Integrand size = 27, antiderivative size = 138 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {3 a^3 x}{2}+\frac {33 a^3 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {3 a^3 \cos (c+d x)}{d}+\frac {2 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{d}-\frac {7 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d} \]
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Time = 0.15 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2951, 3855, 3852, 8, 3853, 2718, 2715} \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {33 a^3 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {3 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{d}+\frac {2 a^3 \cot (c+d x)}{d}-\frac {a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {7 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac {3 a^3 x}{2} \]
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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 (a^7-5 a^7 \csc (c+d x)-5 a^7 \csc ^2(c+d x)+a^7 \csc ^3(c+d x)+3 a^7 \csc ^4(c+d x)+a^7 \csc ^5(c+d x)+3 a^7 \sin (c+d x)+a^7 \sin ^2(c+d x)\right ) \, dx}{a^4} \\ & = a^3 x+a^3 \int \csc ^3(c+d x) \, dx+a^3 \int \csc ^5(c+d x) \, dx+a^3 \int \sin ^2(c+d x) \, dx+\left (3 a^3\right ) \int \csc ^4(c+d x) \, dx+\left (3 a^3\right ) \int \sin (c+d x) \, dx-\left (5 a^3\right ) \int \csc (c+d x) \, dx-\left (5 a^3\right ) \int \csc ^2(c+d x) \, dx \\ & = a^3 x+\frac {5 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 \cot (c+d x) \csc ^3(c+d x)}{4 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 {1}{4} \left (3 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac {\left (3 a^3\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}+\frac {\left (5 a^3\right ) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d} \\ & = \frac {3 a^3 x}{2}+\frac {9 a^3 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {3 a^3 \cos (c+d x)}{d}+\frac {2 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{d}-\frac {7 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 {a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{8} \left (3 a^3\right ) \int \csc (c+d x) \, dx \\ & = \frac {3 a^3 x}{2}+\frac {33 a^3 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {3 a^3 \cos (c+d x)}{d}+\frac {2 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{d}-\frac {7 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 {a^3 \cos (c+d x) \sin (c+d x)}{2 d} \\ \end{align*}
Time = 7.26 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.56 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (1+\sin (c+d x))^3 \left (96 (c+d x)-192 \cos (c+d x)+96 \cot \left (\frac {1}{2} (c+d x)\right )-14 \csc ^2\left (\frac {1}{2} (c+d x)\right )-\csc ^4\left (\frac {1}{2} (c+d x)\right )+264 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-264 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+14 \sec ^2\left (\frac {1}{2} (c+d x)\right )+\sec ^4\left (\frac {1}{2} (c+d x)\right )+64 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-4 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-16 \sin (2 (c+d x))-96 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{64 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
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Time = 0.38 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.43
method | result | size |
parallelrisch | \(-\frac {\left (2112 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-8 \left (\sec ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-96 \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+600\right ) \left (\csc ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\cos \left (\frac {11 d x}{2}+\frac {11 c}{2}\right )+\cos \left (\frac {9 d x}{2}+\frac {9 c}{2}\right )+21 \cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )+21 \cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+10 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+10 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2320 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (1536 \cos \left (d x +c \right )-4344\right ) \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-768 d x \right ) a^{3}}{512 d}\) | \(198\) |
risch | \(\frac {3 a^{3} x}{2}+\frac {i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {3 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {3 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {a^{3} \left (7 \,{\mathrm e}^{7 i \left (d x +c \right )}-15 \,{\mathrm e}^{5 i \left (d x +c \right )}+40 i {\mathrm e}^{6 i \left (d x +c \right )}-15 \,{\mathrm e}^{3 i \left (d x +c \right )}-72 i {\mathrm e}^{4 i \left (d x +c \right )}+7 \,{\mathrm e}^{i \left (d x +c \right )}+56 i {\mathrm e}^{2 i \left (d x +c \right )}-24 i\right )}{4 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {33 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}-\frac {33 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}\) | \(222\) |
derivativedivides | \(\frac {a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+3 a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(224\) |
default | \(\frac {a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+3 a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(224\) |
norman | \(\frac {\frac {a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{3}}{64 d}-\frac {a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {19 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {11 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {11 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {19 a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {a^{3} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {3 a^{3} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {9 a^{3} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {9 a^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {3 a^{3} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {63 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {477 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {243 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {33 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}\) | \(349\) |
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Time = 0.31 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.67 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {24 \, a^{3} d x \cos \left (d x + c\right )^{4} - 48 \, a^{3} \cos \left (d x + c\right )^{5} - 48 \, a^{3} d x \cos \left (d x + c\right )^{2} + 110 \, a^{3} \cos \left (d x + c\right )^{3} + 24 \, a^{3} d x - 66 \, a^{3} \cos \left (d x + c\right ) + 33 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 33 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 8 \, {\left (a^{3} \cos \left (d x + c\right )^{5} + 4 \, a^{3} \cos \left (d x + c\right )^{3} - 3 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{16 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
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Timed out. \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 0.32 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.51 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {8 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} + 2}{\tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{3} - 16 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a^{3} + a^{3} {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 12 \, a^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{16 \, d} \]
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Time = 0.42 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.75 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 16 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 96 \, {\left (d x + c\right )} a^{3} - 264 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 88 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {64 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} + \frac {550 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 88 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{64 \, d} \]
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Time = 10.30 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.38 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {33\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d}-\frac {3\,a^3\,\mathrm {atan}\left (\frac {9\,a^6}{\frac {99\,a^6}{4}+9\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {99\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {99\,a^6}{4}+9\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}\right )}{d}-\frac {11\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {-38\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+100\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-26\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {417\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{4}-18\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {9\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+2\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^3}{4}}{d\,\left (16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )} \]
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