*b*f), expand=True, _diff=True, _numerical=True)469 assert rubi_test(rubi_integrate(cos(e + f*x)**S(4)/​(b*sin(e + f*x))**(S(5)/​3), x), x, -S(3)*Hypergeometric2F1(S(-3)/​2, S(-1)/​3, S(2)/​3, sin(e + f*x)**S(2))*cos(e + f*x)/​(S(2)*b*f*(b*sin(e + f*x))**(S(2)/​3)*sqrt(cos(e + f*x)**S(2))), expand=True, _diff=True, _numerical=True)470 assert rubi_test(rubi_integrate(cos(e + f*x)**S(2)/​(b*sin(e + f*x))**(S(5)/​3), x), x, -S(3)*Hypergeometric2F1(S(-1)/​2, S(-1)/​3, S(2)/​3, sin(e + f*x)**S(2))*cos(e + f*x)/​(S(2)*b*f*(b*sin(e + f*x))**(S(2)/​3)*sqrt(cos(e + f*x)**S(2))), expand=True, _diff=True, _numerical=True)471 assert rubi_test(rubi_integrate((b*sin(e + f*x))**(S(-5)/​3), x), x, -S(3)*Hypergeometric2F1(S(-1)/​3, S(1)/​2, S(2)/​3, sin(e + f*x)**S(2))*cos(e + f*x)/​(S(2)*b*f*(b*sin(e + f*x))**(S(2)/​3)*sqrt(cos(e + f*x)**S(2))), expand=True, _diff=True, _numerical=True)472 assert rubi_test(rubi_integrate(sec(e + f*x)**S(2)/​(b*sin(e + f*x))**(S(5)/​3), x), x, -S(3)*sqrt(cos(e + f*x)**S(2))*Hypergeometric2F1(S(-1)/​3, S(3)/​2, S(2)/​3, sin(e + f*x)**S(2))*sec(e + f*x)/​(S(2)*b*f*(b*sin(e + f*x))**(S(2)/​3)), expand=True, _diff=True, _numerical=True)473 assert rubi_test(rubi_integrate(sec(e + f*x)**S(4)/​(b*sin(e + f*x))**(S(5)/​3), x), x, -S(3)*sqrt(cos(e + f*x)**S(2))*Hypergeometric2F1(S(-1)/​3, S(5)/​2, S(2)/​3, sin(e + f*x)**S(2))*sec(e + f*x)/​(S(2)*b*f*(b*sin(e + f*x))**(S(2)/​3)), expand=True, _diff=True, _numerical=True)474 assert rubi_test(rubi_integrate(sin(a + b*x)**(S(1)/​3)/​cos(a + b*x)**(S(1)/​3), x), x, -sqrt(S(3))*ArcTan(sqrt(S(3))*(-S(2)*sin(a + b*x)**(S(2)/​3)/​cos(a + b*x)**(S(2)/​3) + S(1))/​S(3))/​(S(2)*b) - log(sin(a + b*x)**(S(2)/​3)/​cos(a + b*x)**(S(2)/​3) + S(1))/​(S(2)*b) + log(sin(a + b*x)**(S(4)/​3)/​cos(a + b*x)**(S(4)/​3) - sin(a + b*x)**(S(2)/​3)/​cos(a + b*x)**(S(2)/​3) + S(1))/​(S(4)*b), expand=True, _diff=True, _numerical=True)475 assert rubi_test(rubi_integrate(sin(a + b*x)**(S(2)/​3)/​cos(a + b*x)**(S(2)/​3), x), x, ArcTan(sin(a + b*x)**(S(1)/​3)/​cos(a + b*x)**(S(1)/​3))/​b - ArcTan(-S(2)*sin(a + b*x)**(S(1)/​3)/​cos(a + b*x)**(S(1)/​3) + sqrt(S(3)))/​(S(2)*b) + ArcTan(S(2)*sin(a + b*x)**(S(1)/​3)/​cos(a + b*x)**(S(1)/​3) + sqrt(S(3)))/​(S(2)*b) + sqrt(S(3))*log(sin(a + b*x)**(S(2)/​3)/​cos(a + b*x)**(S(2)/​3) - sqrt(S(3))*sin(a + b*x)**(S(1)/​3)/​cos(a + b*x)**(S(1)/​3) + S(1))/​(S(4)*b) - sqrt(S(3))*log(sin(a + b*x)**(S(2)/​3)/​cos(a + b*x)**(S(2)/​3) + sqrt(S(3))*sin(a + b*x)**(S(1)/​3)/​cos(a + b*x)**(S(1)/​3) + S(1))/​(S(4)*b), expand=True, _diff=True, _numerical=True)476 assert rubi_test(rubi_integrate(sin(a + b*x)**(S(4)/​3)/​cos(a + b*x)**(S(4)/​3), x), x, ArcTan(cos(a + b*x)**(S(1)/​3)/​sin(a + b*x)**(S(1)/​3))/​b - ArcTan(sqrt(S(3)) - S(2)*cos(a + b*x)**(S(1)/​3)/​sin(a + b*x)**(S(1)/​3))/​(S(2)*b) + ArcTan(sqrt(S(3)) + S(2)*cos(a + b*x)**(S(1)/​3)/​sin(a + b*x)**(S(1)/​3))/​(S(2)*b) + sqrt(S(3))*log(S(1) - sqrt(S(3))*cos(a + b*x)**(S(1)/​3)/​sin(a + b*x)**(S(1)/​3) + cos(a + b*x)**(S(2)/​3)/​sin(a + b*x)**(S(2)/​3))/​(S(4)*b) - sqrt(S(3))*log(S(1) + sqrt(S(3))*cos(a + b*x)**(S(1)/​3)/​sin(a + b*x)**(S(1)/​3) + cos(a + b*x)**(S(2)/​3)/​sin(a + b*x)**(S(2)/​3))/​(S(4)*b) + S(3)*sin(a + b*x)**(S(1)/​3)/​(b*cos(a + b*x)**(S(1)/​3)), expand=True, _diff=True, _numerical=True)477 assert rubi_test(rubi_integrate(sin(a + b*x)**(S(5)/​3)/​cos(a + b*x)**(S(5)/​3), x), x, -sqrt(S(3))*ArcTan(sqrt(S(3))*(S(1) - S(2)*cos(a + b*x)**(S(2)/​3)/​sin(a + b*x)**(S(2)/​3))/​S(3))/​(S(2)*b) - log(S(1) + cos(a + b*x)**(S(2)/​3)/​sin(a + b*x)**(S(2)/​3))/​(S(2)*b) + log(S(1) - cos(a + b*x)**(S(2)/​3)/​sin(a + b*x)**(S(2)/​3) + cos(a + b*x)**(S(4)/​3)/​sin(a + b*x)**(S(4)/​3))/​(S(4)*b) + S(3)*sin(a + b*x)**(S(2)/​3)/​(S(2)*b*cos(a + b*x)**(S(2)/​3)), expand=True, _diff=True, _numerical=True)478 assert rubi_test(rubi_integrate(sin(a + b*x)**(S(7)/​3)/​cos(a + b*x)**(S(7)/​3), x), x, sqrt(S(3))*ArcTan(sqrt(S(3))*(-S(2)*sin(a + b*x)**(S(2)/​3)/​cos(a + b*x)**(S(2)/​3) + S(1))/​S(3))/​(S(2)*b) + log(sin(a + b*x)**(S(2)/​3)/​cos(a + b*x)**(S(2)/​3) + S(1))/​(S(2)*b) - log(sin(a + b*x)**(S(4)/​3)/​cos(a + b*x)**(S(4)/​3) - sin(a + b*x)**(S(2)/​3)/​cos(a + b*x)**(S(2)/​3) + S(1))/​(S(4)*b) + S(3)*sin(a + b*x)**(S(4)/​3)/​(S(4)*b*cos(a + b*x)**(S(4)/​3)), expand=True, _diff=True, _numerical=True)479 assert rubi_test(rubi_integrate(cos(a + b*x)**(S(1)/​3)/​sin(a + b*x)**(S(1)/​3), x), x, sqrt(S(3))*ArcTan(sqrt(S(3))*(S(1) - S(2)*cos(a + b*x)**(S(2)/​3)/​sin(a + b*x)**(S(2)/​3))/​S(3))/​(S(2)*b) + log(S(1) + cos(a + b*x)**(S(2)/​3)/​sin(a + b*x)**(S(2)/​3))/​(S(2)*b) - log(S(1) - cos(a + b*x)**(S(2)/​3)/​sin(a + b*x)**(S(2)/​3) + cos(a + b*x)**(S(4)/​3)/​sin(a + b*x)**(S(4)/​3))/​(S(4)*b), expand=True, _diff=True, _numerical=True)480 assert rubi_test(rubi_integrate(cos(a + b*x)**(S(2)/​3)/​sin(a + b*x)**(S(2)/​3), x), x, -ArcTan(cos(a + b*x)**(S(1)/​3)/​sin(a + b*x)**(S(1)/​3))/​b + ArcTan(sqrt(S(3)) - S(2)*cos(a + b*x)**(S(1)/​3)/​sin(a + b*x)**(S(1)/​3))/​(S(2)*b) - ArcTan(sqrt(S(3)) + S(2)*cos(a + b*x)**(S(1)/​3)/​sin(a + b*x)**(S(1)/​3))/​(S(2)*b) - sqrt(S(3))*log(S(1) - sqrt(S(3))*cos(a + b*x)**(S(1)/​3)/​sin(a + b*x)**(S(1)/​3) + cos(a + b*x)**(S(2)/​3)/​sin(a + b*x)**(S(2)/​3))/​(S(4)*b) + sqrt(S(3))*log(S(1) + sqrt(S(3))*cos(a + b*x)**(S(1)/​3)/​sin(a + b*x)**(S(1)/​3) + cos(a + b*x)**(S(2)/​3)/​sin(a + b*x)**(S(2)/​3))/​(S(4)*b), expand=True, _diff=True, _numerical=True)481 assert rubi_test(rubi_integrate(cos(a + b*x)**(S(4)/​3)/​sin(a + b*x)**(S(4)/​3), x), x, -ArcTan(sin(a + b*x)**(S(1)/​3)/​cos(a + b*x)**(S(1)/​3))/​b + ArcTan(-S(2)*sin(a + b*x)**(S(1)/​3)/​cos(a + b*x)**(S(1)/​3) + sqrt(S(3)))/​(S(2)*b) - ArcTan(S(2)*sin(a + b*x)**(S(1)/​3)/​cos(a + b*x)**(S(1)/​3) + sqrt(S(3)))/​(S(2)*b) - sqrt(S(3))*log(sin(a + b*x)**(S(2)/​3)/​cos(a + b*x)**(S(2)/​3) - sqrt(S(3))*sin(a + b*x)**(S(1)/​3)/​cos(a + b*x)**(S(1)/​3) + S(1))/​(S(4)*b) + sqrt(S(3))*log(sin(a + b*x)**(S(2)/​3)/​cos(a + b*x)**(S(2)/​3) + sqrt(S(3))*sin(a + b*x)**(S(1)/​3)/​cos(a + b*x)**(S(1)/​3) + S(1))/​(S(4)*b) - S(3)*cos(a + b*x)**(S(1)/​3)/​(b*sin(a + b*x)**(S(1)/​3)), expand=True, _diff=True, _numerical=True)482 assert rubi_test(rubi_integrate(cos(a + b*x)**(S(5)/​3)/​sin(a + b*x)**(S(5)/​3), x), x, sqrt(S(3))*ArcTan(sqrt(S(3))*(-S(2)*sin(a + b*x)**(S(2)/​3)/​cos(a + b*x)**(S(2)/​3) + S(1))/​S(3))/​(S(2)*b) + log(sin(a + b*x)**(S(2)/​3)/​cos(a + b*x)**(S(2)/​3) + S(1))/​(S(2)*b) - log(sin(a + b*x)**(S(4)/​3)/​cos(a + b*x)**(S(4)/​3) - sin(a + b*x)**(S(2)/​3)/​cos(a + b*x)**(S(2)/​3) + S(1))/​(S(4)*b) - S(3)*cos(a + b*x)**(S(2)/​3)/​(S(2)*b*sin(a + b*x)**(S(2)/​3)), expand=True, _diff=True, _numerical=True)483 assert rubi_test(rubi_integrate(cos(a + b*x)**(S(7)/​3)/​sin(a + b*x)**(S(7)/​3), x), x, -sqrt(S(3))*ArcTan(sqrt(S(3))*(S(1) - S(2)*cos(a + b*x)**(S(2)/​3)/​sin(a + b*x)**(S(2)/​3))/​S(3))/​(S(2)*b) - log(S(1) + cos(a + b*x)**(S(2)/​3)/​sin(a + b*x)**(S(2)/​3))/​(S(2)*b) + log(S(1) - cos(a + b*x)**(S(2)/​3)/​sin(a + b*x)**(S(2)/​3) + cos(a + b*x)**(S(4)/​3)/​sin(a + b*x)**(S(4)/​3))/​(S(4)*b) - S(3)*cos(a + b*x)**(S(4)/​3)/​(S(4)*b*sin(a + b*x)**(S(4)/​3)), expand=True, _diff=True, _numerical=True)484 assert rubi_test(rubi_integrate(cos(x)**(S(2)/​3)/​sin(x)**(S(8)/​3), x), x, -S(3)*cos(x)**(S(5)/​3)/​(S(5)*sin(x)**(S(5)/​3)), expand=True, _diff=True, _numerical=True)485 assert rubi_test(rubi_integrate(sin(x)**(S(2)/​3)/​cos(x)**(S(8)/​3), x), x, S(3)*sin(x)**(S(5)/​3)/​(S(5)*cos(x)**(S(5)/​3)), expand=True, _diff=True, _numerical=True)486 assert rubi_test(rubi_integrate(sin(e + f*x)**m*cos(e + f*x)**n, x), x, (cos(e + f*x)**S(2))**(-n/​S(2) + S(1)/​2)*Hypergeometric2F1(m/​S(2) + S(1)/​2, -n/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(e + f*x)**S(2))*sin(e + f*x)**(m + S(1))*cos(e + f*x)**(n + S(-1))/​(f*(m + S(1))), expand=True, _diff=True, _numerical=True)487 assert rubi_test(rubi_integrate((d*cos(e + f*x))**n*sin(e + f*x)**m, x), x, -(d*cos(e + f*x))**(n + S(1))*(sin(e + f*x)**S(2))**(-m/​S(2) + S(1)/​2)*Hypergeometric2F1(-m/​S(2) + S(1)/​2, n/​S(2) + S(1)/​2, n/​S(2) + S(3)/​2, cos(e + f*x)**S(2))*sin(e + f*x)**(m + S(-1))/​(d*f*(n + S(1))), expand=True, _diff=True, _numerical=True)488 assert rubi_test(rubi_integrate((b*sin(e + f*x))**m*cos(e + f*x)**n, x), x, (b*sin(e + f*x))**(m + S(1))*(cos(e + f*x)**S(2))**(-n/​S(2) + S(1)/​2)*Hypergeometric2F1(m/​S(2) + S(1)/​2, -n/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(e + f*x)**S(2))*cos(e + f*x)**(n + S(-1))/​(b*f*(m + S(1))), expand=True, _diff=True, _numerical=True)489 assert rubi_test(rubi_integrate((b*sin(e + f*x))**m*(d*cos(e + f*x))**n, x), x, d*(b*sin(e + f*x))**(m + S(1))*(d*cos(e + f*x))**(n + S(-1))*(cos(e + f*x)**S(2))**(-n/​S(2) + S(1)/​2)*Hypergeometric2F1(m/​S(2) + S(1)/​2, -n/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(e + f*x)**S(2))/​(b*f*(m + S(1))), expand=True, _diff=True, _numerical=True)490 assert rubi_test(rubi_integrate((c*sin(a + b*x))**m*cos(a + b*x)**S(5), x), x, (c*sin(a + b*x))**(m + S(1))/​(b*c*(m + S(1))) - S(2)*(c*sin(a + b*x))**(m + S(3))/​(b*c**S(3)*(m + S(3))) + (c*sin(a + b*x))**(m + S(5))/​(b*c**S(5)*(m + S(5))), expand=True, _diff=True, _numerical=True)491 assert rubi_test(rubi_integrate((c*sin(a + b*x))**m*cos(a + b*x)**S(3), x), x, (c*sin(a + b*x))**(m + S(1))/​(b*c*(m + S(1))) - (c*sin(a + b*x))**(m + S(3))/​(b*c**S(3)*(m + S(3))), expand=True, _diff=True, _numerical=True)492 assert rubi_test(rubi_integrate((c*sin(a + b*x))**m*cos(a + b*x), x), x, (c*sin(a + b*x))**(m + S(1))/​(b*c*(m + S(1))), expand=True, _diff=True, _numerical=True)493 assert rubi_test(rubi_integrate((c*sin(a + b*x))**m*sec(a + b*x), x), x, (c*sin(a + b*x))**(m + S(1))*Hypergeometric2F1(S(1), m/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(a + b*x)**S(2))/​(b*c*(m + S(1))), expand=True, _diff=True, _numerical=True)494 assert rubi_test(rubi_integrate((c*sin(a + b*x))**m*sec(a + b*x)**S(3), x), x, (c*sin(a + b*x))**(m + S(1))*Hypergeometric2F1(S(2), m/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(a + b*x)**S(2))/​(b*c*(m + S(1))), expand=True, _diff=True, _numerical=True)495 assert rubi_test(rubi_integrate((c*sin(a + b*x))**m*cos(a + b*x)**S(4), x), x, (c*sin(a + b*x))**(m + S(1))*Hypergeometric2F1(S(-3)/​2, m/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(a + b*x)**S(2))*cos(a + b*x)/​(b*c*(m + S(1))*sqrt(cos(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)496 assert rubi_test(rubi_integrate((c*sin(a + b*x))**m*cos(a + b*x)**S(2), x), x, (c*sin(a + b*x))**(m + S(1))*Hypergeometric2F1(S(-1)/​2, m/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(a + b*x)**S(2))*cos(a + b*x)/​(b*c*(m + S(1))*sqrt(cos(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)497 assert rubi_test(rubi_integrate((c*sin(a + b*x))**m, x), x, (c*sin(a + b*x))**(m + S(1))*Hypergeometric2F1(S(1)/​2, m/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(a + b*x)**S(2))*cos(a + b*x)/​(b*c*(m + S(1))*sqrt(cos(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)498 assert rubi_test(rubi_integrate((c*sin(a + b*x))**m*sec(a + b*x)**S(2), x), x, (c*sin(a + b*x))**(m + S(1))*sqrt(cos(a + b*x)**S(2))*Hypergeometric2F1(S(3)/​2, m/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(a + b*x)**S(2))*sec(a + b*x)/​(b*c*(m + S(1))), expand=True, _diff=True, _numerical=True)499 assert rubi_test(rubi_integrate((c*sin(a + b*x))**m*sec(a + b*x)**S(4), x), x, (c*sin(a + b*x))**(m + S(1))*sqrt(cos(a + b*x)**S(2))*Hypergeometric2F1(S(5)/​2, m/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(a + b*x)**S(2))*sec(a + b*x)/​(b*c*(m + S(1))), expand=True, _diff=True, _numerical=True)500 assert rubi_test(rubi_integrate((c*sin(a + b*x))**m*(d*cos(a + b*x))**(S(3)/​2), x), x, d*(c*sin(a + b*x))**(m + S(1))*sqrt(d*cos(a + b*x))*Hypergeometric2F1(S(-1)/​4, m/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(a + b*x)**S(2))/​(b*c*(m + S(1))*(cos(a + b*x)**S(2))**(S(1)/​4)), expand=True, _diff=True, _numerical=True)501 assert rubi_test(rubi_integrate((c*sin(a + b*x))**m*sqrt(d*cos(a + b*x)), x), x, d*(c*sin(a + b*x))**(m + S(1))*(cos(a + b*x)**S(2))**(S(1)/​4)*Hypergeometric2F1(S(1)/​4, m/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(a + b*x)**S(2))/​(b*c*sqrt(d*cos(a + b*x))*(m + S(1))), expand=True, _diff=True, _numerical=True)502 assert rubi_test(rubi_integrate((c*sin(a + b*x))**m/​sqrt(d*cos(a + b*x)), x), x, d*(c*sin(a + b*x))**(m + S(1))*(cos(a + b*x)**S(2))**(S(3)/​4)*Hypergeometric2F1(S(3)/​4, m/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(a + b*x)**S(2))/​(b*c*(d*cos(a + b*x))**(S(3)/​2)*(m + S(1))), expand=True, _diff=True, _numerical=True)503 assert rubi_test(rubi_integrate((c*sin(a + b*x))**m/​(d*cos(a + b*x))**(S(3)/​2), x), x, (c*sin(a + b*x))**(m + S(1))*(cos(a + b*x)**S(2))**(S(1)/​4)*Hypergeometric2F1(S(5)/​4, m/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(a + b*x)**S(2))/​(b*c*d*sqrt(d*cos(a + b*x))*(m + S(1))), expand=True, _diff=True, _numerical=True)504 assert rubi_test(rubi_integrate((c*sin(a + b*x))**m/​(d*cos(a + b*x))**(S(5)/​2), x), x, (c*sin(a + b*x))**(m + S(1))*(cos(a + b*x)**S(2))**(S(3)/​4)*Hypergeometric2F1(S(7)/​4, m/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(a + b*x)**S(2))/​(b*c*d*(d*cos(a + b*x))**(S(3)/​2)*(m + S(1))), expand=True, _diff=True, _numerical=True)505 assert rubi_test(rubi_integrate((d*cos(a + b*x))**n*sin(a + b*x)**S(5), x), x, -(d*cos(a + b*x))**(n + S(1))/​(b*d*(n + S(1))) + S(2)*(d*cos(a + b*x))**(n + S(3))/​(b*d**S(3)*(n + S(3))) - (d*cos(a + b*x))**(n + S(5))/​(b*d**S(5)*(n + S(5))), expand=True, _diff=True, _numerical=True)506 assert rubi_test(rubi_integrate((d*cos(a + b*x))**n*sin(a + b*x)**S(3), x), x, -(d*cos(a + b*x))**(n + S(1))/​(b*d*(n + S(1))) + (d*cos(a + b*x))**(n + S(3))/​(b*d**S(3)*(n + S(3))), expand=True, _diff=True, _numerical=True)507 assert rubi_test(rubi_integrate((d*cos(a + b*x))**n*sin(a + b*x), x), x, -(d*cos(a + b*x))**(n + S(1))/​(b*d*(n + S(1))), expand=True, _diff=True, _numerical=True)508 assert rubi_test(rubi_integrate((d*cos(a + b*x))**n*csc(a + b*x), x), x, -(d*cos(a + b*x))**(n + S(1))*Hypergeometric2F1(S(1), n/​S(2) + S(1)/​2, n/​S(2) + S(3)/​2, cos(a + b*x)**S(2))/​(b*d*(n + S(1))), expand=True, _diff=True, _numerical=True)509 assert rubi_test(rubi_integrate((d*cos(a + b*x))**n*csc(a + b*x)**S(3), x), x, -(d*cos(a + b*x))**(n + S(1))*Hypergeometric2F1(S(2), n/​S(2) + S(1)/​2, n/​S(2) + S(3)/​2, cos(a + b*x)**S(2))/​(b*d*(n + S(1))), expand=True, _diff=True, _numerical=True)510 assert rubi_test(rubi_integrate((d*cos(a + b*x))**n*csc(a + b*x)**S(5), x), x, -(d*cos(a + b*x))**(n + S(1))*Hypergeometric2F1(S(3), n/​S(2) + S(1)/​2, n/​S(2) + S(3)/​2, cos(a + b*x)**S(2))/​(b*d*(n + S(1))), expand=True, _diff=True, _numerical=True)511 assert rubi_test(rubi_integrate((d*cos(a + b*x))**n*sin(a + b*x)**S(4), x), x, -(d*cos(a + b*x))**(n + S(1))*Hypergeometric2F1(S(-3)/​2, n/​S(2) + S(1)/​2, n/​S(2) + S(3)/​2, cos(a + b*x)**S(2))*sin(a + b*x)/​(b*d*(n + S(1))*sqrt(sin(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)512 assert rubi_test(rubi_integrate((d*cos(a + b*x))**n*sin(a + b*x)**S(2), x), x, -(d*cos(a + b*x))**(n + S(1))*Hypergeometric2F1(S(-1)/​2, n/​S(2) + S(1)/​2, n/​S(2) + S(3)/​2, cos(a + b*x)**S(2))*sin(a + b*x)/​(b*d*(n + S(1))*sqrt(sin(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)513 assert rubi_test(rubi_integrate((d*cos(a + b*x))**n, x), x, -(d*cos(a + b*x))**(n + S(1))*Hypergeometric2F1(S(1)/​2, n/​S(2) + S(1)/​2, n/​S(2) + S(3)/​2, cos(a + b*x)**S(2))*sin(a + b*x)/​(b*d*(n + S(1))*sqrt(sin(a + b*x)**S(2))), expand=True, _diff=True, _numerical=True)514 assert rubi_test(rubi_integrate((d*cos(a + b*x))**n*csc(a + b*x)**S(2), x), x, -(d*cos(a + b*x))**(n + S(1))*sqrt(sin(a + b*x)**S(2))*Hypergeometric2F1(S(3)/​2, n/​S(2) + S(1)/​2, n/​S(2) + S(3)/​2, cos(a + b*x)**S(2))*csc(a + b*x)/​(b*d*(n + S(1))), expand=True, _diff=True, _numerical=True)515 assert rubi_test(rubi_integrate((d*cos(a + b*x))**n*csc(a + b*x)**S(4), x), x, -(d*cos(a + b*x))**(n + S(1))*sqrt(sin(a + b*x)**S(2))*Hypergeometric2F1(S(5)/​2, n/​S(2) + S(1)/​2, n/​S(2) + S(3)/​2, cos(a + b*x)**S(2))*csc(a + b*x)/​(b*d*(n + S(1))), expand=True, _diff=True, _numerical=True)516 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(5)/​2)*(d*cos(a + b*x))**n, x), x, -c*(c*sin(a + b*x))**(S(3)/​2)*(d*cos(a + b*x))**(n + S(1))*Hypergeometric2F1(S(-3)/​4, n/​S(2) + S(1)/​2, n/​S(2) + S(3)/​2, cos(a + b*x)**S(2))/​(b*d*(n + S(1))*(sin(a + b*x)**S(2))**(S(3)/​4)), expand=True, _diff=True, _numerical=True)517 assert rubi_test(rubi_integrate((c*sin(a + b*x))**(S(3)/​2)*(d*cos(a + b*x))**n, x), x, -c*sqrt(c*sin(a + b*x))*(d*cos(a + b*x))**(n + S(1))*Hypergeometric2F1(S(-1)/​4, n/​S(2) + S(1)/​2, n/​S(2) + S(3)/​2, cos(a + b*x)**S(2))/​(b*d*(n + S(1))*(sin(a + b*x)**S(2))**(S(1)/​4)), expand=True, _diff=True, _numerical=True)518 assert rubi_test(rubi_integrate(sqrt(c*sin(a + b*x))*(d*cos(a + b*x))**n, x), x, -c*(d*cos(a + b*x))**(n + S(1))*(sin(a + b*x)**S(2))**(S(1)/​4)*Hypergeometric2F1(S(1)/​4, n/​S(2) + S(1)/​2, n/​S(2) + S(3)/​2, cos(a + b*x)**S(2))/​(b*d*sqrt(c*sin(a + b*x))*(n + S(1))), expand=True, _diff=True, _numerical=True)519 assert rubi_test(rubi_integrate((d*cos(a + b*x))**n/​sqrt(c*sin(a + b*x)), x), x, -c*(d*cos(a + b*x))**(n + S(1))*(sin(a + b*x)**S(2))**(S(3)/​4)*Hypergeometric2F1(S(3)/​4, n/​S(2) + S(1)/​2, n/​S(2) + S(3)/​2, cos(a + b*x)**S(2))/​(b*d*(c*sin(a + b*x))**(S(3)/​2)*(n + S(1))), expand=True, _diff=True, _numerical=True)520 assert rubi_test(rubi_integrate((d*cos(a + b*x))**n/​(c*sin(a + b*x))**(S(3)/​2), x), x, -(d*cos(a + b*x))**(n + S(1))*(sin(a + b*x)**S(2))**(S(1)/​4)*Hypergeometric2F1(S(5)/​4, n/​S(2) + S(1)/​2, n/​S(2) + S(3)/​2, cos(a + b*x)**S(2))/​(b*c*d*sqrt(c*sin(a + b*x))*(n + S(1))), expand=True, _diff=True, _numerical=True)521 assert rubi_test(rubi_integrate(sqrt(b*sec(e + f*x))*sin(e + f*x)**S(7), x), x, S(2)*b**S(7)/​(S(13)*f*(b*sec(e + f*x))**(S(13)/​2)) - S(2)*b**S(5)/​(S(3)*f*(b*sec(e + f*x))**(S(9)/​2)) + S(6)*b**S(3)/​(S(5)*f*(b*sec(e + f*x))**(S(5)/​2)) - S(2)*b/​(f*sqrt(b*sec(e + f*x))), expand=True, _diff=True, _numerical=True)522 assert rubi_test(rubi_integrate(sqrt(b*sec(e + f*x))*sin(e + f*x)**S(5), x), x, -S(2)*b**S(5)/​(S(9)*f*(b*sec(e + f*x))**(S(9)/​2)) + S(4)*b**S(3)/​(S(5)*f*(b*sec(e + f*x))**(S(5)/​2)) - S(2)*b/​(f*sqrt(b*sec(e + f*x))), expand=True, _diff=True, _numerical=True)523 assert rubi_test(rubi_integrate(sqrt(b*sec(e + f*x))*sin(e + f*x)**S(3), x), x, S(2)*b**S(3)/​(S(5)*f*(b*sec(e + f*x))**(S(5)/​2)) - S(2)*b/​(f*sqrt(b*sec(e + f*x))), expand=True, _diff=True, _numerical=True)524 assert rubi_test(rubi_integrate(sqrt(b*sec(e + f*x))*sin(e + f*x), x), x, -S(2)*b/​(f*sqrt(b*sec(e + f*x))), expand=True, _diff=True, _numerical=True)525 assert rubi_test(rubi_integrate(sqrt(b*sec(e + f*x))*csc(e + f*x), x), x, sqrt(b)*ArcTan(sqrt(b*sec(e + f*x))/​sqrt(b))/​f - sqrt(b)*atanh(sqrt(b*sec(e + f*x))/​sqrt(b))/​f, expand=True, _diff=True, _numerical=True)526 assert rubi_test(rubi_integrate(sqrt(b*sec(e + f*x))*csc(e + f*x)**S(3), x), x, S(3)*sqrt(b)*ArcTan(sqrt(b*sec(e + f*x))/​sqrt(b))/​(S(4)*f) - S(3)*sqrt(b)*atanh(sqrt(b*sec(e + f*x))/​sqrt(b))/​(S(4)*f) - (b*sec(e + f*x))**(S(3)/​2)*cot(e + f*x)**S(2)/​(S(2)*b*f), expand=True, _diff=True, _numerical=True)527 assert rubi_test(rubi_integrate(sqrt(b*sec(e + f*x))*csc(e + f*x)**S(5), x), x, S(21)*sqrt(b)*ArcTan(sqrt(b*sec(e + f*x))/​sqrt(b))/​(S(32)*f) - S(21)*sqrt(b)*atanh(sqrt(b*sec(e + f*x))/​sqrt(b))/​(S(32)*f) - S(7)*(b*sec(e + f*x))**(S(3)/​2)*cot(e + f*x)**S(2)/​(S(16)*b*f) - (b*sec(e + f*x))**(S(7)/​2)*cot(e + f*x)**S(4)/​(S(4)*b**S(3)*f), expand=True, _diff=True, _numerical=True)528 assert rubi_test(rubi_integrate(sqrt(b*sec(e + f*x))*sin(e + f*x)**S(6), x), x, -S(2)*b*sin(e + f*x)**S(5)/​(S(11)*f*sqrt(b*sec(e + f*x))) - S(20)*b*sin(e + f*x)**S(3)/​(S(77)*f*sqrt(b*sec(e + f*x))) - S(40)*b*sin(e + f*x)/​(S(77)*f*sqrt(b*sec(e + f*x))) + S(80)*sqrt(b*sec(e + f*x))*EllipticF(e/​S(2) + f*x/​S(2), S(2))*sqrt(cos(e + f*x))/​(S(77)*f), expand=True, _diff=True, _numerical=True)529 assert rubi_test(rubi_integrate(sqrt(b*sec(e + f*x))*sin(e + f*x)**S(4), x), x, -S(2)*b*sin(e + f*x)**S(3)/​(S(7)*f*sqrt(b*sec(e + f*x))) - S(4)*b*sin(e + f*x)/​(S(7)*f*sqrt(b*sec(e + f*x))) + S(8)*sqrt(b*sec(e + f*x))*EllipticF(e/​S(2) + f*x/​S(2), S(2))*sqrt(cos(e + f*x))/​(S(7)*f), expand=True, _diff=True, _numerical=True)530 assert rubi_test(rubi_integrate(sqrt(b*sec(e + f*x))*sin(e + f*x)**S(2), x), x, -S(2)*b*sin(e + f*x)/​(S(3)*f*sqrt(b*sec(e + f*x))) + S(4)*sqrt(b*sec(e + f*x))*EllipticF(e/​S(2) + f*x/​S(2), S(2))*sqrt(cos(e + f*x))/​(S(3)*f), expand=True, _diff=True, _numerical=True)531 assert rubi_test(rubi_integrate(sqrt(b*sec(e + f*x)), x), x, S(2)*sqrt(b*sec(e + f*x))*EllipticF(e/​S(2) + f*x/​S(2), S(2))*sqrt(cos(e + f*x))/​f, expand=True, _diff=True, _numerical=True)532 assert rubi_test(rubi_integrate(sqrt(b*sec(e + f*x))*csc(e + f*x)**S(2), x), x, -b*csc(e + f*x)/​(f*sqrt(b*sec(e + f*x))) + sqrt(b*sec(e + f*x))*EllipticF(e/​S(2) + f*x/​S(2), S(2))*sqrt(cos(e + f*x))/​f, expand=True, _diff=True, _numerical=True)533 assert rubi_test(rubi_integrate(sqrt(b*sec(e + f*x))*csc(e + f*x)**S(4), x), x, -b*csc(e + f*x)**S(3)/​(S(3)*f*sqrt(b*sec(e + f*x))) - S(5)*b*csc(e + f*x)/​(S(6)*f*sqrt(b*sec(e + f*x))) + S(5)*sqrt(b*sec(e + f*x))*EllipticF(e/​S(2) + f*x/​S(2), S(2))*sqrt(cos(e + f*x))/​(S(6)*f), expand=True, _diff=True, _numerical=True)534 assert rubi_test(rubi_integrate(sqrt(b*sec(e + f*x))*csc(e + f*x)**S(6), x), x, -b*csc(e + f*x)**S(5)/​(S(5)*f*sqrt(b*sec(e + f*x))) - S(3)*b*csc(e + f*x)**S(3)/​(S(10)*f*sqrt(b*sec(e + f*x))) - S(3)*b*csc(e + f*x)/​(S(4)*f*sqrt(b*sec(e + f*x))) + S(3)*sqrt(b*sec(e + f*x))*EllipticF(e/​S(2) + f*x/​S(2), S(2))*sqrt(cos(e + f*x))/​(S(4)*f), expand=True, _diff=True, _numerical=True)535 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)**S(7), x), x, S(2)*b**S(7)/​(S(11)*f*(b*sec(e + f*x))**(S(11)/​2)) - S(6)*b**S(5)/​(S(7)*f*(b*sec(e + f*x))**(S(7)/​2)) + S(2)*b**S(3)/​(f*(b*sec(e + f*x))**(S(3)/​2)) + S(2)*b*sqrt(b*sec(e + f*x))/​f, expand=True, _diff=True, _numerical=True)536 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)**S(5), x), x, -S(2)*b**S(5)/​(S(7)*f*(b*sec(e + f*x))**(S(7)/​2)) + S(4)*b**S(3)/​(S(3)*f*(b*sec(e + f*x))**(S(3)/​2)) + S(2)*b*sqrt(b*sec(e + f*x))/​f, expand=True, _diff=True, _numerical=True)537 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)**S(3), x), x, S(2)*b**S(3)/​(S(3)*f*(b*sec(e + f*x))**(S(3)/​2)) + S(2)*b*sqrt(b*sec(e + f*x))/​f, expand=True, _diff=True, _numerical=True)538 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x), x), x, S(2)*b*sqrt(b*sec(e + f*x))/​f, expand=True, _diff=True, _numerical=True)539 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(3)/​2)*csc(e + f*x), x), x, -b**(S(3)/​2)*ArcTan(sqrt(b*sec(e + f*x))/​sqrt(b))/​f - b**(S(3)/​2)*atanh(sqrt(b*sec(e + f*x))/​sqrt(b))/​f + S(2)*b*sqrt(b*sec(e + f*x))/​f, expand=True, _diff=True, _numerical=True)540 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(3)/​2)*csc(e + f*x)**S(3), x), x, -S(5)*b**(S(3)/​2)*ArcTan(sqrt(b*sec(e + f*x))/​sqrt(b))/​(S(4)*f) - S(5)*b**(S(3)/​2)*atanh(sqrt(b*sec(e + f*x))/​sqrt(b))/​(S(4)*f) + S(5)*b*sqrt(b*sec(e + f*x))/​(S(2)*f) - (b*sec(e + f*x))**(S(5)/​2)*cot(e + f*x)**S(2)/​(S(2)*b*f), expand=True, _diff=True, _numerical=True)541 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)**S(6), x), x, S(20)*b**S(3)*sin(e + f*x)**S(3)/​(S(9)*f*(b*sec(e + f*x))**(S(3)/​2)) + S(8)*b**S(3)*sin(e + f*x)/​(S(3)*f*(b*sec(e + f*x))**(S(3)/​2)) - S(16)*b**S(2)*EllipticE(e/​S(2) + f*x/​S(2), S(2))/​(S(3)*f*sqrt(b*sec(e + f*x))*sqrt(cos(e + f*x))) + S(2)*b*sqrt(b*sec(e + f*x))*sin(e + f*x)**S(5)/​f, expand=True, _diff=True, _numerical=True)542 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)**S(4), x), x, S(12)*b**S(3)*sin(e + f*x)/​(S(5)*f*(b*sec(e + f*x))**(S(3)/​2)) - S(24)*b**S(2)*EllipticE(e/​S(2) + f*x/​S(2), S(2))/​(S(5)*f*sqrt(b*sec(e + f*x))*sqrt(cos(e + f*x))) + S(2)*b*sqrt(b*sec(e + f*x))*sin(e + f*x)**S(3)/​f, expand=True, _diff=True, _numerical=True)543 