Integrand size = 27, antiderivative size = 98 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {5 a^3 x}{2}-\frac {5 a^3 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {3 a^3 \cos (c+d x)}{d}-\frac {3 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d} \]
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Time = 0.12 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2951, 3855, 3852, 8, 3853, 2718, 2715} \[ \int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {5 a^3 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {3 a^3 \cos (c+d x)}{d}-\frac {3 a^3 \cot (c+d x)}{d}+\frac {a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {5 a^3 x}{2} \]
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Rule 8
Rule 2715
Rule 2718
Rule 2951
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (-2 a^5+2 a^5 \csc (c+d x)+3 a^5 \csc ^2(c+d x)+a^5 \csc ^3(c+d x)-3 a^5 \sin (c+d x)-a^5 \sin ^2(c+d x)\right ) \, dx}{a^2} \\ & = -2 a^3 x+a^3 \int \csc ^3(c+d x) \, dx-a^3 \int \sin ^2(c+d x) \, dx+\left (2 a^3\right ) \int \csc (c+d x) \, dx+\left (3 a^3\right ) \int \csc ^2(c+d x) \, dx-\left (3 a^3\right ) \int \sin (c+d x) \, dx \\ & = -2 a^3 x-\frac {2 a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {3 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d}-\frac {1}{2} a^3 \int 1 \, dx+\frac {1}{2} a^3 \int \csc (c+d x) \, dx-\frac {\left (3 a^3\right ) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d} \\ & = -\frac {5 a^3 x}{2}-\frac {5 a^3 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {3 a^3 \cos (c+d x)}{d}-\frac {3 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d} \\ \end{align*}
Time = 0.77 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.14 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \left (-20 c-20 d x+24 \cos (c+d x)-12 \cot \left (\frac {1}{2} (c+d x)\right )-\csc ^2\left (\frac {1}{2} (c+d x)\right )-20 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+20 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\sec ^2\left (\frac {1}{2} (c+d x)\right )+2 \sin (2 (c+d x))+12 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{8 d} \]
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Time = 0.29 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.33
method | result | size |
derivativedivides | \(\frac {a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{3} \left (\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \left (-\cot \left (d x +c \right )-d x -c \right )+a^{3} \left (-\frac {\cos ^{3}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(130\) |
default | \(\frac {a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{3} \left (\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \left (-\cot \left (d x +c \right )-d x -c \right )+a^{3} \left (-\frac {\cos ^{3}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(130\) |
parallelrisch | \(-\frac {3 \left (-\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}+\left (-\frac {\left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24}+\frac {1}{6}\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {23 \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+\frac {\cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )}{23}+\frac {\cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )}{23}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )}{96}+\left (\cos \left (d x +c \right )-\frac {13}{12}\right ) \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {5 d x}{6}\right ) a^{3}}{d}\) | \(136\) |
risch | \(-\frac {5 a^{3} x}{2}-\frac {i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {3 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {3 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {a^{3} \left ({\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}-6 i {\mathrm e}^{2 i \left (d x +c \right )}+6 i\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}+\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}\) | \(171\) |
norman | \(\frac {-\frac {a^{3}}{8 d}-\frac {3 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}-\frac {2 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {3 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {5 a^{3} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {15 a^{3} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {15 a^{3} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {5 a^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {21 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {41 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {85 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {5 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(274\) |
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Time = 0.31 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.62 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {10 \, a^{3} d x \cos \left (d x + c\right )^{2} - 12 \, a^{3} \cos \left (d x + c\right )^{3} - 10 \, a^{3} d x + 10 \, a^{3} \cos \left (d x + c\right ) + 5 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 5 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, {\left (a^{3} \cos \left (d x + c\right )^{3} + 5 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
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Timed out. \[ \int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.27 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} - 12 \, {\left (d x + c + \frac {1}{\tan \left (d x + c\right )}\right )} a^{3} + a^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + \log \left (\cos \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 6 \, a^{3} {\left (2 \, \cos \left (d x + c\right ) - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{4 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (90) = 180\).
Time = 0.74 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.88 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 20 \, {\left (d x + c\right )} a^{3} + 20 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {10 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 20 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 27 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 16 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{2}}}{8 \, d} \]
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Time = 9.76 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.64 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {5\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}-\frac {10\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {47\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}+8\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-23\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+6\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^3}{2}}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}+\frac {5\,a^3\,\mathrm {atan}\left (\frac {25\,a^6}{25\,a^6+25\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {25\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{25\,a^6+25\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d} \]
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