Integrand size = 13, antiderivative size = 86 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^3} \, dx=-\frac {13 \text {arctanh}(\cos (x))}{2 a^3}+\frac {152 \cot (x)}{15 a^3}-\frac {13 \cot (x) \csc (x)}{2 a^3}+\frac {\cot (x) \csc (x)}{5 (a+a \sin (x))^3}+\frac {11 \cot (x) \csc (x)}{15 a (a+a \sin (x))^2}+\frac {76 \cot (x) \csc (x)}{15 \left (a^3+a^3 \sin (x)\right )} \]
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Time = 0.16 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2845, 3057, 2827, 3853, 3855, 3852, 8} \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^3} \, dx=-\frac {13 \text {arctanh}(\cos (x))}{2 a^3}+\frac {152 \cot (x)}{15 a^3}-\frac {13 \cot (x) \csc (x)}{2 a^3}+\frac {76 \cot (x) \csc (x)}{15 \left (a^3 \sin (x)+a^3\right )}+\frac {11 \cot (x) \csc (x)}{15 a (a \sin (x)+a)^2}+\frac {\cot (x) \csc (x)}{5 (a \sin (x)+a)^3} \]
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Rule 8
Rule 2827
Rule 2845
Rule 3057
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\cot (x) \csc (x)}{5 (a+a \sin (x))^3}+\frac {\int \frac {\csc ^3(x) (7 a-4 a \sin (x))}{(a+a \sin (x))^2} \, dx}{5 a^2} \\ & = \frac {\cot (x) \csc (x)}{5 (a+a \sin (x))^3}+\frac {11 \cot (x) \csc (x)}{15 a (a+a \sin (x))^2}+\frac {\int \frac {\csc ^3(x) \left (43 a^2-33 a^2 \sin (x)\right )}{a+a \sin (x)} \, dx}{15 a^4} \\ & = \frac {\cot (x) \csc (x)}{5 (a+a \sin (x))^3}+\frac {11 \cot (x) \csc (x)}{15 a (a+a \sin (x))^2}+\frac {76 \cot (x) \csc (x)}{15 \left (a^3+a^3 \sin (x)\right )}+\frac {\int \csc ^3(x) \left (195 a^3-152 a^3 \sin (x)\right ) \, dx}{15 a^6} \\ & = \frac {\cot (x) \csc (x)}{5 (a+a \sin (x))^3}+\frac {11 \cot (x) \csc (x)}{15 a (a+a \sin (x))^2}+\frac {76 \cot (x) \csc (x)}{15 \left (a^3+a^3 \sin (x)\right )}-\frac {152 \int \csc ^2(x) \, dx}{15 a^3}+\frac {13 \int \csc ^3(x) \, dx}{a^3} \\ & = -\frac {13 \cot (x) \csc (x)}{2 a^3}+\frac {\cot (x) \csc (x)}{5 (a+a \sin (x))^3}+\frac {11 \cot (x) \csc (x)}{15 a (a+a \sin (x))^2}+\frac {76 \cot (x) \csc (x)}{15 \left (a^3+a^3 \sin (x)\right )}+\frac {13 \int \csc (x) \, dx}{2 a^3}+\frac {152 \text {Subst}(\int 1 \, dx,x,\cot (x))}{15 a^3} \\ & = -\frac {13 \text {arctanh}(\cos (x))}{2 a^3}+\frac {152 \cot (x)}{15 a^3}-\frac {13 \cot (x) \csc (x)}{2 a^3}+\frac {\cot (x) \csc (x)}{5 (a+a \sin (x))^3}+\frac {11 \cot (x) \csc (x)}{15 a (a+a \sin (x))^2}+\frac {76 \cot (x) \csc (x)}{15 \left (a^3+a^3 \sin (x)\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(247\) vs. \(2(86)=172\).
