Integrand size = 13, antiderivative size = 65 \[ \int \frac {\csc ^2(x)}{(a+a \sin (x))^3} \, dx=\frac {3 \text {arctanh}(\cos (x))}{a^3}-\frac {24 \cot (x)}{5 a^3}+\frac {\cot (x)}{5 (a+a \sin (x))^3}+\frac {3 \cot (x)}{5 a (a+a \sin (x))^2}+\frac {3 \cot (x)}{a^3+a^3 \sin (x)} \]
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Time = 0.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2845, 3057, 2827, 3852, 8, 3855} \[ \int \frac {\csc ^2(x)}{(a+a \sin (x))^3} \, dx=\frac {3 \text {arctanh}(\cos (x))}{a^3}-\frac {24 \cot (x)}{5 a^3}+\frac {3 \cot (x)}{a^3 \sin (x)+a^3}+\frac {3 \cot (x)}{5 a (a \sin (x)+a)^2}+\frac {\cot (x)}{5 (a \sin (x)+a)^3} \]
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Rule 8
Rule 2827
Rule 2845
Rule 3057
Rule 3852
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\cot (x)}{5 (a+a \sin (x))^3}+\frac {\int \frac {\csc ^2(x) (6 a-3 a \sin (x))}{(a+a \sin (x))^2} \, dx}{5 a^2} \\ & = \frac {\cot (x)}{5 (a+a \sin (x))^3}+\frac {3 \cot (x)}{5 a (a+a \sin (x))^2}+\frac {\int \frac {\csc ^2(x) \left (27 a^2-18 a^2 \sin (x)\right )}{a+a \sin (x)} \, dx}{15 a^4} \\ & = \frac {\cot (x)}{5 (a+a \sin (x))^3}+\frac {3 \cot (x)}{5 a (a+a \sin (x))^2}+\frac {3 \cot (x)}{a^3+a^3 \sin (x)}+\frac {\int \csc ^2(x) \left (72 a^3-45 a^3 \sin (x)\right ) \, dx}{15 a^6} \\ & = \frac {\cot (x)}{5 (a+a \sin (x))^3}+\frac {3 \cot (x)}{5 a (a+a \sin (x))^2}+\frac {3 \cot (x)}{a^3+a^3 \sin (x)}-\frac {3 \int \csc (x) \, dx}{a^3}+\frac {24 \int \csc ^2(x) \, dx}{5 a^3} \\ & = \frac {3 \text {arctanh}(\cos (x))}{a^3}+\frac {\cot (x)}{5 (a+a \sin (x))^3}+\frac {3 \cot (x)}{5 a (a+a \sin (x))^2}+\frac {3 \cot (x)}{a^3+a^3 \sin (x)}-\frac {24 \text {Subst}(\int 1 \, dx,x,\cot (x))}{5 a^3} \\ & = \frac {3 \text {arctanh}(\cos (x))}{a^3}-\frac {24 \cot (x)}{5 a^3}+\frac {\cot (x)}{5 (a+a \sin (x))^3}+\frac {3 \cot (x)}{5 a (a+a \sin (x))^2}+\frac {3 \cot (x)}{a^3+a^3 \sin (x)} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(206\) vs. \(2(65)=130\).
Time = 1.13 (sec) , antiderivative size = 206, normalized size of antiderivative = 3.17 \[ \int \frac {\csc ^2(x)}{(a+a \sin (x))^3} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (4 \sin \left (\frac {x}{2}\right )-2 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+16 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2-8 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3+76 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^4-5 \cot \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5+30 \log \left (\cos \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5-30 \log \left (\sin \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5+5 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5 \tan \left (\frac {x}{2}\right )\right )}{10 (a+a \sin (x))^3} \]
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Time = 0.54 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.17
method | result | size |
default | \(\frac {\tan \left (\frac {x}{2}\right )-\frac {16}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {8}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {16}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {16}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {24}{\tan \left (\frac {x}{2}\right )+1}-\frac {1}{\tan \left (\frac {x}{2}\right )}-6 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{2 a^{3}}\) | \(76\) |
risch | \(-\frac {2 \left (-160 \,{\mathrm e}^{4 i x}+75 i {\mathrm e}^{5 i x}+189 \,{\mathrm e}^{2 i x}-200 i {\mathrm e}^{3 i x}-24+105 i {\mathrm e}^{i x}+15 \,{\mathrm e}^{6 i x}\right )}{5 \left ({\mathrm e}^{2 i x}-1\right ) \left ({\mathrm e}^{i x}+i\right )^{5} a^{3}}-\frac {3 \ln \left ({\mathrm e}^{i x}-1\right )}{a^{3}}+\frac {3 \ln \left ({\mathrm e}^{i x}+1\right )}{a^{3}}\) | \(99\) |
norman | \(\frac {-\frac {53 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}-\frac {1}{2 a}+\frac {\tan ^{7}\left (\frac {x}{2}\right )}{2 a}-\frac {20 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}-\frac {125 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{2 a}-\frac {73 \tan \left (\frac {x}{2}\right )}{5 a}-\frac {167 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{2 a}}{\tan \left (\frac {x}{2}\right ) a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {3 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{3}}\) | \(100\) |