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)**S(2), x), x, -S(4)*b**S(2)*EllipticE(e/​S(2) + f*x/​S(2), S(2))/​(f*sqrt(b*sec(e + f*x))*sqrt(cos(e + f*x))) + S(2)*b*sqrt(b*sec(e + f*x))*sin(e + f*x)/​f, expand=True, _diff=True, _numerical=True)544 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(3)/​2), x), x, -S(2)*b**S(2)*EllipticE(e/​S(2) + f*x/​S(2), S(2))/​(f*sqrt(b*sec(e + f*x))*sqrt(cos(e + f*x))) + S(2)*b*sqrt(b*sec(e + f*x))*sin(e + f*x)/​f, expand=True, _diff=True, _numerical=True)545 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(3)/​2)*csc(e + f*x)**S(2), x), x, -S(3)*b**S(2)*EllipticE(e/​S(2) + f*x/​S(2), S(2))/​(f*sqrt(b*sec(e + f*x))*sqrt(cos(e + f*x))) + S(3)*b*sqrt(b*sec(e + f*x))*sin(e + f*x)/​f - b*sqrt(b*sec(e + f*x))*csc(e + f*x)/​f, expand=True, _diff=True, _numerical=True)546 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(3)/​2)*csc(e + f*x)**S(4), x), x, -S(7)*b**S(2)*EllipticE(e/​S(2) + f*x/​S(2), S(2))/​(S(2)*f*sqrt(b*sec(e + f*x))*sqrt(cos(e + f*x))) + S(7)*b*sqrt(b*sec(e + f*x))*sin(e + f*x)/​(S(2)*f) - b*sqrt(b*sec(e + f*x))*csc(e + f*x)**S(3)/​(S(3)*f) - S(7)*b*sqrt(b*sec(e + f*x))*csc(e + f*x)/​(S(6)*f), expand=True, _diff=True, _numerical=True)547 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(5)/​2)*sin(e + f*x)**S(7), x), x, S(2)*b**S(7)/​(S(9)*f*(b*sec(e + f*x))**(S(9)/​2)) - S(6)*b**S(5)/​(S(5)*f*(b*sec(e + f*x))**(S(5)/​2)) + S(6)*b**S(3)/​(f*sqrt(b*sec(e + f*x))) + S(2)*b*(b*sec(e + f*x))**(S(3)/​2)/​(S(3)*f), expand=True, _diff=True, _numerical=True)548 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(5)/​2)*sin(e + f*x)**S(5), x), x, -S(2)*b**S(5)/​(S(5)*f*(b*sec(e + f*x))**(S(5)/​2)) + S(4)*b**S(3)/​(f*sqrt(b*sec(e + f*x))) + S(2)*b*(b*sec(e + f*x))**(S(3)/​2)/​(S(3)*f), expand=True, _diff=True, _numerical=True)549 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(5)/​2)*sin(e + f*x)**S(3), x), x, S(2)*b**S(3)/​(f*sqrt(b*sec(e + f*x))) + S(2)*b*(b*sec(e + f*x))**(S(3)/​2)/​(S(3)*f), expand=True, _diff=True, _numerical=True)550 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(5)/​2)*sin(e + f*x), x), x, S(2)*b*(b*sec(e + f*x))**(S(3)/​2)/​(S(3)*f), expand=True, _diff=True, _numerical=True)551 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(5)/​2)*csc(e + f*x), x), x, b**(S(5)/​2)*ArcTan(sqrt(b*sec(e + f*x))/​sqrt(b))/​f - b**(S(5)/​2)*atanh(sqrt(b*sec(e + f*x))/​sqrt(b))/​f + S(2)*b*(b*sec(e + f*x))**(S(3)/​2)/​(S(3)*f), expand=True, _diff=True, _numerical=True)552 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(5)/​2)*csc(e + f*x)**S(3), x), x, S(7)*b**(S(5)/​2)*ArcTan(sqrt(b*sec(e + f*x))/​sqrt(b))/​(S(4)*f) - S(7)*b**(S(5)/​2)*atanh(sqrt(b*sec(e + f*x))/​sqrt(b))/​(S(4)*f) + S(7)*b*(b*sec(e + f*x))**(S(3)/​2)/​(S(6)*f) - (b*sec(e + f*x))**(S(7)/​2)*cot(e + f*x)**S(2)/​(S(2)*b*f), expand=True, _diff=True, _numerical=True)553 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(5)/​2)*csc(e + f*x)**S(5), x), x, S(77)*b**(S(5)/​2)*ArcTan(sqrt(b*sec(e + f*x))/​sqrt(b))/​(S(32)*f) - S(77)*b**(S(5)/​2)*atanh(sqrt(b*sec(e + f*x))/​sqrt(b))/​(S(32)*f) + S(77)*b*(b*sec(e + f*x))**(S(3)/​2)/​(S(48)*f) - S(11)*(b*sec(e + f*x))**(S(7)/​2)*cot(e + f*x)**S(2)/​(S(16)*b*f) - (b*sec(e + f*x))**(S(11)/​2)*cot(e + f*x)**S(4)/​(S(4)*b**S(3)*f), expand=True, _diff=True, _numerical=True)554 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(5)/​2)*sin(e + f*x)**S(6), x), x, S(20)*b**S(3)*sin(e + f*x)**S(3)/​(S(21)*f*sqrt(b*sec(e + f*x))) + S(40)*b**S(3)*sin(e + f*x)/​(S(21)*f*sqrt(b*sec(e + f*x))) - S(80)*b**S(2)*sqrt(b*sec(e + f*x))*EllipticF(e/​S(2) + f*x/​S(2), S(2))*sqrt(cos(e + f*x))/​(S(21)*f) + S(2)*b*(b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)**S(5)/​(S(3)*f), expand=True, _diff=True, _numerical=True)555 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(5)/​2)*sin(e + f*x)**S(4), x), x, S(4)*b**S(3)*sin(e + f*x)/​(S(3)*f*sqrt(b*sec(e + f*x))) - S(8)*b**S(2)*sqrt(b*sec(e + f*x))*EllipticF(e/​S(2) + f*x/​S(2), S(2))*sqrt(cos(e + f*x))/​(S(3)*f) + S(2)*b*(b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)**S(3)/​(S(3)*f), expand=True, _diff=True, _numerical=True)556 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(5)/​2)*sin(e + f*x)**S(2), x), x, -S(4)*b**S(2)*sqrt(b*sec(e + f*x))*EllipticF(e/​S(2) + f*x/​S(2), S(2))*sqrt(cos(e + f*x))/​(S(3)*f) + S(2)*b*(b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)/​(S(3)*f), expand=True, _diff=True, _numerical=True)557 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(5)/​2), x), x, S(2)*b**S(2)*sqrt(b*sec(e + f*x))*EllipticF(e/​S(2) + f*x/​S(2), S(2))*sqrt(cos(e + f*x))/​(S(3)*f) + S(2)*b*(b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)/​(S(3)*f), expand=True, _diff=True, _numerical=True)558 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(5)/​2)*csc(e + f*x)**S(2), x), x, S(5)*b**S(2)*sqrt(b*sec(e + f*x))*EllipticF(e/​S(2) + f*x/​S(2), S(2))*sqrt(cos(e + f*x))/​(S(3)*f) + S(5)*b*(b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)/​(S(3)*f) - b*(b*sec(e + f*x))**(S(3)/​2)*csc(e + f*x)/​f, expand=True, _diff=True, _numerical=True)559 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(5)/​2)*csc(e + f*x)**S(4), x), x, S(5)*b**S(2)*sqrt(b*sec(e + f*x))*EllipticF(e/​S(2) + f*x/​S(2), S(2))*sqrt(cos(e + f*x))/​(S(2)*f) + S(5)*b*(b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)/​(S(2)*f) - b*(b*sec(e + f*x))**(S(3)/​2)*csc(e + f*x)**S(3)/​(S(3)*f) - S(3)*b*(b*sec(e + f*x))**(S(3)/​2)*csc(e + f*x)/​(S(2)*f), expand=True, _diff=True, _numerical=True)560 assert rubi_test(rubi_integrate(sin(e + f*x)**S(7)/​sqrt(b*sec(e + f*x)), x), x, S(2)*b**S(7)/​(S(15)*f*(b*sec(e + f*x))**(S(15)/​2)) - S(6)*b**S(5)/​(S(11)*f*(b*sec(e + f*x))**(S(11)/​2)) + S(6)*b**S(3)/​(S(7)*f*(b*sec(e + f*x))**(S(7)/​2)) - S(2)*b/​(S(3)*f*(b*sec(e + f*x))**(S(3)/​2)), expand=True, _diff=True, _numerical=True)561 assert rubi_test(rubi_integrate(sin(e + f*x)**S(5)/​sqrt(b*sec(e + f*x)), x), x, -S(2)*b**S(5)/​(S(11)*f*(b*sec(e + f*x))**(S(11)/​2)) + S(4)*b**S(3)/​(S(7)*f*(b*sec(e + f*x))**(S(7)/​2)) - S(2)*b/​(S(3)*f*(b*sec(e + f*x))**(S(3)/​2)), expand=True, _diff=True, _numerical=True)562 assert rubi_test(rubi_integrate(sin(e + f*x)**S(3)/​sqrt(b*sec(e + f*x)), x), x, S(2)*b**S(3)/​(S(7)*f*(b*sec(e + f*x))**(S(7)/​2)) - S(2)*b/​(S(3)*f*(b*sec(e + f*x))**(S(3)/​2)), expand=True, _diff=True, _numerical=True)563 assert rubi_test(rubi_integrate(sin(e + f*x)/​sqrt(b*sec(e + f*x)), x), x, -S(2)*b/​(S(3)*f*(b*sec(e + f*x))**(S(3)/​2)), expand=True, _diff=True, _numerical=True)564 assert rubi_test(rubi_integrate(csc(e + f*x)/​sqrt(b*sec(e + f*x)), x), x, -ArcTan(sqrt(b*sec(e + f*x))/​sqrt(b))/​(sqrt(b)*f) - atanh(sqrt(b*sec(e + f*x))/​sqrt(b))/​(sqrt(b)*f), expand=True, _diff=True, _numerical=True)565 assert rubi_test(rubi_integrate(csc(e + f*x)**S(3)/​sqrt(b*sec(e + f*x)), x), x, -sqrt(b*sec(e + f*x))*cot(e + f*x)**S(2)/​(S(2)*b*f) - ArcTan(sqrt(b*sec(e + f*x))/​sqrt(b))/​(S(4)*sqrt(b)*f) - atanh(sqrt(b*sec(e + f*x))/​sqrt(b))/​(S(4)*sqrt(b)*f), expand=True, _diff=True, _numerical=True)566 assert rubi_test(rubi_integrate(csc(e + f*x)**S(5)/​sqrt(b*sec(e + f*x)), x), x, -S(5)*sqrt(b*sec(e + f*x))*cot(e + f*x)**S(2)/​(S(16)*b*f) - (b*sec(e + f*x))**(S(5)/​2)*cot(e + f*x)**S(4)/​(S(4)*b**S(3)*f) - S(5)*ArcTan(sqrt(b*sec(e + f*x))/​sqrt(b))/​(S(32)*sqrt(b)*f) - S(5)*atanh(sqrt(b*sec(e + f*x))/​sqrt(b))/​(S(32)*sqrt(b)*f), expand=True, _diff=True, _numerical=True)567 assert rubi_test(rubi_integrate(sin(e + f*x)**S(6)/​sqrt(b*sec(e + f*x)), x), x, -S(2)*b*sin(e + f*x)**S(5)/​(S(13)*f*(b*sec(e + f*x))**(S(3)/​2)) - S(20)*b*sin(e + f*x)**S(3)/​(S(117)*f*(b*sec(e + f*x))**(S(3)/​2)) - S(8)*b*sin(e + f*x)/​(S(39)*f*(b*sec(e + f*x))**(S(3)/​2)) + S(16)*EllipticE(e/​S(2) + f*x/​S(2), S(2))/​(S(39)*f*sqrt(b*sec(e + f*x))*sqrt(cos(e + f*x))), expand=True, _diff=True, _numerical=True)568 assert rubi_test(rubi_integrate(sin(e + f*x)**S(4)/​sqrt(b*sec(e + f*x)), x), x, -S(2)*b*sin(e + f*x)**S(3)/​(S(9)*f*(b*sec(e + f*x))**(S(3)/​2)) - S(4)*b*sin(e + f*x)/​(S(15)*f*(b*sec(e + f*x))**(S(3)/​2)) + S(8)*EllipticE(e/​S(2) + f*x/​S(2), S(2))/​(S(15)*f*sqrt(b*sec(e + f*x))*sqrt(cos(e + f*x))), expand=True, _diff=True, _numerical=True)569 assert rubi_test(rubi_integrate(sin(e + f*x)**S(2)/​sqrt(b*sec(e + f*x)), x), x, -S(2)*b*sin(e + f*x)/​(S(5)*f*(b*sec(e + f*x))**(S(3)/​2)) + S(4)*EllipticE(e/​S(2) + f*x/​S(2), S(2))/​(S(5)*f*sqrt(b*sec(e + f*x))*sqrt(cos(e + f*x))), expand=True, _diff=True, _numerical=True)570 assert rubi_test(rubi_integrate(S(1)/​sqrt(b*sec(e + f*x)), x), x, S(2)*EllipticE(e/​S(2) + f*x/​S(2), S(2))/​(f*sqrt(b*sec(e + f*x))*sqrt(cos(e + f*x))), expand=True, _diff=True, _numerical=True)571 assert rubi_test(rubi_integrate(csc(e + f*x)**S(2)/​sqrt(b*sec(e + f*x)), x), x, -b*csc(e + f*x)/​(f*(b*sec(e + f*x))**(S(3)/​2)) - EllipticE(e/​S(2) + f*x/​S(2), S(2))/​(f*sqrt(b*sec(e + f*x))*sqrt(cos(e + f*x))), expand=True, _diff=True, _numerical=True)572 assert rubi_test(rubi_integrate(csc(e + f*x)**S(4)/​sqrt(b*sec(e + f*x)), x), x, -b*csc(e + f*x)**S(3)/​(S(3)*f*(b*sec(e + f*x))**(S(3)/​2)) - b*csc(e + f*x)/​(S(2)*f*(b*sec(e + f*x))**(S(3)/​2)) - EllipticE(e/​S(2) + f*x/​S(2), S(2))/​(S(2)*f*sqrt(b*sec(e + f*x))*sqrt(cos(e + f*x))), expand=True, _diff=True, _numerical=True)573 assert rubi_test(rubi_integrate(csc(e + f*x)**S(6)/​sqrt(b*sec(e + f*x)), x), x, -b*csc(e + f*x)**S(5)/​(S(5)*f*(b*sec(e + f*x))**(S(3)/​2)) - S(7)*b*csc(e + f*x)**S(3)/​(S(30)*f*(b*sec(e + f*x))**(S(3)/​2)) - S(7)*b*csc(e + f*x)/​(S(20)*f*(b*sec(e + f*x))**(S(3)/​2)) - S(7)*EllipticE(e/​S(2) + f*x/​S(2), S(2))/​(S(20)*f*sqrt(b*sec(e + f*x))*sqrt(cos(e + f*x))), expand=True, _diff=True, _numerical=True)574 assert rubi_test(rubi_integrate(sin(e + f*x)**S(7)/​(b*sec(e + f*x))**(S(3)/​2), x), x, S(2)*b**S(7)/​(S(17)*f*(b*sec(e + f*x))**(S(17)/​2)) - S(6)*b**S(5)/​(S(13)*f*(b*sec(e + f*x))**(S(13)/​2)) + S(2)*b**S(3)/​(S(3)*f*(b*sec(e + f*x))**(S(9)/​2)) - S(2)*b/​(S(5)*f*(b*sec(e + f*x))**(S(5)/​2)), expand=True, _diff=True, _numerical=True)575 assert rubi_test(rubi_integrate(sin(e + f*x)**S(5)/​(b*sec(e + f*x))**(S(3)/​2), x), x, -S(2)*b**S(5)/​(S(13)*f*(b*sec(e + f*x))**(S(13)/​2)) + S(4)*b**S(3)/​(S(9)*f*(b*sec(e + f*x))**(S(9)/​2)) - S(2)*b/​(S(5)*f*(b*sec(e + f*x))**(S(5)/​2)), expand=True, _diff=True, _numerical=True)576 assert rubi_test(rubi_integrate(sin(e + f*x)**S(3)/​(b*sec(e + f*x))**(S(3)/​2), x), x, S(2)*b**S(3)/​(S(9)*f*(b*sec(e + f*x))**(S(9)/​2)) - S(2)*b/​(S(5)*f*(b*sec(e + f*x))**(S(5)/​2)), expand=True, _diff=True, _numerical=True)577 assert rubi_test(rubi_integrate(sin(e + f*x)/​(b*sec(e + f*x))**(S(3)/​2), x), x, -S(2)*b/​(S(5)*f*(b*sec(e + f*x))**(S(5)/​2)), expand=True, _diff=True, _numerical=True)578 assert rubi_test(rubi_integrate(csc(e + f*x)/​(b*sec(e + f*x))**(S(3)/​2), x), x, S(2)/​(b*f*sqrt(b*sec(e + f*x))) + ArcTan(sqrt(b*sec(e + f*x))/​sqrt(b))/​(b**(S(3)/​2)*f) - atanh(sqrt(b*sec(e + f*x))/​sqrt(b))/​(b**(S(3)/​2)*f), expand=True, _diff=True, _numerical=True)579 assert rubi_test(rubi_integrate(csc(e + f*x)**S(3)/​(b*sec(e + f*x))**(S(3)/​2), x), x, -(b*sec(e + f*x))**(S(3)/​2)*cot(e + f*x)**S(2)/​(S(2)*b**S(3)*f) - ArcTan(sqrt(b*sec(e + f*x))/​sqrt(b))/​(S(4)*b**(S(3)/​2)*f) + atanh(sqrt(b*sec(e + f*x))/​sqrt(b))/​(S(4)*b**(S(3)/​2)*f), expand=True, _diff=True, _numerical=True)580 assert rubi_test(rubi_integrate(csc(e + f*x)**S(5)/​(b*sec(e + f*x))**(S(3)/​2), x), x, -(b*sec(e + f*x))**(S(3)/​2)*cot(e + f*x)**S(4)/​(S(4)*b**S(3)*f) - S(3)*(b*sec(e + f*x))**(S(3)/​2)*cot(e + f*x)**S(2)/​(S(16)*b**S(3)*f) - S(3)*ArcTan(sqrt(b*sec(e + f*x))/​sqrt(b))/​(S(32)*b**(S(3)/​2)*f) + S(3)*atanh(sqrt(b*sec(e + f*x))/​sqrt(b))/​(S(32)*b**(S(3)/​2)*f), expand=True, _diff=True, _numerical=True)581 assert rubi_test(rubi_integrate(sin(e + f*x)**S(4)/​(b*sec(e + f*x))**(S(3)/​2), x), x, -S(2)*b*sin(e + f*x)**S(3)/​(S(11)*f*(b*sec(e + f*x))**(S(5)/​2)) - S(12)*b*sin(e + f*x)/​(S(77)*f*(b*sec(e + f*x))**(S(5)/​2)) + S(8)*sin(e + f*x)/​(S(77)*b*f*sqrt(b*sec(e + f*x))) + S(8)*sqrt(b*sec(e + f*x))*EllipticF(e/​S(2) + f*x/​S(2), S(2))*sqrt(cos(e + f*x))/​(S(77)*b**S(2)*f), expand=True, _diff=True, _numerical=True)582 assert rubi_test(rubi_integrate(sin(e + f*x)**S(2)/​(b*sec(e + f*x))**(S(3)/​2), x), x, -S(2)*b*sin(e + f*x)/​(S(7)*f*(b*sec(e + f*x))**(S(5)/​2)) + S(4)*sin(e + f*x)/​(S(21)*b*f*sqrt(b*sec(e + f*x))) + S(4)*sqrt(b*sec(e + f*x))*EllipticF(e/​S(2) + f*x/​S(2), S(2))*sqrt(cos(e + f*x))/​(S(21)*b**S(2)*f), expand=True, _diff=True, _numerical=True)583 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(-3)/​2), x), x, S(2)*sin(e + f*x)/​(S(3)*b*f*sqrt(b*sec(e + f*x))) + S(2)*sqrt(b*sec(e + f*x))*EllipticF(e/​S(2) + f*x/​S(2), S(2))*sqrt(cos(e + f*x))/​(S(3)*b**S(2)*f), expand=True, _diff=True, _numerical=True)584 assert rubi_test(rubi_integrate(csc(e + f*x)**S(2)/​(b*sec(e + f*x))**(S(3)/​2), x), x, -csc(e + f*x)/​(b*f*sqrt(b*sec(e + f*x))) - sqrt(b*sec(e + f*x))*EllipticF(e/​S(2) + f*x/​S(2), S(2))*sqrt(cos(e + f*x))/​(b**S(2)*f), expand=True, _diff=True, _numerical=True)585 assert rubi_test(rubi_integrate(csc(e + f*x)**S(4)/​(b*sec(e + f*x))**(S(3)/​2), x), x, -csc(e + f*x)**S(3)/​(S(3)*b*f*sqrt(b*sec(e + f*x))) + csc(e + f*x)/​(S(6)*b*f*sqrt(b*sec(e + f*x))) - sqrt(b*sec(e + f*x))*EllipticF(e/​S(2) + f*x/​S(2), S(2))*sqrt(cos(e + f*x))/​(S(6)*b**S(2)*f), expand=True, _diff=True, _numerical=True)586 assert rubi_test(rubi_integrate(csc(e + f*x)**S(6)/​(b*sec(e + f*x))**(S(3)/​2), x), x, -csc(e + f*x)**S(5)/​(S(5)*b*f*sqrt(b*sec(e + f*x))) + csc(e + f*x)**S(3)/​(S(30)*b*f*sqrt(b*sec(e + f*x))) + csc(e + f*x)/​(S(12)*b*f*sqrt(b*sec(e + f*x))) - sqrt(b*sec(e + f*x))*EllipticF(e/​S(2) + f*x/​S(2), S(2))*sqrt(cos(e + f*x))/​(S(12)*b**S(2)*f), expand=True, _diff=True, _numerical=True)587 assert rubi_test(rubi_integrate(sin(e + f*x)**S(7)/​(b*sec(e + f*x))**(S(5)/​2), x), x, S(2)*b**S(7)/​(S(19)*f*(b*sec(e + f*x))**(S(19)/​2)) - S(2)*b**S(5)/​(S(5)*f*(b*sec(e + f*x))**(S(15)/​2)) + S(6)*b**S(3)/​(S(11)*f*(b*sec(e + f*x))**(S(11)/​2)) - S(2)*b/​(S(7)*f*(b*sec(e + f*x))**(S(7)/​2)), expand=True, _diff=True, _numerical=True)588 assert rubi_test(rubi_integrate(sin(e + f*x)**S(5)/​(b*sec(e + f*x))**(S(5)/​2), x), x, -S(2)*b**S(5)/​(S(15)*f*(b*sec(e + f*x))**(S(15)/​2)) + S(4)*b**S(3)/​(S(11)*f*(b*sec(e + f*x))**(S(11)/​2)) - S(2)*b/​(S(7)*f*(b*sec(e + f*x))**(S(7)/​2)), expand=True, _diff=True, _numerical=True)589 assert rubi_test(rubi_integrate(sin(e + f*x)**S(3)/​(b*sec(e + f*x))**(S(5)/​2), x), x, S(2)*b**S(3)/​(S(11)*f*(b*sec(e + f*x))**(S(11)/​2)) - S(2)*b/​(S(7)*f*(b*sec(e + f*x))**(S(7)/​2)), expand=True, _diff=True, _numerical=True)590 assert rubi_test(rubi_integrate(sin(e + f*x)/​(b*sec(e + f*x))**(S(5)/​2), x), x, -S(2)*b/​(S(7)*f*(b*sec(e + f*x))**(S(7)/​2)), expand=True, _diff=True, _numerical=True)591 assert rubi_test(rubi_integrate(csc(e + f*x)/​(b*sec(e + f*x))**(S(5)/​2), x), x, S(2)/​(S(3)*b*f*(b*sec(e + f*x))**(S(3)/​2)) - ArcTan(sqrt(b*sec(e + f*x))/​sqrt(b))/​(b**(S(5)/​2)*f) - atanh(sqrt(b*sec(e + f*x))/​sqrt(b))/​(b**(S(5)/​2)*f), expand=True, _diff=True, _numerical=True)592 assert rubi_test(rubi_integrate(csc(e + f*x)**S(3)/​(b*sec(e + f*x))**(S(5)/​2), x), x, -sqrt(b*sec(e + f*x))*cot(e + f*x)**S(2)/​(S(2)*b**S(3)*f) + S(3)*ArcTan(sqrt(b*sec(e + f*x))/​sqrt(b))/​(S(4)*b**(S(5)/​2)*f) + S(3)*atanh(sqrt(b*sec(e + f*x))/​sqrt(b))/​(S(4)*b**(S(5)/​2)*f), expand=True, _diff=True, _numerical=True)593 assert rubi_test(rubi_integrate(csc(e + f*x)**S(5)/​(b*sec(e + f*x))**(S(5)/​2), x), x, -sqrt(b*sec(e + f*x))*cot(e + f*x)**S(4)/​(S(4)*b**S(3)*f) - sqrt(b*sec(e + f*x))*cot(e + f*x)**S(2)/​(S(16)*b**S(3)*f) + S(3)*ArcTan(sqrt(b*sec(e + f*x))/​sqrt(b))/​(S(32)*b**(S(5)/​2)*f) + S(3)*atanh(sqrt(b*sec(e + f*x))/​sqrt(b))/​(S(32)*b**(S(5)/​2)*f), expand=True, _diff=True, _numerical=True)594 assert rubi_test(rubi_integrate(sin(e + f*x)**S(4)/​(b*sec(e + f*x))**(S(5)/​2), x), x, -S(2)*b*sin(e + f*x)**S(3)/​(S(13)*f*(b*sec(e + f*x))**(S(7)/​2)) - S(4)*b*sin(e + f*x)/​(S(39)*f*(b*sec(e + f*x))**(S(7)/​2)) + S(8)*sin(e + f*x)/​(S(195)*b*f*(b*sec(e + f*x))**(S(3)/​2)) + S(8)*EllipticE(e/​S(2) + f*x/​S(2), S(2))/​(S(65)*b**S(2)*f*sqrt(b*sec(e + f*x))*sqrt(cos(e + f*x))), expand=True, _diff=True, _numerical=True)595 assert rubi_test(rubi_integrate(sin(e + f*x)**S(2)/​(b*sec(e + f*x))**(S(5)/​2), x), x, -S(2)*b*sin(e + f*x)/​(S(9)*f*(b*sec(e + f*x))**(S(7)/​2)) + S(4)*sin(e + f*x)/​(S(45)*b*f*(b*sec(e + f*x))**(S(3)/​2)) + S(4)*EllipticE(e/​S(2) + f*x/​S(2), S(2))/​(S(15)*b**S(2)*f*sqrt(b*sec(e + f*x))*sqrt(cos(e + f*x))), expand=True, _diff=True, _numerical=True)596 assert rubi_test(rubi_integrate((b*sec(e + f*x))**(S(-5)/​2), x), x, S(2)*sin(e + f*x)/​(S(5)*b*f*(b*sec(e + f*x))**(S(3)/​2)) + S(6)*EllipticE(e/​S(2) + f*x/​S(2), S(2))/​(S(5)*b**S(2)*f*sqrt(b*sec(e + f*x))*sqrt(cos(e + f*x))), expand=True, _diff=True, _numerical=True)597 assert rubi_test(rubi_integrate(csc(e + f*x)**S(2)/​(b*sec(e + f*x))**(S(5)/​2), x), x, -csc(e + f*x)/​(b*f*(b*sec(e + f*x))**(S(3)/​2)) - S(3)*EllipticE(e/​S(2) + f*x/​S(2), S(2))/​(b**S(2)*f*sqrt(b*sec(e + f*x))*sqrt(cos(e + f*x))), expand=True, _diff=True, _numerical=True)598 assert rubi_test(rubi_integrate(csc(e + f*x)**S(4)/​(b*sec(e + f*x))**(S(5)/​2), x), x, -csc(e + f*x)**S(3)/​(S(3)*b*f*(b*sec(e + f*x))**(S(3)/​2)) + csc(e + f*x)/​(S(2)*b*f*(b*sec(e + f*x))**(S(3)/​2)) + EllipticE(e/​S(2) + f*x/​S(2), S(2))/​(S(2)*b**S(2)*f*sqrt(b*sec(e + f*x))*sqrt(cos(e + f*x))), expand=True, _diff=True, _numerical=True)599 assert rubi_test(rubi_integrate(csc(e + f*x)**S(6)/​(b*sec(e + f*x))**(S(5)/​2), x), x, -csc(e + f*x)**S(5)/​(S(5)*b*f*(b*sec(e + f*x))**(S(3)/​2)) + csc(e + f*x)**S(3)/​(S(10)*b*f*(b*sec(e + f*x))**(S(3)/​2)) + S(3)*csc(e + f*x)/​(S(20)*b*f*(b*sec(e + f*x))**(S(3)/​2)) + S(3)*EllipticE(e/​S(2) + f*x/​S(2), S(2))/​(S(20)*b**S(2)*f*sqrt(b*sec(e + f*x))*sqrt(cos(e + f*x))), expand=True, _diff=True, _numerical=True)600 assert rubi_test(rubi_integrate(sin(e + f*x)**(S(9)/​2)/​sqrt(b*sec(e + f*x)), x), x, -b*sin(e + f*x)**(S(7)/​2)/​(S(5)*f*(b*sec(e + f*x))**(S(3)/​2)) - S(7)*b*sin(e + f*x)**(S(3)/​2)/​(S(30)*f*(b*sec(e + f*x))**(S(3)/​2)) + S(7)*sqrt(b*sec(e + f*x))*EllipticE(-Pi/​S(4) + e + f*x, S(2))*sqrt(sin(e + f*x))*cos(e + f*x)/​(S(20)*b*f*sqrt(sin(S(2)*e + S(2)*f*x))), expand=True, _diff=True, _numerical=True)601 assert rubi_test(rubi_integrate(sin(e + f*x)**(S(5)/​2)/​sqrt(b*sec(e + f*x)), x), x, -b*sin(e + f*x)**(S(3)/​2)/​(S(3)*f*(b*sec(e + f*x))**(S(3)/​2)) + sqrt(b*sec(e + f*x))*EllipticE(-Pi/​S(4) + e + f*x, S(2))*sqrt(sin(e + f*x))*cos(e + f*x)/​(S(2)*b*f*sqrt(sin(S(2)*e + S(2)*f*x))), expand=True, _diff=True, _numerical=True)602 assert rubi_test(rubi_integrate(sqrt(sin(e + f*x))/​sqrt(b*sec(e + f*x)), x), x, sqrt(b*sec(e + f*x))*EllipticE(-Pi/​S(4) + e + f*x, S(2))*sqrt(sin(e + f*x))*cos(e + f*x)/​(b*f*sqrt(sin(S(2)*e + S(2)*f*x))), expand=True, _diff=True, _numerical=True)603 assert rubi_test(rubi_integrate(S(1)/​(sqrt(b*sec(e + f*x))*sin(e + f*x)**(S(3)/​2)), x), x, -S(2)*b/​(f*(b*sec(e + f*x))**(S(3)/​2)*sqrt(sin(e + f*x))) - S(2)*sqrt(b*sec(e + f*x))*EllipticE(-Pi/​S(4) + e + f*x, S(2))*sqrt(sin(e + f*x))*cos(e + f*x)/​(b*f*sqrt(sin(S(2)*e + S(2)*f*x))), expand=True, _diff=True, _numerical=True)604 assert rubi_test(rubi_integrate(S(1)/​(sqrt(b*sec(e + f*x))*sin(e + f*x)**(S(7)/​2)), x), x, -S(4)*b/​(S(5)*f*(b*sec(e + f*x))**(S(3)/​2)*sqrt(sin(e + f*x))) - S(2)*b/​(S(5)*f*(b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)**(S(5)/​2)) - S(4)*sqrt(b*sec(e + f*x))*EllipticE(-Pi/​S(4) + e + f*x, S(2))*sqrt(sin(e + f*x))*cos(e + f*x)/​(S(5)*b*f*sqrt(sin(S(2)*e + S(2)*f*x))), expand=True, _diff=True, _numerical=True)605 assert rubi_test(rubi_integrate(sin(e + f*x)**(S(3)/​2)/​sqrt(b*sec(e + f*x)), x), x, -b*sqrt(sin(e + f*x))/​(S(2)*f*(b*sec(e + f*x))**(S(3)/​2)) + sqrt(S(2))*sqrt(cos(e + f*x)/​b)*sqrt(b*sec(e + f*x))*ArcTan(-sqrt(S(2))*sqrt(b)*sqrt(cos(e + f*x)/​b)/​sqrt(sin(e + f*x)) + S(1))/​(S(8)*sqrt(b)*f) - sqrt(S(2))*sqrt(cos(e + f*x)/​b)*sqrt(b*sec(e + f*x))*ArcTan(sqrt(S(2))*sqrt(b)*sqrt(cos(e + f*x)/​b)/​sqrt(sin(e + f*x)) + S(1))/​(S(8)*sqrt(b)*f) - sqrt(S(2))*sqrt(cos(e + f*x)/​b)*sqrt(b*sec(e + f*x))*log(-sqrt(S(2))*sqrt(b)*sqrt(cos(e + f*x)/​b)/​sqrt(sin(e + f*x)) + cot(e + f*x) + S(1))/​(S(16)*sqrt(b)*f) + sqrt(S(2))*sqrt(cos(e + f*x)/​b)*sqrt(b*sec(e + f*x))*log(sqrt(S(2))*sqrt(b)*sqrt(cos(e + f*x)/​b)/​sqrt(sin(e + f*x)) + cot(e + f*x) + S(1))/​(S(16)*sqrt(b)*f), expand=True, _diff=True, _numerical=True)606 assert rubi_test(rubi_integrate(S(1)/​(sqrt(b*sec(e + f*x))*sqrt(sin(e + f*x))), x), x, sqrt(S(2))*sqrt(cos(e + f*x)/​b)*sqrt(b*sec(e + f*x))*ArcTan(-sqrt(S(2))*sqrt(b)*sqrt(cos(e + f*x)/​b)/​sqrt(sin(e + f*x)) + S(1))/​(S(2)*sqrt(b)*f) - sqrt(S(2))*sqrt(cos(e + f*x)/​b)*sqrt(b*sec(e + f*x))*ArcTan(sqrt(S(2))*sqrt(b)*sqrt(cos(e + f*x)/​b)/​sqrt(sin(e + f*x)) + S(1))/​(S(2)*sqrt(b)*f) - sqrt(S(2))*sqrt(cos(e + f*x)/​b)*sqrt(b*sec(e + f*x))*log(-sqrt(S(2))*sqrt(b)*sqrt(cos(e + f*x)/​b)/​sqrt(sin(e + f*x)) + cot(e + f*x) + S(1))/​(S(4)*sqrt(b)*f) + sqrt(S(2))*sqrt(cos(e + f*x)/​b)*sqrt(b*sec(e + f*x))*log(sqrt(S(2))*sqrt(b)*sqrt(cos(e + f*x)/​b)/​sqrt(sin(e + f*x)) + cot(e + f*x) + S(1))/​(S(4)*sqrt(b)*f), expand=True, _diff=True, _numerical=True)607 assert rubi_test(rubi_integrate(S(1)/​(sqrt(b*sec(e + f*x))*sin(e + f*x)**(S(5)/​2)), x), x, -S(2)*b/​(S(3)*f*(b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)**(S(3)/​2)), expand=True, _diff=True, _numerical=True)608 assert rubi_test(rubi_integrate(S(1)/​(sqrt(b*sec(e + f*x))*sin(e + f*x)**(S(9)/​2)), x), x, -S(8)*b/​(S(21)*f*(b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)**(S(3)/​2)) - S(2)*b/​(S(7)*f*(b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)**(S(7)/​2)), expand=True, _diff=True, _numerical=True)609 assert rubi_test(rubi_integrate(S(1)/​(sqrt(b*sec(e + f*x))*sin(e + f*x)**(S(13)/​2)), x), x, -S(64)*b/​(S(231)*f*(b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)**(S(3)/​2)) - S(16)*b/​(S(77)*f*(b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)**(S(7)/​2)) - S(2)*b/​(S(11)*f*(b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)**(S(11)/​2)), expand=True, _diff=True, _numerical=True)610 assert rubi_test(rubi_integrate(S(1)/​(sqrt(b*sec(e + f*x))*sin(e + f*x)**(S(17)/​2)), x), x, -S(256)*b/​(S(1155)*f*(b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)**(S(3)/​2)) - S(64)*b/​(S(385)*f*(b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)**(S(7)/​2)) - S(8)*b/​(S(55)*f*(b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)**(S(11)/​2)) - S(2)*b/​(S(15)*f*(b*sec(e + f*x))**(S(3)/​2)*sin(e + f*x)**(S(15)/​2)), expand=True, _diff=True, _numerical=True)611 assert rubi_test(rubi_integrate((c*sin(a + b*x))**m*(d*sec(a + b*x))**(S(5)/​2), x), x, d**S(2)*(c*sin(a + b*x))**(m + S(1))*sqrt(d*sec(a + b*x))*(cos(a + b*x)**S(2))**(S(3)/​4)*Hypergeometric2F1(S(7)/​4, m/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(a + b*x)**S(2))*sec(a + b*x)/​(b*c*(m + S(1))), expand=True, _diff=True, _numerical=True)612 assert rubi_test(rubi_integrate((c*sin(a + b*x))**m*(d*sec(a + b*x))**(S(3)/​2), x), x, d*(c*sin(a + b*x))**(m + S(1))*sqrt(d*sec(a + b*x))*(cos(a + b*x)**S(2))**(S(1)/​4)*Hypergeometric2F1(S(5)/​4, m/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(a + b*x)**S(2))/​(b*c*(m + S(1))), expand=True, _diff=True, _numerical=True)613 assert rubi_test(rubi_integrate((c*sin(a + b*x))**m*sqrt(d*sec(a + b*x)), x), x, (c*sin(a + b*x))**(m + S(1))*sqrt(d*sec(a + b*x))*(cos(a + b*x)**S(2))**(S(3)/​4)*Hypergeometric2F1(S(3)/​4, m/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(a + b*x)**S(2))*sec(a + b*x)/​(b*c*(m + S(1))), expand=True, _diff=True, _numerical=True)614 assert rubi_test(rubi_integrate((c*sin(a + b*x))**m/​sqrt(d*sec(a + b*x)), x), x, (c*sin(a + b*x))**(m + S(1))*sqrt(d*sec(a + b*x))*(cos(a + b*x)**S(2))**(S(1)/​4)*Hypergeometric2F1(S(1)/​4, m/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(a + b*x)**S(2))/​(b*c*d*(m + S(1))), expand=True, _diff=True, _numerical=True)615 assert rubi_test(rubi_integrate((c*sin(a + b*x))**m/​(d*sec(a + b*x))**(S(3)/​2), x), x, (c*sin(a + b*x))**(m + S(1))*sqrt(d*sec(a + b*x))*Hypergeometric2F1(S(-1)/​4, m/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(a + b*x)**S(2))*cos(a + b*x)/​(b*c*d**S(2)*(m + S(1))*(cos(a + b*x)**S(2))**(S(1)/​4)), expand=True, _diff=True, _numerical=True)616 assert rubi_test(rubi_integrate(sin(e + f*x)**m*sec(e + f*x)**n, x), x, (cos(e + f*x)**S(2))**(n/​S(2) + S(1)/​2)*Hypergeometric2F1(m/​S(2) + S(1)/​2, n/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(e + f*x)**S(2))*sin(e + f*x)**(m + S(1))*sec(e + f*x)**(n + S(1))/​(f*(m + S(1))), expand=True, _diff=True, _numerical=True)617 assert rubi_test(rubi_integrate((a*sin(e + f*x))**m*sec(e + f*x)**n, x), x, (a*sin(e + f*x))**(m + S(1))*(cos(e + f*x)**S(2))**(n/​S(2) + S(1)/​2)*Hypergeometric2F1(m/​S(2) + S(1)/​2, n/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(e + f*x)**S(2))*sec(e + f*x)**(n + S(1))/​(a*f*(m + S(1))), expand=True, _diff=True, _numerical=True)618 assert rubi_test(rubi_integrate((b*sec(e + f*x))**n*sin(e + f*x)**m, x), x, -b*(b*sec(e + f*x))**(n + S(-1))*(sin(e + f*x)**S(2))**(-m/​S(2) + S(1)/​2)*Hypergeometric2F1(-m/​S(2) + S(1)/​2, -n/​S(2) + S(1)/​2, -n/​S(2) + S(3)/​2, cos(e + f*x)**S(2))*sin(e + f*x)**(m + S(-1))/​(f*(-n + S(1))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((b*sec(e + f*x))**n*sin(e + f*x)**m, x), x, -(b*sec(e + f*x))**n*(sin(e + f*x)**S(2))**(-m/​S(2) + S(1)/​2)*Hypergeometric2F1(-m/​S(2) + S(1)/​2, -n/​S(2) + S(1)/​2, -n/​S(2) + S(3)/​2, cos(e + f*x)**S(2))*sin(e + f*x)**(m + S(-1))*cos(e + f*x)/​(f*(-n + S(1))), expand=True, _diff=True, _numerical=True)619 assert rubi_test(rubi_integrate((a*sin(e + f*x))**m*(b*sec(e + f*x))**n, x), x, (a*sin(e + f*x))**(m + S(1))*(b*sec(e + f*x))**(n + S(1))*(cos(e + f*x)**S(2))**(n/​S(2) + S(1)/​2)*Hypergeometric2F1(m/​S(2) + S(1)/​2, n/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(e + f*x)**S(2))/​(a*b*f*(m + S(1))), expand=True, _diff=True, _numerical=True) or rubi_test(rubi_integrate((a*sin(e + f*x))**m*(b*sec(e + f*x))**n, x), x, (a*sin(e + f*x))**(m + S(1))*(b*sec(e + f*x))**n*(cos(e + f*x)**S(2))**(n/​S(2) + S(1)/​2)*Hypergeometric2F1(m/​S(2) + S(1)/​2, n/​S(2) + S(1)/​2, m/​S(2) + S(3)/​2, sin(e + f*x)**S(2))*sec(e + f*x)/​(a*f*(m + S(1))), expand=True, _diff=True, _numerical=True)620 assert rubi_test(rubi_integrate((b*sec(e + f*x))**n*sin(e + f*x)**S(5), x), x, -b**S(5)*(b*sec(e + f*x))**(n + S(-5))/​(f*(-n + S(5))) + S(2)*b**S(3)*(b*sec(e + f*x))**(n + S(-3))/​(f*(-n + S(3))) - b*(b*sec(e + f*x))**(n + S(-1))/​(f*(-n + S(1))), expand=True, _diff=True, _numerical=True)621 assert rubi_test(rubi_integrate((b*sec(e + f*x))**n*sin(e + f*x)**S(3), x), x, b**S(3)*(b*sec(e + f*x))**(n + S(-3))/​(f*(-n + S(3))) - b*(b*sec(e + f*x))**(n + S(-1))/​(f*(-n + S(1))), expand=True, _diff=True, _numerical=True)622 assert rubi_test(rubi_integrate((b*sec(e + f*x))**n*sin(e + f*x), x), x, -b*(b*sec(e + f*x))**(n + S(-1))/​(f*(-n + S(1))), expand=True, _diff=True, _numerical=True)623 assert rubi_test(rubi_integrate((b*sec(e + f*x))**n*csc(e + f*x), x), x, -(b*sec(e + f*x))**(n + S(1))*Hypergeometric2F1(S(1), n/​S(2) + S(1)/​2, n/​S(2) + S(3)/​2, sec(e + f*x)**S(2))/​(b*f*(n + S(1))), expand=True, _diff=True, _numerical=True)624 # long time assert rubi_test(rubi_integrate((b*sec(e + f*x))**n*csc(e + f*x)**S(3), x), x, (b*sec(e + f*x))**(n + S(3))*Hypergeometric2F1(S(2), n/​S(2) + S(3)/​2, n/​S(2) + S(5)/​2, sec(e + f*x)**S(2))/​(b**S(3)*f*(n + S(3))), expand=True, _diff=True, _numerical=True)625 assert rubi_test(rubi_integrate((b*sec(e + f*x))**n*sin(e + f*x)**S(6), x), x, -(b*sec(e + f*x))**n*Hypergeometric2F1(S(-5)/​2, -n/​S(2) + S(1)/​2, -n/​S(2) + S(3)/​2, cos(e + f*x)**S(2))*sin(e + f*x)*cos(e + f*x)/​(f*(-n + S(1))*sqrt(sin(e + f*x)**S(2))), expand=True, _diff=True, _numerical=True)626 assert rubi_test(rubi_integrate((b*sec(e + f*x))**n*sin(e + f*x)**S(4), x), x, -(b*sec(e + f*x))**n*Hypergeometric2F1(S(-3)/​2, -n/​S(2) + S(1)/​2, -n/​S(2) + S(3)/​2, cos(e + f*x)**S(2))*sin(e + f*x)*cos(e + f*x)/​(f*(-n + S(1))*sqrt(sin(e + f*x)**S(2))), expand=True, _diff=True, _numerical=True)627 assert rubi_test(rubi_integrate((b*sec(e + f*x))**n*sin(e + f*x)**S(2), x), x, -(b*sec(e + f*x))**n*Hypergeometric2F1(S(-1)/​2, -n/​S(2) + S(1)/​2, -n/​S(2) + S(3)/​2, cos(e + f*x)**S(2))*sin(e + f*x)*cos(e + f*x)/​(f*(-n + S(1))*sqrt(sin(e + f*x)**S(2))), expand=True, _diff=True, _numerical=True)628 assert rubi_test(rubi_integrate((b*sec(e + f*x))**n, x), x, -b*(b*sec(e + f*x))**(n + S(-1))*Hypergeometric2F1(S(1)/​2, -n/​S(2) + S(1)/​2, -n/​S(2) + S(3)/​2, cos(e + f*x)**S(2))*sin(e + f*x)/​(f*(-n + S(1))*sqrt(sin(e + f*x)**S(2))), expand=True, _diff=True, _numerical=True)629 assert rubi_test(rubi_integrate((b*sec(e + f*x))**n*csc(e + f*x)**S(2), x), x, -(b*sec(e + f*x))**n*sqrt(sin(e + f*x)**S(2))*Hypergeometric2F1(S(3)/​2, -n/​S(2) + S(1)/​2, -n/​S(2) + S(3)/​2, cos(e + f*x)**S(2))*cot(e + f*x)/​(f*(-n + S(1))), expand=True, _diff=True, _numerical=True)630 assert rubi_test(rubi_integrate((b*sec(e + f*x))**n*csc(e + f*x)**S(4), x), x, -(b*sec(e + f*x))**n*sqrt(sin(e + f*x)**S(2))*Hypergeometric2F1(S(5)/​2, -n/​S(2) + S(1)/​2, -n/​S(2) + S(3)/​2, cos(e + f*x)**S(2))*cot(e + f*x)/​(f*(-n + S(1))), expand=True, _diff=True, _numerical=True)631 assert rubi_test(rubi_integrate((b*sec(a + b*x))**n*(c*sin(a + b*x))**(S(3)/​2), x), x, -c*(b*sec(a + b*x))**n*sqrt(c*sin(a + b*x))*Hypergeometric2F1(S(-1)/​4, -n/​S(2) + S(1)/​2, -n/​S(2) + S(3)/​2, cos(a + b*x)**S(2))*cos(a + b*x)/​(b*(-n + S(1))*(sin(a + b*x)**S(2))**(S(1)/​4)), expand=True, _diff=True, _numerical=True)632 assert rubi_test(rubi_integrate((b*sec(a + b*x))**n*sqrt(c*sin(a + b*x)), x), x, -c*(b*sec(a + b*x))**n*(sin(a + b*x)**S(2))**(S(1)/​4)*Hypergeometric2F1(S(1)/​4, -n/​S(2) + S(1)/​2, -n/​S(2) + S(3)/​2, cos(a + b*x)**S(2))*cos(a + b*x)/​(b*sqrt(c*sin(a + b*x))*(-n + S(1))), expand=True, _diff=True, _numerical=True)633 assert rubi_test(rubi_integrate((b*sec(a + b*x))**n/​sqrt(c*sin(a + b*x)), x), x, -c*(b*sec(a + b*x))**n*(sin(a + b*x)**S(2))**(S(3)/​4)*Hypergeometric2F1(S(3)/​4, -n/​S(2) + S(1)/​2, -n/​S(2) + S(3)/​2, cos(a + b*x)**S(2))*cos(a + b*x)/​(b*(c*sin(a + b*x))**(S(3)/​2)*(-n + S(1))), expand=True, _diff=True, _numerical=True)634 assert rubi_test(rubi_integrate((b*sec(a + b*x))**n/​(c*sin(a + b*x))**(S(3)/​2), x), x, -(b*sec(a + b*x))**n*(sin(a + b*x)**S(2))**(S(1)/​4)*Hypergeometric2F1(S(5)/​4, -n/​S(2) + S(1)/​2, -n/​S(2) + S(3)/​2, cos(a + b*x)**S(2))*cos(a + b*x)/​(b*c*sqrt(c*sin(a + b*x))*(-n + S(1))), expand=True, _diff=True, _numerical=True)635 assert rubi_test(rubi_integrate(sqrt(d*csc(e + f*x))*sin(e + f*x)**S(4), x), x, -S(2)*d**S(3)*cos(e + f*x)/​(S(7)*f*(d*csc(e + f*x))**(S(5)/​2)) - S(10)*d*cos(e + f*x)/​(S(21)*f*sqrt(d*csc(e + f*x))) + S(10)*sqrt(d*csc(e + f*x))*EllipticF(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))*sqrt(sin(e + f*x))/​(S(21)*f), expand=True, _diff=True, _numerical=True)636 assert rubi_test(rubi_integrate(sqrt(d*csc(e + f*x))*sin(e + f*x)**S(3), x), x, -S(2)*d**S(2)*cos(e + f*x)/​(S(5)*f*(d*csc(e + f*x))**(S(3)/​2)) + S(6)*d*EllipticE(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))/​(S(5)*f*sqrt(d*csc(e + f*x))*sqrt(sin(e + f*x))), expand=True, _diff=True, _numerical=True)637 assert rubi_test(rubi_integrate(sqrt(d*csc(e + f*x))*sin(e + f*x)**S(2), x), x, -S(2)*d*cos(e + f*x)/​(S(3)*f*sqrt(d*csc(e + f*x))) + S(2)*sqrt(d*csc(e + f*x))*EllipticF(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))*sqrt(sin(e + f*x))/​(S(3)*f), expand=True, _diff=True, _numerical=True)638 assert rubi_test(rubi_integrate(sqrt(d*csc(e + f*x))*sin(e + f*x), x), x, S(2)*d*EllipticE(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))/​(f*sqrt(d*csc(e + f*x))*sqrt(sin(e + f*x))), expand=True, _diff=True, _numerical=True)639 assert rubi_test(rubi_integrate(sqrt(d*csc(e + f*x)), x), x, S(2)*sqrt(d*csc(e + f*x))*EllipticF(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))*sqrt(sin(e + f*x))/​f, expand=True, _diff=True, _numerical=True)640 assert