Time = 1.26 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.87 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^3} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (-48 \sin \left (\frac {x}{2}\right )-15 \left (1+\cot \left (\frac {x}{2}\right )\right )^5 \sin ^3\left (\frac {x}{2}\right )+24 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )-272 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2+136 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3-1712 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^4+180 \cot \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5-780 \log \left (\cos \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5+780 \log \left (\sin \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5-180 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5 \tan \left (\frac {x}{2}\right )+15 \cos ^3\left (\frac {x}{2}\right ) \left (1+\tan \left (\frac {x}{2}\right )\right )^5\right )}{120 a^3 (1+\sin (x))^3} \]
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Time = 0.65 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.98
method | result | size |
parallelrisch | \(\frac {-883+390 \left (-1+\cos \left (2 x \right )\right ) \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )-96 \left (\sec ^{5}\left (x \right )\right )+96 \tan \left (x \right ) \left (\sec ^{4}\left (x \right )\right )-104 \left (\sec ^{3}\left (x \right )\right )+152 \tan \left (x \right ) \left (\sec ^{2}\left (x \right )\right )+780 \cos \left (x \right )-520 \sec \left (x \right )+608 \tan \left (x \right )+883 \cos \left (2 x \right )-608 \sin \left (2 x \right )}{60 a^{3} \left (-1+\cos \left (2 x \right )\right )}\) | \(84\) |
default | \(\frac {\frac {\left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}-6 \tan \left (\frac {x}{2}\right )-\frac {1}{2 \tan \left (\frac {x}{2}\right )^{2}}+\frac {6}{\tan \left (\frac {x}{2}\right )}+26 \ln \left (\tan \left (\frac {x}{2}\right )\right )+\frac {32}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {16}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {112}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {40}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {80}{\tan \left (\frac {x}{2}\right )+1}}{4 a^{3}}\) | \(94\) |
risch | \(\frac {975 i {\mathrm e}^{7 i x}+195 \,{\mathrm e}^{8 i x}-3575 i {\mathrm e}^{5 i x}-2275 \,{\mathrm e}^{6 i x}+3805 i {\mathrm e}^{3 i x}+4329 \,{\mathrm e}^{4 i x}-1325 i {\mathrm e}^{i x}-2673 \,{\mathrm e}^{2 i x}+304}{15 \left ({\mathrm e}^{2 i x}-1\right )^{2} \left ({\mathrm e}^{i x}+i\right )^{5} a^{3}}+\frac {13 \ln \left ({\mathrm e}^{i x}-1\right )}{2 a^{3}}-\frac {13 \ln \left ({\mathrm e}^{i x}+1\right )}{2 a^{3}}\) | \(114\) |
norman | \(\frac {\frac {39 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}-\frac {1}{8 a}+\frac {7 \tan \left (\frac {x}{2}\right )}{8 a}-\frac {7 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{8 a}+\frac {\tan ^{9}\left (\frac {x}{2}\right )}{8 a}+\frac {251 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {649 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{6 a}+\frac {883 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{30 a}+\frac {1013 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{6 a}}{\tan \left (\frac {x}{2}\right )^{2} a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {13 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{2 a^{3}}\) | \(122\) |
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Leaf count of result is larger than twice the leaf count of optimal. 276 vs. \(2 (74) = 148\).
Time = 0.30 (sec) , antiderivative size = 276, normalized size of antiderivative = 3.21 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^3} \, dx=\frac {608 \, \cos \left (x\right )^{5} - 826 \, \cos \left (x\right )^{4} - 2174 \, \cos \left (x\right )^{3} + 784 \, \cos \left (x\right )^{2} - 195 \, {\left (\cos \left (x\right )^{5} + 3 \, \cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{3} - 7 \, \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) + 2 \, \cos \left (x\right ) + 4\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 195 \, {\left (\cos \left (x\right )^{5} + 3 \, \cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{3} - 7 \, \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) + 2 \, \cos \left (x\right ) + 4\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 2 \, {\left (304 \, \cos \left (x\right )^{4} + 717 \, \cos \left (x\right )^{3} - 370 \, \cos \left (x\right )^{2} - 762 \, \cos \left (x\right ) + 6\right )} \sin \left (x\right ) + 1536 \, \cos \left (x\right ) + 12}{60 \, {\left (a^{3} \cos \left (x\right )^{5} + 3 \, a^{3} \cos \left (x\right )^{4} - 3 \, a^{3} \cos \left (x\right )^{3} - 7 \, a^{3} \cos \left (x\right )^{2} + 2 \, a^{3} \cos \left (x\right ) + 4 \, a^{3} + {\left (a^{3} \cos \left (x\right )^{4} - 2 \, a^{3} \cos \left (x\right )^{3} - 5 \, a^{3} \cos \left (x\right )^{2} + 2 \, a^{3} \cos \left (x\right ) + 4 \, a^{3}\right )} \sin \left (x\right )\right )}} \]
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\[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^3} \, dx=\frac {\int \frac {\csc ^{3}{\left (x \right )}}{\sin ^{3}{\left (x \right )} + 3 \sin ^{2}{\left (x \right )} + 3 \sin {\left (x \right )} + 1}\, dx}{a^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (74) = 148\).
Time = 0.21 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.43 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^3} \, dx=\frac {\frac {105 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {2782 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {9410 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {13645 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {9285 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {2580 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - 15}{120 \, {\left (\frac {a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {5 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {10 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {10 \, a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {5 \, a^{3} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {a^{3} \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}}\right )}} - \frac {\frac {12 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}}{8 \, a^{3}} + \frac {13 \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{2 \, a^{3}} \]
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Time = 0.31 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.27 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^3} \, dx=\frac {13 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{2 \, a^{3}} - \frac {78 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 12 \, \tan \left (\frac {1}{2} \, x\right ) + 1}{8 \, a^{3} \tan \left (\frac {1}{2} \, x\right )^{2}} + \frac {a^{3} \tan \left (\frac {1}{2} \, x\right )^{2} - 12 \, a^{3} \tan \left (\frac {1}{2} \, x\right )}{8 \, a^{6}} + \frac {2 \, {\left (150 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 525 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 745 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 485 \, \tan \left (\frac {1}{2} \, x\right ) + 127\right )}}{15 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \]
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Time = 6.08 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.13 \[ \int \frac {\csc ^3(x)}{(a+a \sin (x))^3} \, dx=\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8\,a^3}-\frac {3\,\mathrm {tan}\left (\frac {x}{2}\right )}{2\,a^3}+\frac {13\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{2\,a^3}+\frac {\frac {43\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6}{2}+\frac {619\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5}{8}+\frac {2729\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{24}+\frac {941\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{12}+\frac {1391\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{60}+\frac {7\,\mathrm {tan}\left (\frac {x}{2}\right )}{8}-\frac {1}{8}}{a^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \]
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