Integrand size = 29, antiderivative size = 72 \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\text {arctanh}(\cos (c+d x))}{2 a d}-\frac {\cot (c+d x)}{a d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d} \]
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Time = 0.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2918, 3852, 3853, 3855} \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\text {arctanh}(\cos (c+d x))}{2 a d}-\frac {\cot ^3(c+d x)}{3 a d}-\frac {\cot (c+d x)}{a d}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d} \]
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Rule 2918
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \csc ^3(c+d x) \, dx}{a}+\frac {\int \csc ^4(c+d x) \, dx}{a} \\ & = \frac {\cot (c+d x) \csc (c+d x)}{2 a d}-\frac {\int \csc (c+d x) \, dx}{2 a}-\frac {\text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a d} \\ & = \frac {\text {arctanh}(\cos (c+d x))}{2 a d}-\frac {\cot (c+d x)}{a d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d} \\ \end{align*}
Time = 1.08 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.75 \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \left (\csc \left (\frac {1}{2} (c+d x)\right )+\sec \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (-12 \cos (c+d x) (-1+\sin (c+d x))-4 \left (\cos (3 (c+d x))+3 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sin ^3(c+d x)\right )\right )}{192 a d (1+\sin (c+d x))} \]
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Time = 0.21 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.31
method | result | size |
parallelrisch | \(\frac {-\left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )+3 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-9 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d a}\) | \(94\) |
derivativedivides | \(\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{8 d a}\) | \(96\) |
default | \(\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {3}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{8 d a}\) | \(96\) |
risch | \(-\frac {3 \,{\mathrm e}^{5 i \left (d x +c \right )}-12 i {\mathrm e}^{2 i \left (d x +c \right )}+4 i-3 \,{\mathrm e}^{i \left (d x +c \right )}}{3 a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d a}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d a}\) | \(100\) |
norman | \(\frac {-\frac {1}{24 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{12 d a}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d a}+\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d a}-\frac {\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{12 d a}+\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d a}-\frac {3 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a d}\) | \(166\) |
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Time = 0.29 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.65 \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {8 \, \cos \left (d x + c\right )^{3} - 3 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 3 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 6 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 12 \, \cos \left (d x + c\right )}{12 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \]
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\[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {\cos ^{2}{\left (c + d x \right )} \csc ^{4}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (66) = 132\).
Time = 0.20 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.12 \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a} - \frac {12 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {{\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {9 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{a \sin \left (d x + c\right )^{3}}}{24 \, d} \]
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Time = 0.40 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.78 \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {12 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} - \frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{3}} - \frac {22 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
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Time = 9.91 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.65 \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,a\,d}+\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {1}{3}\right )}{8\,a\,d} \]
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