rubi_test(rubi_integrate(sqrt(d*csc(e + f*x))*csc(e + f*x), x), x, -S(2)*d*EllipticE(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))/​(f*sqrt(d*csc(e + f*x))*sqrt(sin(e + f*x))) - S(2)*sqrt(d*csc(e + f*x))*cos(e + f*x)/​f, expand=True, _diff=True, _numerical=True)641 assert rubi_test(rubi_integrate(sqrt(d*csc(e + f*x))*csc(e + f*x)**S(2), x), x, S(2)*sqrt(d*csc(e + f*x))*EllipticF(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))*sqrt(sin(e + f*x))/​(S(3)*f) - S(2)*(d*csc(e + f*x))**(S(3)/​2)*cos(e + f*x)/​(S(3)*d*f), expand=True, _diff=True, _numerical=True)642 assert rubi_test(rubi_integrate(sqrt(d*csc(e + f*x))*csc(e + f*x)**S(3), x), x, -S(6)*d*EllipticE(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))/​(S(5)*f*sqrt(d*csc(e + f*x))*sqrt(sin(e + f*x))) - S(6)*sqrt(d*csc(e + f*x))*cos(e + f*x)/​(S(5)*f) - S(2)*(d*csc(e + f*x))**(S(5)/​2)*cos(e + f*x)/​(S(5)*d**S(2)*f), expand=True, _diff=True, _numerical=True)643 assert rubi_test(rubi_integrate((d*csc(e + f*x))**(S(3)/​2)*sin(e + f*x)**S(5), x), x, -S(2)*d**S(4)*cos(e + f*x)/​(S(7)*f*(d*csc(e + f*x))**(S(5)/​2)) - S(10)*d**S(2)*cos(e + f*x)/​(S(21)*f*sqrt(d*csc(e + f*x))) + S(10)*d*sqrt(d*csc(e + f*x))*EllipticF(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))*sqrt(sin(e + f*x))/​(S(21)*f), expand=True, _diff=True, _numerical=True)644 assert rubi_test(rubi_integrate((d*csc(e + f*x))**(S(3)/​2)*sin(e + f*x)**S(4), x), x, -S(2)*d**S(3)*cos(e + f*x)/​(S(5)*f*(d*csc(e + f*x))**(S(3)/​2)) + S(6)*d**S(2)*EllipticE(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))/​(S(5)*f*sqrt(d*csc(e + f*x))*sqrt(sin(e + f*x))), expand=True, _diff=True, _numerical=True)645 assert rubi_test(rubi_integrate((d*csc(e + f*x))**(S(3)/​2)*sin(e + f*x)**S(3), x), x, -S(2)*d**S(2)*cos(e + f*x)/​(S(3)*f*sqrt(d*csc(e + f*x))) + S(2)*d*sqrt(d*csc(e + f*x))*EllipticF(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))*sqrt(sin(e + f*x))/​(S(3)*f), expand=True, _diff=True, _numerical=True)646 assert rubi_test(rubi_integrate((d*csc(e + f*x))**(S(3)/​2)*sin(e + f*x)**S(2), x), x, S(2)*d**S(2)*EllipticE(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))/​(f*sqrt(d*csc(e + f*x))*sqrt(sin(e + f*x))), expand=True, _diff=True, _numerical=True)647 assert rubi_test(rubi_integrate((d*csc(e + f*x))**(S(3)/​2)*sin(e + f*x), x), x, S(2)*d*sqrt(d*csc(e + f*x))*EllipticF(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))*sqrt(sin(e + f*x))/​f, expand=True, _diff=True, _numerical=True)648 assert rubi_test(rubi_integrate((d*csc(e + f*x))**(S(3)/​2), x), x, -S(2)*d**S(2)*EllipticE(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))/​(f*sqrt(d*csc(e + f*x))*sqrt(sin(e + f*x))) - S(2)*d*sqrt(d*csc(e + f*x))*cos(e + f*x)/​f, expand=True, _diff=True, _numerical=True)649 assert rubi_test(rubi_integrate((d*csc(e + f*x))**(S(3)/​2)*csc(e + f*x), x), x, S(2)*d*sqrt(d*csc(e + f*x))*EllipticF(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))*sqrt(sin(e + f*x))/​(S(3)*f) - S(2)*(d*csc(e + f*x))**(S(3)/​2)*cos(e + f*x)/​(S(3)*f), expand=True, _diff=True, _numerical=True)650 assert rubi_test(rubi_integrate((d*csc(e + f*x))**(S(3)/​2)*csc(e + f*x)**S(2), x), x, -S(6)*d**S(2)*EllipticE(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))/​(S(5)*f*sqrt(d*csc(e + f*x))*sqrt(sin(e + f*x))) - S(6)*d*sqrt(d*csc(e + f*x))*cos(e + f*x)/​(S(5)*f) - S(2)*(d*csc(e + f*x))**(S(5)/​2)*cos(e + f*x)/​(S(5)*d*f), expand=True, _diff=True, _numerical=True)651 assert rubi_test(rubi_integrate(sin(e + f*x)**S(3)/​sqrt(d*csc(e + f*x)), x), x, -S(2)*d**S(2)*cos(e + f*x)/​(S(7)*f*(d*csc(e + f*x))**(S(5)/​2)) - S(10)*cos(e + f*x)/​(S(21)*f*sqrt(d*csc(e + f*x))) + S(10)*sqrt(d*csc(e + f*x))*EllipticF(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))*sqrt(sin(e + f*x))/​(S(21)*d*f), expand=True, _diff=True, _numerical=True)652 assert rubi_test(rubi_integrate(sin(e + f*x)**S(2)/​sqrt(d*csc(e + f*x)), x), x, -S(2)*d*cos(e + f*x)/​(S(5)*f*(d*csc(e + f*x))**(S(3)/​2)) + S(6)*EllipticE(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))/​(S(5)*f*sqrt(d*csc(e + f*x))*sqrt(sin(e + f*x))), expand=True, _diff=True, _numerical=True)653 assert rubi_test(rubi_integrate(sin(e + f*x)/​sqrt(d*csc(e + f*x)), x), x, -S(2)*cos(e + f*x)/​(S(3)*f*sqrt(d*csc(e + f*x))) + S(2)*sqrt(d*csc(e + f*x))*EllipticF(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))*sqrt(sin(e + f*x))/​(S(3)*d*f), expand=True, _diff=True, _numerical=True)654 assert rubi_test(rubi_integrate(S(1)/​sqrt(d*csc(e + f*x)), x), x, S(2)*EllipticE(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))/​(f*sqrt(d*csc(e + f*x))*sqrt(sin(e + f*x))), expand=True, _diff=True, _numerical=True)655 assert rubi_test(rubi_integrate(csc(e + f*x)/​sqrt(d*csc(e + f*x)), x), x, S(2)*sqrt(d*csc(e + f*x))*EllipticF(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))*sqrt(sin(e + f*x))/​(d*f), expand=True, _diff=True, _numerical=True)656 assert rubi_test(rubi_integrate(csc(e + f*x)**S(2)/​sqrt(d*csc(e + f*x)), x), x, -S(2)*EllipticE(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))/​(f*sqrt(d*csc(e + f*x))*sqrt(sin(e + f*x))) - S(2)*sqrt(d*csc(e + f*x))*cos(e + f*x)/​(d*f), expand=True, _diff=True, _numerical=True)657 assert rubi_test(rubi_integrate(csc(e + f*x)**S(3)/​sqrt(d*csc(e + f*x)), x), x, S(2)*sqrt(d*csc(e + f*x))*EllipticF(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))*sqrt(sin(e + f*x))/​(S(3)*d*f) - S(2)*(d*csc(e + f*x))**(S(3)/​2)*cos(e + f*x)/​(S(3)*d**S(2)*f), expand=True, _diff=True, _numerical=True)658 assert rubi_test(rubi_integrate(sin(e + f*x)**S(2)/​(d*csc(e + f*x))**(S(3)/​2), x), x, -S(2)*d*cos(e + f*x)/​(S(7)*f*(d*csc(e + f*x))**(S(5)/​2)) - S(10)*cos(e + f*x)/​(S(21)*d*f*sqrt(d*csc(e + f*x))) + S(10)*sqrt(d*csc(e + f*x))*EllipticF(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))*sqrt(sin(e + f*x))/​(S(21)*d**S(2)*f), expand=True, _diff=True, _numerical=True)659 assert rubi_test(rubi_integrate(sin(e + f*x)/​(d*csc(e + f*x))**(S(3)/​2), x), x, -S(2)*cos(e + f*x)/​(S(5)*f*(d*csc(e + f*x))**(S(3)/​2)) + S(6)*EllipticE(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))/​(S(5)*d*f*sqrt(d*csc(e + f*x))*sqrt(sin(e + f*x))), expand=True, _diff=True, _numerical=True)660 assert rubi_test(rubi_integrate((d*csc(e + f*x))**(S(-3)/​2), x), x, -S(2)*cos(e + f*x)/​(S(3)*d*f*sqrt(d*csc(e + f*x))) + S(2)*sqrt(d*csc(e + f*x))*EllipticF(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))*sqrt(sin(e + f*x))/​(S(3)*d**S(2)*f), expand=True, _diff=True, _numerical=True)661 assert rubi_test(rubi_integrate(csc(e + f*x)/​(d*csc(e + f*x))**(S(3)/​2), x), x, S(2)*EllipticE(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))/​(d*f*sqrt(d*csc(e + f*x))*sqrt(sin(e + f*x))), expand=True, _diff=True, _numerical=True)662 assert rubi_test(rubi_integrate(csc(e + f*x)**S(2)/​(d*csc(e + f*x))**(S(3)/​2), x), x, S(2)*sqrt(d*csc(e + f*x))*EllipticF(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))*sqrt(sin(e + f*x))/​(d**S(2)*f), expand=True, _diff=True, _numerical=True)663 assert rubi_test(rubi_integrate(csc(e + f*x)**S(3)/​(d*csc(e + f*x))**(S(3)/​2), x), x, -S(2)*EllipticE(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))/​(d*f*sqrt(d*csc(e + f*x))*sqrt(sin(e + f*x))) - S(2)*sqrt(d*csc(e + f*x))*cos(e + f*x)/​(d**S(2)*f), expand=True, _diff=True, _numerical=True)664 assert rubi_test(rubi_integrate(csc(e + f*x)**S(4)/​(d*csc(e + f*x))**(S(3)/​2), x), x, S(2)*sqrt(d*csc(e + f*x))*EllipticF(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))*sqrt(sin(e + f*x))/​(S(3)*d**S(2)*f) - S(2)*(d*csc(e + f*x))**(S(3)/​2)*cos(e + f*x)/​(S(3)*d**S(3)*f), expand=True, _diff=True, _numerical=True)665 assert rubi_test(rubi_integrate(csc(e + f*x)**S(5)/​(d*csc(e + f*x))**(S(3)/​2), x), x, -S(6)*EllipticE(-Pi/​S(4) + e/​S(2) + f*x/​S(2), S(2))/​(S(5)*d*f*sqrt(d*csc(e + f*x))*sqrt(sin(e + f*x))) - S(6)*sqrt(d*csc(e + f*x))*cos(e + f*x)/​(S(5)*d**S(2)*f) - S(2)*(d*csc(e + f*x))**(S(5)/​2)*cos(e + f*x)/​(S(5)*d**S(4)*f), expand=True, _diff=True, _numerical=True)...

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