Integrand size = 13, antiderivative size = 42 \[ \int \frac {\csc ^3(x)}{a+a \sin (x)} \, dx=-\frac {3 \text {arctanh}(\cos (x))}{2 a}+\frac {2 \cot (x)}{a}-\frac {3 \cot (x) \csc (x)}{2 a}+\frac {\cot (x) \csc (x)}{a+a \sin (x)} \]
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Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2847, 2827, 3853, 3855, 3852, 8} \[ \int \frac {\csc ^3(x)}{a+a \sin (x)} \, dx=-\frac {3 \text {arctanh}(\cos (x))}{2 a}+\frac {2 \cot (x)}{a}-\frac {3 \cot (x) \csc (x)}{2 a}+\frac {\cot (x) \csc (x)}{a \sin (x)+a} \]
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Rule 8
Rule 2827
Rule 2847
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\cot (x) \csc (x)}{a+a \sin (x)}-\frac {\int \csc ^3(x) (-3 a+2 a \sin (x)) \, dx}{a^2} \\ & = \frac {\cot (x) \csc (x)}{a+a \sin (x)}-\frac {2 \int \csc ^2(x) \, dx}{a}+\frac {3 \int \csc ^3(x) \, dx}{a} \\ & = -\frac {3 \cot (x) \csc (x)}{2 a}+\frac {\cot (x) \csc (x)}{a+a \sin (x)}+\frac {3 \int \csc (x) \, dx}{2 a}+\frac {2 \text {Subst}(\int 1 \, dx,x,\cot (x))}{a} \\ & = -\frac {3 \text {arctanh}(\cos (x))}{2 a}+\frac {2 \cot (x)}{a}-\frac {3 \cot (x) \csc (x)}{2 a}+\frac {\cot (x) \csc (x)}{a+a \sin (x)} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.98 \[ \int \frac {\csc ^3(x)}{a+a \sin (x)} \, dx=\frac {4 \cot \left (\frac {x}{2}\right )-\csc ^2\left (\frac {x}{2}\right )-12 \log \left (\cos \left (\frac {x}{2}\right )\right )+12 \log \left (\sin \left (\frac {x}{2}\right )\right )+\sec ^2\left (\frac {x}{2}\right )-\frac {16 \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}-4 \tan \left (\frac {x}{2}\right )}{8 a} \]
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Time = 0.48 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.29
method | result | size |
default | \(\frac {\frac {\left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}-2 \tan \left (\frac {x}{2}\right )-\frac {1}{2 \tan \left (\frac {x}{2}\right )^{2}}+\frac {2}{\tan \left (\frac {x}{2}\right )}+6 \ln \left (\tan \left (\frac {x}{2}\right )\right )+\frac {8}{\tan \left (\frac {x}{2}\right )+1}}{4 a}\) | \(54\) |
parallelrisch | \(\frac {\left (-3+3 \cos \left (2 x \right )\right ) \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+6 \cos \left (x \right )-4 \sec \left (x \right )+4 \tan \left (x \right )+3 \cos \left (2 x \right )-4 \sin \left (2 x \right )-3}{2 a \left (-1+\cos \left (2 x \right )\right )}\) | \(57\) |
norman | \(\frac {\frac {3 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}-\frac {1}{8 a}+\frac {3 \tan \left (\frac {x}{2}\right )}{8 a}-\frac {3 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{8 a}+\frac {\tan ^{5}\left (\frac {x}{2}\right )}{8 a}}{\tan \left (\frac {x}{2}\right )^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )}+\frac {3 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{2 a}\) | \(75\) |
risch | \(\frac {3 \,{\mathrm e}^{4 i x}-5 \,{\mathrm e}^{2 i x}+3 i {\mathrm e}^{3 i x}+4-i {\mathrm e}^{i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2} \left ({\mathrm e}^{i x}+i\right ) a}-\frac {3 \ln \left ({\mathrm e}^{i x}+1\right )}{2 a}+\frac {3 \ln \left ({\mathrm e}^{i x}-1\right )}{2 a}\) | \(83\) |
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Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (38) = 76\).
Time = 0.27 (sec) , antiderivative size = 134, normalized size of antiderivative = 3.19 \[ \int \frac {\csc ^3(x)}{a+a \sin (x)} \, dx=\frac {8 \, \cos \left (x\right )^{3} + 6 \, \cos \left (x\right )^{2} - 3 \, {\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) - \cos \left (x\right ) - 1\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 3 \, {\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) - \cos \left (x\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 2 \, {\left (4 \, \cos \left (x\right )^{2} + \cos \left (x\right ) - 2\right )} \sin \left (x\right ) - 6 \, \cos \left (x\right ) - 4}{4 \, {\left (a \cos \left (x\right )^{3} + a \cos \left (x\right )^{2} - a \cos \left (x\right ) + {\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right ) - a\right )}} \]
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\[ \int \frac {\csc ^3(x)}{a+a \sin (x)} \, dx=\frac {\int \frac {\csc ^{3}{\left (x \right )}}{\sin {\left (x \right )} + 1}\, dx}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (38) = 76\).
Time = 0.21 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.31 \[ \int \frac {\csc ^3(x)}{a+a \sin (x)} \, dx=-\frac {\frac {4 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}}{8 \, a} + \frac {\frac {3 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {20 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1}{8 \, {\left (\frac {a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}\right )}} + \frac {3 \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{2 \, a} \]
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none
Time = 0.33 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.74 \[ \int \frac {\csc ^3(x)}{a+a \sin (x)} \, dx=\frac {3 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{2 \, a} + \frac {a \tan \left (\frac {1}{2} \, x\right )^{2} - 4 \, a \tan \left (\frac {1}{2} \, x\right )}{8 \, a^{2}} + \frac {2}{a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}} - \frac {18 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 4 \, \tan \left (\frac {1}{2} \, x\right ) + 1}{8 \, a \tan \left (\frac {1}{2} \, x\right )^{2}} \]
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Time = 5.89 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.64 \[ \int \frac {\csc ^3(x)}{a+a \sin (x)} \, dx=\frac {10\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\frac {3\,\mathrm {tan}\left (\frac {x}{2}\right )}{2}-\frac {1}{2}}{4\,a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+4\,a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}-\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{2\,a}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8\,a}+\frac {3\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{2\,a} \]
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