Integrand size = 29, antiderivative size = 131 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {3 a^3 x}{8}-\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {3 a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{d}-\frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {11 a^3 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {3 a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d} \]
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Time = 0.17 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2951, 3855, 3852, 8, 2718, 2715, 2713} \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{d}-\frac {a^3 \cos ^5(c+d x)}{5 d}+\frac {a^3 \cos ^3(c+d x)}{d}+\frac {3 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot (c+d x)}{d}-\frac {3 a^3 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac {11 a^3 \sin (c+d x) \cos (c+d x)}{8 d}-\frac {3 a^3 x}{8} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2718
Rule 2951
Rule 3852
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (a^7+3 a^7 \csc (c+d x)+a^7 \csc ^2(c+d x)-5 a^7 \sin (c+d x)-5 a^7 \sin ^2(c+d x)+a^7 \sin ^3(c+d x)+3 a^7 \sin ^4(c+d x)+a^7 \sin ^5(c+d x)\right ) \, dx}{a^4} \\ & = a^3 x+a^3 \int \csc ^2(c+d x) \, dx+a^3 \int \sin ^3(c+d x) \, dx+a^3 \int \sin ^5(c+d x) \, dx+\left (3 a^3\right ) \int \csc (c+d x) \, dx+\left (3 a^3\right ) \int \sin ^4(c+d x) \, dx-\left (5 a^3\right ) \int \sin (c+d x) \, dx-\left (5 a^3\right ) \int \sin ^2(c+d x) \, dx \\ & = a^3 x-\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {5 a^3 \cos (c+d x)}{d}+\frac {5 a^3 \cos (c+d x) \sin (c+d x)}{2 d}-\frac {3 a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac {1}{4} \left (9 a^3\right ) \int \sin ^2(c+d x) \, dx-\frac {1}{2} \left (5 a^3\right ) \int 1 \, dx-\frac {a^3 \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}-\frac {a^3 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {a^3 \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {3 a^3 x}{2}-\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {3 a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{d}-\frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {11 a^3 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {3 a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac {1}{8} \left (9 a^3\right ) \int 1 \, dx \\ & = -\frac {3 a^3 x}{8}-\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {3 a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{d}-\frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {11 a^3 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {3 a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d} \\ \end{align*}
Time = 6.70 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.13 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {(a+a \sin (c+d x))^3 \left (-60 (c+d x)+580 \cos (c+d x)+30 \cos (3 (c+d x))-2 \cos (5 (c+d x))-80 \cot \left (\frac {1}{2} (c+d x)\right )-480 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+480 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+80 \sin (2 (c+d x))+15 \sin (4 (c+d x))+80 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{160 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
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Time = 0.34 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98
method | result | size |
parallelrisch | \(\frac {19 \left (\frac {48 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{19}+\left (\cos \left (d x +c \right )-\frac {11 \cos \left (2 d x +2 c \right )}{19}+\frac {3 \cos \left (3 d x +3 c \right )}{19}-\frac {3 \cos \left (4 d x +4 c \right )}{38}-\frac {51}{38}\right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {8 \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )}{19}-\frac {6 d x}{19}+\frac {58 \cos \left (d x +c \right )}{19}+\frac {3 \cos \left (3 d x +3 c \right )}{19}-\frac {\cos \left (5 d x +5 c \right )}{95}+\frac {16}{5}\right ) a^{3}}{16 d}\) | \(128\) |
derivativedivides | \(\frac {-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+3 a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )}{d}\) | \(150\) |
default | \(\frac {-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+3 a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )}{d}\) | \(150\) |
risch | \(-\frac {3 a^{3} x}{8}-\frac {i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{4 d}+\frac {29 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{16 d}+\frac {29 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{16 d}+\frac {i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{4 d}-\frac {2 i a^{3}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}-\frac {a^{3} \cos \left (5 d x +5 c \right )}{80 d}+\frac {3 a^{3} \sin \left (4 d x +4 c \right )}{32 d}+\frac {3 a^{3} \cos \left (3 d x +3 c \right )}{16 d}\) | \(191\) |
norman | \(\frac {\frac {28 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{3}}{2 d}+\frac {3 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {3 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {3 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {3 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {3 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}-\frac {15 a^{3} x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {15 a^{3} x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {15 a^{3} x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {15 a^{3} x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {3 a^{3} x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {10 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {40 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {36 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {38 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{5 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {3 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(344\) |
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Time = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.12 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {30 \, a^{3} \cos \left (d x + c\right )^{5} - 5 \, a^{3} \cos \left (d x + c\right )^{3} + 60 \, a^{3} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 60 \, a^{3} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 15 \, a^{3} \cos \left (d x + c\right ) + {\left (8 \, a^{3} \cos \left (d x + c\right )^{5} - 40 \, a^{3} \cos \left (d x + c\right )^{3} + 15 \, a^{3} d x - 120 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{40 \, d \sin \left (d x + c\right )} \]
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Timed out. \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 0.33 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.08 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {32 \, a^{3} \cos \left (d x + c\right )^{5} - 80 \, {\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right ) - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} + 80 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} + 2}{\tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{3}}{160 \, d} \]
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Time = 0.38 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.73 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {15 \, {\left (d x + c\right )} a^{3} - 120 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 20 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {20 \, {\left (6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \frac {2 \, {\left (55 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 200 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 10 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 720 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 800 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 10 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 560 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 55 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 152 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{40 \, d} \]
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Time = 10.01 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.72 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {3\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {-\frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{2}+20\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+72\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-10\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+80\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+56\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+\frac {76\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}-a^3}{d\,\left (2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}+\frac {3\,a^3\,\mathrm {atan}\left (\frac {9\,a^6}{16\,\left (\frac {9\,a^6}{2}+\frac {9\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}-\frac {9\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\left (\frac {9\,a^6}{2}+\frac {9\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}\right )}{4\,d}+\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d} \]
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