Integrand size = 13, antiderivative size = 90 \[ \int \frac {\sin ^5(x)}{(a+a \sin (x))^3} \, dx=\frac {13 x}{2 a^3}+\frac {152 \cos (x)}{15 a^3}-\frac {13 \cos (x) \sin (x)}{2 a^3}+\frac {\cos (x) \sin ^4(x)}{5 (a+a \sin (x))^3}+\frac {11 \cos (x) \sin ^3(x)}{15 a (a+a \sin (x))^2}+\frac {76 \cos (x) \sin ^2(x)}{15 \left (a^3+a^3 \sin (x)\right )} \]
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Time = 0.14 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2844, 3056, 2813} \[ \int \frac {\sin ^5(x)}{(a+a \sin (x))^3} \, dx=\frac {13 x}{2 a^3}+\frac {152 \cos (x)}{15 a^3}+\frac {76 \sin ^2(x) \cos (x)}{15 \left (a^3 \sin (x)+a^3\right )}-\frac {13 \sin (x) \cos (x)}{2 a^3}+\frac {\sin ^4(x) \cos (x)}{5 (a \sin (x)+a)^3}+\frac {11 \sin ^3(x) \cos (x)}{15 a (a \sin (x)+a)^2} \]
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Rule 2813
Rule 2844
Rule 3056
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.12 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.89 \[ \int \frac {\sin ^5(x)}{(a+a \sin (x))^3} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (-24 \sin \left (\frac {x}{2}\right )+12 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+184 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2-92 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3-1016 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^4+390 x \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5+180 \cos (x) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5-15 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5 \sin (2 x)\right )}{60 (a+a \sin (x))^3} \]
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Time = 0.61 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.63
method | result | size |
parallelrisch | \(\frac {-48 \tan \left (x \right ) \left (\sec ^{4}\left (x \right )\right )+48 \left (\sec ^{5}\left (x \right )\right )+196 \tan \left (x \right ) \left (\sec ^{2}\left (x \right )\right )-220 \left (\sec ^{3}\left (x \right )\right )-508 \tan \left (x \right )+180 \cos \left (x \right )-15 \sin \left (2 x \right )+600 \sec \left (x \right )+390 x +608}{60 a^{3}}\) | \(57\) |
risch | \(\frac {13 x}{2 a^{3}}+\frac {i {\mathrm e}^{2 i x}}{8 a^{3}}+\frac {3 \,{\mathrm e}^{i x}}{2 a^{3}}+\frac {3 \,{\mathrm e}^{-i x}}{2 a^{3}}-\frac {i {\mathrm e}^{-2 i x}}{8 a^{3}}+\frac {70 i {\mathrm e}^{3 i x}+20 \,{\mathrm e}^{4 i x}-\frac {194 i {\mathrm e}^{i x}}{3}-\frac {298 \,{\mathrm e}^{2 i x}}{3}+\frac {254}{15}}{\left ({\mathrm e}^{i x}+i\right )^{5} a^{3}}\) | \(97\) |
default | \(\frac {\frac {8}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {4}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {6}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {12}{\tan \left (\frac {x}{2}\right )+1}+\frac {4 \left (\frac {\left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{4}+\frac {3 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}-\frac {\tan \left (\frac {x}{2}\right )}{4}+\frac {3}{2}\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{2}}+13 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a^{3}}\) | \(100\) |
norman | \(\frac {\frac {13 x \left (\tan ^{15}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {455 x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {1313 x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {304}{15 a}+\frac {65 x \tan \left (\frac {x}{2}\right )}{2 a}+\frac {455 x \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {3491 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {239 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}+\frac {1556 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {4717 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {1755 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {845 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {65 x \left (\tan ^{14}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {195 x \left (\tan ^{13}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {1755 x \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {2015 x \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {1300 \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {572 \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {65 \left (\tan ^{13}\left (\frac {x}{2}\right )\right )}{a}+\frac {13 \left (\tan ^{14}\left (\frac {x}{2}\right )\right )}{a}+\frac {265 \tan \left (\frac {x}{2}\right )}{3 a}+\frac {884 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}+\frac {2015 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {195 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {3793 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {845 x \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {13 x}{2 a}+\frac {3913 \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{5 a}+\frac {1512 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{a}+\frac {4888 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {1313 x \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{2 a}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{5} a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}\) | \(365\) |
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Time = 0.27 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.61 \[ \int \frac {\sin ^5(x)}{(a+a \sin (x))^3} \, dx=\frac {15 \, \cos \left (x\right )^{5} + {\left (195 \, x + 449\right )} \cos \left (x\right )^{3} + 60 \, \cos \left (x\right )^{4} + {\left (585 \, x - 358\right )} \cos \left (x\right )^{2} - 6 \, {\left (65 \, x + 128\right )} \cos \left (x\right ) - {\left (15 \, \cos \left (x\right )^{4} - {\left (195 \, x - 404\right )} \cos \left (x\right )^{2} - 45 \, \cos \left (x\right )^{3} + 6 \, {\left (65 \, x + 127\right )} \cos \left (x\right ) + 780 \, x - 6\right )} \sin \left (x\right ) - 780 \, x - 6}{30 \, {\left (a^{3} \cos \left (x\right )^{3} + 3 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3} + {\left (a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3}\right )} \sin \left (x\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2259 vs. \(2 (95) = 190\).
Time = 11.37 (sec) , antiderivative size = 2259, normalized size of antiderivative = 25.10 \[ \int \frac {\sin ^5(x)}{(a+a \sin (x))^3} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (78) = 156\).
Time = 0.31 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.80 \[ \int \frac {\sin ^5(x)}{(a+a \sin (x))^3} \, dx=\frac {\frac {1325 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {2673 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {3805 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {4329 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {3575 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {2275 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {975 \, \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} + \frac {195 \, \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + 304}{15 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {12 \, a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {20 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {26 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {26 \, a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {20 \, a^{3} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {12 \, a^{3} \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} + \frac {5 \, a^{3} \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + \frac {a^{3} \sin \left (x\right )^{9}}{{\left (\cos \left (x\right ) + 1\right )}^{9}}\right )}} + \frac {13 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{3}} \]
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Time = 0.31 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.98 \[ \int \frac {\sin ^5(x)}{(a+a \sin (x))^3} \, dx=\frac {13 \, x}{2 \, a^{3}} + \frac {\tan \left (\frac {1}{2} \, x\right )^{3} + 6 \, \tan \left (\frac {1}{2} \, x\right )^{2} - \tan \left (\frac {1}{2} \, x\right ) + 6}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{2} a^{3}} + \frac {2 \, {\left (90 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 405 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 665 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 445 \, \tan \left (\frac {1}{2} \, x\right ) + 107\right )}}{15 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \]
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Time = 6.35 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.03 \[ \int \frac {\sin ^5(x)}{(a+a \sin (x))^3} \, dx=\frac {13\,x}{2\,a^3}+\frac {13\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8+65\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7+\frac {455\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6}{3}+\frac {715\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5}{3}+\frac {1443\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{5}+\frac {761\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{3}+\frac {891\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}+\frac {265\,\mathrm {tan}\left (\frac {x}{2}\right )}{3}+\frac {304}{15}}{a^3\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}^2\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \]
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