Integrand size = 11, antiderivative size = 58 \[ \int \frac {\csc (x)}{(a+a \sin (x))^3} \, dx=-\frac {\text {arctanh}(\cos (x))}{a^3}+\frac {\cos (x)}{5 (a+a \sin (x))^3}+\frac {7 \cos (x)}{15 a (a+a \sin (x))^2}+\frac {22 \cos (x)}{15 \left (a^3+a^3 \sin (x)\right )} \]
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Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2845, 3057, 12, 3855} \[ \int \frac {\csc (x)}{(a+a \sin (x))^3} \, dx=-\frac {\text {arctanh}(\cos (x))}{a^3}+\frac {22 \cos (x)}{15 \left (a^3 \sin (x)+a^3\right )}+\frac {7 \cos (x)}{15 a (a \sin (x)+a)^2}+\frac {\cos (x)}{5 (a \sin (x)+a)^3} \]
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Rule 12
Rule 2845
Rule 3057
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (x)}{5 (a+a \sin (x))^3}+\frac {\int \frac {\csc (x) (5 a-2 a \sin (x))}{(a+a \sin (x))^2} \, dx}{5 a^2} \\ & = \frac {\cos (x)}{5 (a+a \sin (x))^3}+\frac {7 \cos (x)}{15 a (a+a \sin (x))^2}+\frac {\int \frac {\csc (x) \left (15 a^2-7 a^2 \sin (x)\right )}{a+a \sin (x)} \, dx}{15 a^4} \\ & = \frac {\cos (x)}{5 (a+a \sin (x))^3}+\frac {7 \cos (x)}{15 a (a+a \sin (x))^2}+\frac {22 \cos (x)}{15 \left (a^3+a^3 \sin (x)\right )}+\frac {\int 15 a^3 \csc (x) \, dx}{15 a^6} \\ & = \frac {\cos (x)}{5 (a+a \sin (x))^3}+\frac {7 \cos (x)}{15 a (a+a \sin (x))^2}+\frac {22 \cos (x)}{15 \left (a^3+a^3 \sin (x)\right )}+\frac {\int \csc (x) \, dx}{a^3} \\ & = -\frac {\text {arctanh}(\cos (x))}{a^3}+\frac {\cos (x)}{5 (a+a \sin (x))^3}+\frac {7 \cos (x)}{15 a (a+a \sin (x))^2}+\frac {22 \cos (x)}{15 \left (a^3+a^3 \sin (x)\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(160\) vs. \(2(58)=116\).
Time = 0.72 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.76 \[ \int \frac {\csc (x)}{(a+a \sin (x))^3} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (-6 \sin \left (\frac {x}{2}\right )+3 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )-14 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2+7 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3-44 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^4-15 \log \left (\cos \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5+15 \log \left (\sin \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5\right )}{15 (a+a \sin (x))^3} \]
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Time = 0.51 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.93
method | result | size |
parallelrisch | \(\frac {12 \left (\sec ^{5}\left (x \right )\right )-12 \tan \left (x \right ) \left (\sec ^{4}\left (x \right )\right )+5 \left (\sec ^{3}\left (x \right )\right )-11 \tan \left (x \right ) \left (\sec ^{2}\left (x \right )\right )+15 \sec \left (x \right )-22 \tan \left (x \right )+15 \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+32}{15 a^{3}}\) | \(54\) |
default | \(\frac {\frac {8}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {20}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {6}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {6}{\tan \left (\frac {x}{2}\right )+1}+\ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{3}}\) | \(61\) |
norman | \(\frac {\frac {6 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}+\frac {64}{15 a}+\frac {18 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}+\frac {74 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {46 \tan \left (\frac {x}{2}\right )}{3 a}}{a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {\ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{3}}\) | \(71\) |
risch | \(\frac {10 i {\mathrm e}^{3 i x}+2 \,{\mathrm e}^{4 i x}-\frac {38 i {\mathrm e}^{i x}}{3}-\frac {58 \,{\mathrm e}^{2 i x}}{3}+\frac {44}{15}}{\left ({\mathrm e}^{i x}+i\right )^{5} a^{3}}-\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{a^{3}}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{a^{3}}\) | \(74\) |
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Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (52) = 104\).
Time = 0.29 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.90 \[ \int \frac {\csc (x)}{(a+a \sin (x))^3} \, dx=\frac {44 \, \cos \left (x\right )^{3} - 58 \, \cos \left (x\right )^{2} - 15 \, {\left (\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 2 \, {\left (22 \, \cos \left (x\right )^{2} + 51 \, \cos \left (x\right ) - 3\right )} \sin \left (x\right ) - 108 \, \cos \left (x\right ) - 6}{30 \, {\left (a^{3} \cos \left (x\right )^{3} + 3 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3} + {\left (a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3}\right )} \sin \left (x\right )\right )}} \]
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\[ \int \frac {\csc (x)}{(a+a \sin (x))^3} \, dx=\frac {\int \frac {\csc {\left (x \right )}}{\sin ^{3}{\left (x \right )} + 3 \sin ^{2}{\left (x \right )} + 3 \sin {\left (x \right )} + 1}\, dx}{a^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (52) = 104\).
Time = 0.21 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.47 \[ \int \frac {\csc (x)}{(a+a \sin (x))^3} \, dx=\frac {2 \, {\left (\frac {115 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {185 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {135 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {45 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 32\right )}}{15 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {10 \, a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}\right )}} + \frac {\log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.97 \[ \int \frac {\csc (x)}{(a+a \sin (x))^3} \, dx=\frac {\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a^{3}} + \frac {2 \, {\left (45 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 135 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 185 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 115 \, \tan \left (\frac {1}{2} \, x\right ) + 32\right )}}{15 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \]
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Time = 6.21 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.93 \[ \int \frac {\csc (x)}{(a+a \sin (x))^3} \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a^3}+\frac {6\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+18\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+\frac {74\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{3}+\frac {46\,\mathrm {tan}\left (\frac {x}{2}\right )}{3}+\frac {64}{15}}{a^3\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \]
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