Optimal. Leaf size=90 \[ \frac {13 x}{2 a^3}+\frac {152 \cos (x)}{15 a^3}+\frac {76 \sin ^2(x) \cos (x)}{15 \left (a^3 \sin (x)+a^3\right )}-\frac {13 \sin (x) \cos (x)}{2 a^3}+\frac {\sin ^4(x) \cos (x)}{5 (a \sin (x)+a)^3}+\frac {11 \sin ^3(x) \cos (x)}{15 a (a \sin (x)+a)^2} \]
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Rubi [A] time = 0.21, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2765, 2977, 2734} \[ \frac {13 x}{2 a^3}+\frac {152 \cos (x)}{15 a^3}+\frac {76 \sin ^2(x) \cos (x)}{15 \left (a^3 \sin (x)+a^3\right )}-\frac {13 \sin (x) \cos (x)}{2 a^3}+\frac {\sin ^4(x) \cos (x)}{5 (a \sin (x)+a)^3}+\frac {11 \sin ^3(x) \cos (x)}{15 a (a \sin (x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 2765
Rule 2977
Rubi steps
\begin {align*} \int \frac {\sin ^5(x)}{(a+a \sin (x))^3} \, dx &=\frac {\cos (x) \sin ^4(x)}{5 (a+a \sin (x))^3}-\frac {\int \frac {\sin ^3(x) (4 a-7 a \sin (x))}{(a+a \sin (x))^2} \, dx}{5 a^2}\\ &=\frac {\cos (x) \sin ^4(x)}{5 (a+a \sin (x))^3}+\frac {11 \cos (x) \sin ^3(x)}{15 a (a+a \sin (x))^2}-\frac {\int \frac {\sin ^2(x) \left (33 a^2-43 a^2 \sin (x)\right )}{a+a \sin (x)} \, dx}{15 a^4}\\ &=\frac {\cos (x) \sin ^4(x)}{5 (a+a \sin (x))^3}+\frac {11 \cos (x) \sin ^3(x)}{15 a (a+a \sin (x))^2}+\frac {76 \cos (x) \sin ^2(x)}{15 \left (a^3+a^3 \sin (x)\right )}-\frac {\int \sin (x) \left (152 a^3-195 a^3 \sin (x)\right ) \, dx}{15 a^6}\\ &=\frac {13 x}{2 a^3}+\frac {152 \cos (x)}{15 a^3}-\frac {13 \cos (x) \sin (x)}{2 a^3}+\frac {\cos (x) \sin ^4(x)}{5 (a+a \sin (x))^3}+\frac {11 \cos (x) \sin ^3(x)}{15 a (a+a \sin (x))^2}+\frac {76 \cos (x) \sin ^2(x)}{15 \left (a^3+a^3 \sin (x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 170, normalized size = 1.89 \[ \frac {\left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \left (-24 \sin \left (\frac {x}{2}\right )+390 x \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^5+180 \cos (x) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^5-15 \sin (2 x) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^5-1016 \sin \left (\frac {x}{2}\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^4-92 \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^3+184 \sin \left (\frac {x}{2}\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^2+12 \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )\right )}{60 (a \sin (x)+a)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 145, normalized size = 1.61 \[ \frac {15 \, \cos \relax (x)^{5} + {\left (195 \, x + 449\right )} \cos \relax (x)^{3} + 60 \, \cos \relax (x)^{4} + {\left (585 \, x - 358\right )} \cos \relax (x)^{2} - 6 \, {\left (65 \, x + 128\right )} \cos \relax (x) - {\left (15 \, \cos \relax (x)^{4} - {\left (195 \, x - 404\right )} \cos \relax (x)^{2} - 45 \, \cos \relax (x)^{3} + 6 \, {\left (65 \, x + 127\right )} \cos \relax (x) + 780 \, x - 6\right )} \sin \relax (x) - 780 \, x - 6}{30 \, {\left (a^{3} \cos \relax (x)^{3} + 3 \, a^{3} \cos \relax (x)^{2} - 2 \, a^{3} \cos \relax (x) - 4 \, a^{3} + {\left (a^{3} \cos \relax (x)^{2} - 2 \, a^{3} \cos \relax (x) - 4 \, a^{3}\right )} \sin \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 88, normalized size = 0.98 \[ \frac {13 \, x}{2 \, a^{3}} + \frac {\tan \left (\frac {1}{2} \, x\right )^{3} + 6 \, \tan \left (\frac {1}{2} \, x\right )^{2} - \tan \left (\frac {1}{2} \, x\right ) + 6}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{2} a^{3}} + \frac {2 \, {\left (90 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 405 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 665 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 445 \, \tan \left (\frac {1}{2} \, x\right ) + 107\right )}}{15 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 152, normalized size = 1.69 \[ \frac {\tan ^{3}\left (\frac {x}{2}\right )}{a^{3} \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+\frac {6 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a^{3} \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}-\frac {\tan \left (\frac {x}{2}\right )}{a^{3} \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+\frac {6}{a^{3} \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+\frac {13 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a^{3}}+\frac {8}{5 a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {4}{a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {4}{3 a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {6}{a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {12}{a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.77, size = 252, normalized size = 2.80 \[ \frac {\frac {1325 \, \sin \relax (x)}{\cos \relax (x) + 1} + \frac {2673 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {3805 \, \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {4329 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {3575 \, \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}} + \frac {2275 \, \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} + \frac {975 \, \sin \relax (x)^{7}}{{\left (\cos \relax (x) + 1\right )}^{7}} + \frac {195 \, \sin \relax (x)^{8}}{{\left (\cos \relax (x) + 1\right )}^{8}} + 304}{15 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \relax (x)}{\cos \relax (x) + 1} + \frac {12 \, a^{3} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {20 \, a^{3} \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {26 \, a^{3} \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {26 \, a^{3} \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}} + \frac {20 \, a^{3} \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} + \frac {12 \, a^{3} \sin \relax (x)^{7}}{{\left (\cos \relax (x) + 1\right )}^{7}} + \frac {5 \, a^{3} \sin \relax (x)^{8}}{{\left (\cos \relax (x) + 1\right )}^{8}} + \frac {a^{3} \sin \relax (x)^{9}}{{\left (\cos \relax (x) + 1\right )}^{9}}\right )}} + \frac {13 \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.67, size = 93, normalized size = 1.03 \[ \frac {13\,x}{2\,a^3}+\frac {13\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8+65\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7+\frac {455\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6}{3}+\frac {715\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5}{3}+\frac {1443\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{5}+\frac {761\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{3}+\frac {891\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}+\frac {265\,\mathrm {tan}\left (\frac {x}{2}\right )}{3}+\frac {304}{15}}{a^3\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}^2\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 33.59, size = 2259, normalized size = 25.10 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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