Integrand size = 29, antiderivative size = 100 \[ \int \cot ^2(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-a^3 x+\frac {13 a^3 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{d}-\frac {11 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d} \]
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Time = 0.16 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2952, 3554, 8, 2691, 3855, 2687, 30, 3853} \[ \int \cot ^2(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {13 a^3 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a^3 \cot ^3(c+d x)}{d}-\frac {a^3 \cot (c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {11 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-a^3 x \]
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Rule 8
Rule 30
Rule 2687
Rule 2691
Rule 2952
Rule 3554
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 \cot ^2(c+d x)+3 a^3 \cot ^2(c+d x) \csc (c+d x)+3 a^3 \cot ^2(c+d x) \csc ^2(c+d x)+a^3 \cot ^2(c+d x) \csc ^3(c+d x)\right ) \, dx \\ & = a^3 \int \cot ^2(c+d x) \, dx+a^3 \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^2(c+d x) \csc (c+d x) \, dx+\left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^2(c+d x) \, dx \\ & = -\frac {a^3 \cot (c+d x)}{d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {1}{4} a^3 \int \csc ^3(c+d x) \, dx-a^3 \int 1 \, dx-\frac {1}{2} \left (3 a^3\right ) \int \csc (c+d x) \, dx+\frac {\left (3 a^3\right ) \text {Subst}\left (\int x^2 \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -a^3 x+\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{d}-\frac {11 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {1}{8} a^3 \int \csc (c+d x) \, dx \\ & = -a^3 x+\frac {13 a^3 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{d}-\frac {11 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d} \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.33 \[ \int \cot ^2(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \left (-22 \csc ^2\left (\frac {1}{2} (c+d x)\right )+22 \sec ^2\left (\frac {1}{2} (c+d x)\right )+\sec ^4\left (\frac {1}{2} (c+d x)\right )-8 \left (8 c+8 d x-13 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+13 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-8 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )\right )-\csc ^4\left (\frac {1}{2} (c+d x)\right ) (1+4 \sin (c+d x))\right )}{64 d} \]
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Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.96
method | result | size |
parallelrisch | \(-\frac {13 \left (\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {19 \left (\sec ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (d x +c \right )-\frac {11 \cos \left (3 d x +3 c \right )}{19}\right ) \left (\csc ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{832}+\frac {\left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (d x +c \right )}{13}+\frac {8 d x}{13}\right ) a^{3}}{8 d}\) | \(96\) |
risch | \(-a^{3} x +\frac {a^{3} \left (11 \,{\mathrm e}^{7 i \left (d x +c \right )}-19 \,{\mathrm e}^{5 i \left (d x +c \right )}+16 i {\mathrm e}^{6 i \left (d x +c \right )}-19 \,{\mathrm e}^{3 i \left (d x +c \right )}+11 \,{\mathrm e}^{i \left (d x +c \right )}-16 i {\mathrm e}^{2 i \left (d x +c \right )}\right )}{4 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}-\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}\) | \(138\) |
derivativedivides | \(\frac {a^{3} \left (-\cot \left (d x +c \right )-d x -c \right )+3 a^{3} \left (-\frac {\cos ^{3}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )-\frac {a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{\sin \left (d x +c \right )^{3}}+a^{3} \left (-\frac {\cos ^{3}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{3}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{8}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(164\) |
default | \(\frac {a^{3} \left (-\cot \left (d x +c \right )-d x -c \right )+3 a^{3} \left (-\frac {\cos ^{3}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )-\frac {a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{\sin \left (d x +c \right )^{3}}+a^{3} \left (-\frac {\cos ^{3}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{3}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{8}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(164\) |
norman | \(\frac {-\frac {a^{3}}{64 d}-\frac {a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {27 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {5 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {5 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {27 a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {a^{3} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-a^{3} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 a^{3} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 a^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a^{3} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {11 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {75 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {137 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {13 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}\) | \(350\) |
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Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (94) = 188\).
Time = 0.29 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.90 \[ \int \cot ^2(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {16 \, a^{3} d x \cos \left (d x + c\right )^{4} - 32 \, a^{3} d x \cos \left (d x + c\right )^{2} - 22 \, a^{3} \cos \left (d x + c\right )^{3} + 16 \, a^{3} d x + 16 \, a^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 26 \, a^{3} \cos \left (d x + c\right ) - 13 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 13 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{16 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
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Timed out. \[ \int \cot ^2(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.47 \[ \int \cot ^2(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {16 \, {\left (d x + c + \frac {1}{\tan \left (d x + c\right )}\right )} a^{3} + a^{3} {\left (\frac {2 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 12 \, a^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + \log \left (\cos \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {16 \, a^{3}}{\tan \left (d x + c\right )^{3}}}{16 \, d} \]
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Time = 0.37 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.74 \[ \int \cot ^2(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 24 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 192 \, {\left (d x + c\right )} a^{3} - 312 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 24 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {650 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 24 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
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Time = 9.97 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.37 \[ \int \cot ^2(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {2\,a^3\,\mathrm {atan}\left (\frac {8\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+13\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{13\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {13\,a^3\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{8\,d}-\frac {a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d} \]
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