parallelrisch | \(\frac {-36 \left (\tan ^{3}\left (x \right )\right ) \left (\sec ^{2}\left (x \right )\right )+16 \left (\tan ^{2}\left (x \right )\right ) \left (\sec ^{3}\left (x \right )\right )+20 \tan \left (x \right ) \left (\sec ^{4}\left (x \right )\right )-48 \left (\tan ^{3}\left (x \right )\right )+20 \left (\tan ^{2}\left (x \right )\right ) \sec \left (x \right )-30 \ln \left (\csc \left (x \right )-\cot \left (x \right )\right ) \cos \left (2 x \right )+60 \tan \left (x \right ) \sin \left (x \right )+48 \tan \left (x \right ) \cos \left (2 x \right )+30 \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )-48 \tan \left (x \right )-73 \cos \left (2 x \right )+73}{10 a^{3} \left (-1+\cos \left (2 x \right )\right )}\) | \(106\) |
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Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (59) = 118\).
Time = 0.27 (sec) , antiderivative size = 225, normalized size of antiderivative = 3.46 \[ \int \frac {\csc ^2(x)}{(a+a \sin (x))^3} \, dx=\frac {48 \, \cos \left (x\right )^{4} + 114 \, \cos \left (x\right )^{3} - 60 \, \cos \left (x\right )^{2} + 15 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) + 2 \, \cos \left (x\right ) + 4\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 15 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) + 2 \, \cos \left (x\right ) + 4\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 2 \, {\left (24 \, \cos \left (x\right )^{3} - 33 \, \cos \left (x\right )^{2} - 63 \, \cos \left (x\right ) - 1\right )} \sin \left (x\right ) - 124 \, \cos \left (x\right ) + 2}{10 \, {\left (a^{3} \cos \left (x\right )^{4} - 2 \, a^{3} \cos \left (x\right )^{3} - 5 \, a^{3} \cos \left (x\right )^{2} + 2 \, a^{3} \cos \left (x\right ) + 4 \, a^{3} - {\left (a^{3} \cos \left (x\right )^{3} + 3 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3}\right )} \sin \left (x\right )\right )}} \]
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\[ \int \frac {\csc ^2(x)}{(a+a \sin (x))^3} \, dx=\frac {\int \frac {\csc ^{2}{\left (x \right )}}{\sin ^{3}{\left (x \right )} + 3 \sin ^{2}{\left (x \right )} + 3 \sin {\left (x \right )} + 1}\, dx}{a^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (59) = 118\).
Time = 0.20 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.77 \[ \int \frac {\csc ^2(x)}{(a+a \sin (x))^3} \, dx=-\frac {\frac {121 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {410 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {610 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {425 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {125 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + 5}{10 \, {\left (\frac {a^{3} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {5 \, a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {10 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {5 \, a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {a^{3} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}}\right )}} - \frac {3 \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{3}} + \frac {\sin \left (x\right )}{2 \, a^{3} {\left (\cos \left (x\right ) + 1\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.31 \[ \int \frac {\csc ^2(x)}{(a+a \sin (x))^3} \, dx=-\frac {3 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a^{3}} + \frac {\tan \left (\frac {1}{2} \, x\right )}{2 \, a^{3}} + \frac {6 \, \tan \left (\frac {1}{2} \, x\right ) - 1}{2 \, a^{3} \tan \left (\frac {1}{2} \, x\right )} - \frac {4 \, {\left (15 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 50 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 70 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 45 \, \tan \left (\frac {1}{2} \, x\right ) + 12\right )}}{5 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \]
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Time = 6.17 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.98 \[ \int \frac {\csc ^2(x)}{(a+a \sin (x))^3} \, dx=\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{2\,a^3}-\frac {25\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+85\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+122\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+82\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\frac {121\,\mathrm {tan}\left (\frac {x}{2}\right )}{5}+1}{2\,a^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+10\,a^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+20\,a^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+20\,a^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+10\,a^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,a^3\,\mathrm {tan}\left (\frac {x}{2}\right )}-\frac {3\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a^3} \]
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