Optimal. Leaf size=120 \[ \frac {\cos (c+d x)}{a^2 d}-\frac {2 \tan ^5(c+d x)}{5 a^2 d}+\frac {2 \tan ^3(c+d x)}{3 a^2 d}-\frac {2 \tan (c+d x)}{a^2 d}+\frac {2 \sec ^5(c+d x)}{5 a^2 d}-\frac {5 \sec ^3(c+d x)}{3 a^2 d}+\frac {4 \sec (c+d x)}{a^2 d}+\frac {2 x}{a^2} \]
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Rubi [A] time = 0.28, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2875, 2873, 2606, 194, 3473, 8, 2590, 270} \[ \frac {\cos (c+d x)}{a^2 d}-\frac {2 \tan ^5(c+d x)}{5 a^2 d}+\frac {2 \tan ^3(c+d x)}{3 a^2 d}-\frac {2 \tan (c+d x)}{a^2 d}+\frac {2 \sec ^5(c+d x)}{5 a^2 d}-\frac {5 \sec ^3(c+d x)}{3 a^2 d}+\frac {4 \sec (c+d x)}{a^2 d}+\frac {2 x}{a^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 194
Rule 270
Rule 2590
Rule 2606
Rule 2873
Rule 2875
Rule 3473
Rubi steps
\begin {align*} \int \frac {\sin ^3(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \sec (c+d x) (a-a \sin (c+d x))^2 \tan ^5(c+d x) \, dx}{a^4}\\ &=\frac {\int \left (a^2 \sec (c+d x) \tan ^5(c+d x)-2 a^2 \tan ^6(c+d x)+a^2 \sin (c+d x) \tan ^6(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \sec (c+d x) \tan ^5(c+d x) \, dx}{a^2}+\frac {\int \sin (c+d x) \tan ^6(c+d x) \, dx}{a^2}-\frac {2 \int \tan ^6(c+d x) \, dx}{a^2}\\ &=-\frac {2 \tan ^5(c+d x)}{5 a^2 d}+\frac {2 \int \tan ^4(c+d x) \, dx}{a^2}-\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^3}{x^6} \, dx,x,\cos (c+d x)\right )}{a^2 d}+\frac {\operatorname {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac {2 \tan ^3(c+d x)}{3 a^2 d}-\frac {2 \tan ^5(c+d x)}{5 a^2 d}-\frac {2 \int \tan ^2(c+d x) \, dx}{a^2}-\frac {\operatorname {Subst}\left (\int \left (-1+\frac {1}{x^6}-\frac {3}{x^4}+\frac {3}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}+\frac {\operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac {\cos (c+d x)}{a^2 d}+\frac {4 \sec (c+d x)}{a^2 d}-\frac {5 \sec ^3(c+d x)}{3 a^2 d}+\frac {2 \sec ^5(c+d x)}{5 a^2 d}-\frac {2 \tan (c+d x)}{a^2 d}+\frac {2 \tan ^3(c+d x)}{3 a^2 d}-\frac {2 \tan ^5(c+d x)}{5 a^2 d}+\frac {2 \int 1 \, dx}{a^2}\\ &=\frac {2 x}{a^2}+\frac {\cos (c+d x)}{a^2 d}+\frac {4 \sec (c+d x)}{a^2 d}-\frac {5 \sec ^3(c+d x)}{3 a^2 d}+\frac {2 \sec ^5(c+d x)}{5 a^2 d}-\frac {2 \tan (c+d x)}{a^2 d}+\frac {2 \tan ^3(c+d x)}{3 a^2 d}-\frac {2 \tan ^5(c+d x)}{5 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.57, size = 148, normalized size = 1.23 \[ \frac {\sec (c+d x) (400 \sin (c+d x)+480 c \sin (2 (c+d x))+480 d x \sin (2 (c+d x))-796 \sin (2 (c+d x))+304 \sin (3 (c+d x))+(600 c+600 d x-995) \cos (c+d x)+376 \cos (2 (c+d x))-120 c \cos (3 (c+d x))-120 d x \cos (3 (c+d x))+199 \cos (3 (c+d x))-30 \cos (4 (c+d x))+550)}{240 a^2 d (\sin (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 122, normalized size = 1.02 \[ \frac {30 \, d x \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )^{4} - 60 \, d x \cos \left (d x + c\right ) - 62 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (30 \, d x \cos \left (d x + c\right ) + 38 \, \cos \left (d x + c\right )^{2} + 3\right )} \sin \left (d x + c\right ) - 9}{15 \, {\left (a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 151, normalized size = 1.26 \[ \frac {\frac {120 \, {\left (d x + c\right )}}{a^{2}} - \frac {15 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )} a^{2}} + \frac {255 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1170 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1310 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 313}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.44, size = 169, normalized size = 1.41 \[ -\frac {1}{4 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2}{a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {4 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}+\frac {4}{5 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {2}{a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {1}{3 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {5}{2 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {17}{4 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 335, normalized size = 2.79 \[ \frac {4 \, {\left (\frac {\frac {97 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {108 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {27 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {40 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {85 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {60 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + 28}{a^{2} + \frac {4 \, a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {4 \, a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {6 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} + \frac {15 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 15.38, size = 156, normalized size = 1.30 \[ \frac {2\,x}{a^2}-\frac {-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {68\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}-\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {36\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{5}+\frac {144\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {388\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{15}+\frac {112}{15}}{a^2\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^5\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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