Integrand size = 11, antiderivative size = 38 \[ \int \frac {\csc (x)}{(a+a \sin (x))^2} \, dx=-\frac {\text {arctanh}(\cos (x))}{a^2}+\frac {4 \cos (x)}{3 a^2 (1+\sin (x))}+\frac {\cos (x)}{3 (a+a \sin (x))^2} \]
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Time = 0.06 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2845, 3057, 12, 3855} \[ \int \frac {\csc (x)}{(a+a \sin (x))^2} \, dx=-\frac {\text {arctanh}(\cos (x))}{a^2}+\frac {4 \cos (x)}{3 a^2 (\sin (x)+1)}+\frac {\cos (x)}{3 (a \sin (x)+a)^2} \]
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Rule 12
Rule 2845
Rule 3057
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (x)}{3 (a+a \sin (x))^2}+\frac {\int \frac {\csc (x) (3 a-a \sin (x))}{a+a \sin (x)} \, dx}{3 a^2} \\ & = \frac {4 \cos (x)}{3 a^2 (1+\sin (x))}+\frac {\cos (x)}{3 (a+a \sin (x))^2}+\frac {\int 3 a^2 \csc (x) \, dx}{3 a^4} \\ & = \frac {4 \cos (x)}{3 a^2 (1+\sin (x))}+\frac {\cos (x)}{3 (a+a \sin (x))^2}+\frac {\int \csc (x) \, dx}{a^2} \\ & = -\frac {\text {arctanh}(\cos (x))}{a^2}+\frac {4 \cos (x)}{3 a^2 (1+\sin (x))}+\frac {\cos (x)}{3 (a+a \sin (x))^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(129\) vs. \(2(38)=76\).
Time = 0.44 (sec) , antiderivative size = 129, normalized size of antiderivative = 3.39 \[ \int \frac {\csc (x)}{(a+a \sin (x))^2} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {3 x}{2}\right ) \left (8+3 \log \left (\cos \left (\frac {x}{2}\right )\right )-3 \log \left (\sin \left (\frac {x}{2}\right )\right )\right )+\cos \left (\frac {x}{2}\right ) \left (-6-9 \log \left (\cos \left (\frac {x}{2}\right )\right )+9 \log \left (\sin \left (\frac {x}{2}\right )\right )\right )-6 \left (3+2 \log \left (\cos \left (\frac {x}{2}\right )\right )+\cos (x) \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right )-2 \log \left (\sin \left (\frac {x}{2}\right )\right )\right ) \sin \left (\frac {x}{2}\right )\right )}{6 a^2 (1+\sin (x))^2} \]
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Time = 0.60 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.05
method | result | size |
parallelrisch | \(\frac {3 \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+5-4 \tan \left (x \right )+3 \sec \left (x \right )-2 \tan \left (x \right ) \left (\sec ^{2}\left (x \right )\right )+2 \left (\sec ^{3}\left (x \right )\right )}{3 a^{2}}\) | \(40\) |
default | \(\frac {\ln \left (\tan \left (\frac {x}{2}\right )\right )+\frac {4}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {2}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {4}{\tan \left (\frac {x}{2}\right )+1}}{a^{2}}\) | \(41\) |
norman | \(\frac {\frac {4 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}+\frac {10}{3 a}+\frac {6 \tan \left (\frac {x}{2}\right )}{a}}{a \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {\ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{2}}\) | \(49\) |
risch | \(\frac {6 i {\mathrm e}^{i x}+2 \,{\mathrm e}^{2 i x}-\frac {8}{3}}{\left ({\mathrm e}^{i x}+i\right )^{3} a^{2}}-\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{a^{2}}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{a^{2}}\) | \(59\) |
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Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (34) = 68\).
Time = 0.30 (sec) , antiderivative size = 117, normalized size of antiderivative = 3.08 \[ \int \frac {\csc (x)}{(a+a \sin (x))^2} \, dx=-\frac {8 \, \cos \left (x\right )^{2} + 3 \, {\left (\cos \left (x\right )^{2} - {\left (\cos \left (x\right ) + 2\right )} \sin \left (x\right ) - \cos \left (x\right ) - 2\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 3 \, {\left (\cos \left (x\right )^{2} - {\left (\cos \left (x\right ) + 2\right )} \sin \left (x\right ) - \cos \left (x\right ) - 2\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 2 \, {\left (4 \, \cos \left (x\right ) - 1\right )} \sin \left (x\right ) + 10 \, \cos \left (x\right ) + 2}{6 \, {\left (a^{2} \cos \left (x\right )^{2} - a^{2} \cos \left (x\right ) - 2 \, a^{2} - {\left (a^{2} \cos \left (x\right ) + 2 \, a^{2}\right )} \sin \left (x\right )\right )}} \]
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\[ \int \frac {\csc (x)}{(a+a \sin (x))^2} \, dx=\frac {\int \frac {\csc {\left (x \right )}}{\sin ^{2}{\left (x \right )} + 2 \sin {\left (x \right )} + 1}\, dx}{a^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (34) = 68\).
Time = 0.20 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.34 \[ \int \frac {\csc (x)}{(a+a \sin (x))^2} \, dx=\frac {2 \, {\left (\frac {9 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {6 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 5\right )}}{3 \, {\left (a^{2} + \frac {3 \, a^{2} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {3 \, a^{2} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}\right )}} + \frac {\log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{2}} \]
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none
Time = 0.31 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.05 \[ \int \frac {\csc (x)}{(a+a \sin (x))^2} \, dx=\frac {\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a^{2}} + \frac {2 \, {\left (6 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, x\right ) + 5\right )}}{3 \, a^{2} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{3}} \]
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Time = 6.06 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {\csc (x)}{(a+a \sin (x))^2} \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a^2}+\frac {4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+6\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {10}{3}}{a^2\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^3} \]
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