Integrand size = 29, antiderivative size = 116 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {9 a^2 x}{8}-\frac {2 a^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {2 a^2 \cos (c+d x)}{d}+\frac {2 a^2 \cos ^3(c+d x)}{3 d}-\frac {a^2 \cot (c+d x)}{d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^2 \cos (c+d x) \sin ^3(c+d x)}{4 d} \]
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Time = 0.16 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2951, 3855, 3852, 8, 2718, 2715, 2713} \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {2 a^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {2 a^2 \cos ^3(c+d x)}{3 d}+\frac {2 a^2 \cos (c+d x)}{d}-\frac {a^2 \cot (c+d x)}{d}-\frac {a^2 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac {a^2 \sin (c+d x) \cos (c+d x)}{8 d}-\frac {9 a^2 x}{8} \]
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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^6+2 a^6 \csc (c+d x)+a^6 \csc ^2(c+d x)-4 a^6 \sin (c+d x)-a^6 \sin ^2(c+d x)+2 a^6 \sin ^3(c+d x)+a^6 \sin ^4(c+d x)\right ) \, dx}{a^4} \\ & = -a^2 x+a^2 \int \csc ^2(c+d x) \, dx-a^2 \int \sin ^2(c+d x) \, dx+a^2 \int \sin ^4(c+d x) \, dx+\left (2 a^2\right ) \int \csc (c+d x) \, dx+\left (2 a^2\right ) \int \sin ^3(c+d x) \, dx-\left (4 a^2\right ) \int \sin (c+d x) \, dx \\ & = -a^2 x-\frac {2 a^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^2 \cos (c+d x)}{d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{2 d}-\frac {a^2 \cos (c+d x) \sin ^3(c+d x)}{4 d}-\frac {1}{2} a^2 \int 1 \, dx+\frac {1}{4} \left (3 a^2\right ) \int \sin ^2(c+d x) \, dx-\frac {a^2 \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {3 a^2 x}{2}-\frac {2 a^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {2 a^2 \cos (c+d x)}{d}+\frac {2 a^2 \cos ^3(c+d x)}{3 d}-\frac {a^2 \cot (c+d x)}{d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^2 \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac {1}{8} \left (3 a^2\right ) \int 1 \, dx \\ & = -\frac {9 a^2 x}{8}-\frac {2 a^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {2 a^2 \cos (c+d x)}{d}+\frac {2 a^2 \cos ^3(c+d x)}{3 d}-\frac {a^2 \cot (c+d x)}{d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^2 \cos (c+d x) \sin ^3(c+d x)}{4 d} \\ \end{align*}
Time = 0.60 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.72 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \left (240 \cos (c+d x)+16 \cos (3 (c+d x))-3 \left (36 c+36 d x+32 \cot (c+d x)+64 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-64 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-\sin (4 (c+d x))\right )\right )}{96 d} \]
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Time = 0.29 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.99
method | result | size |
parallelrisch | \(\frac {\left (\frac {16}{3}+4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\left (-\frac {33}{2}+\cos \left (d x +c \right )-\cos \left (2 d x +2 c \right )+\cos \left (3 d x +3 c \right )-\frac {\cos \left (4 d x +4 c \right )}{2}\right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}+\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {9 d x}{4}+5 \cos \left (d x +c \right )+\frac {\cos \left (3 d x +3 c \right )}{3}\right ) a^{2}}{2 d}\) | \(115\) |
derivativedivides | \(\frac {a^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )}{d}\) | \(136\) |
default | \(\frac {a^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )}{d}\) | \(136\) |
risch | \(-\frac {9 a^{2} x}{8}+\frac {5 a^{2} {\mathrm e}^{i \left (d x +c \right )}}{4 d}+\frac {5 a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{4 d}-\frac {2 i a^{2}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}-\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}+\frac {a^{2} \sin \left (4 d x +4 c \right )}{32 d}+\frac {a^{2} \cos \left (3 d x +3 c \right )}{6 d}\) | \(138\) |
norman | \(\frac {-\frac {a^{2}}{2 d}-\frac {5 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {11 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {11 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {5 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {9 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}-\frac {9 a^{2} x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {27 a^{2} x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {9 a^{2} x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {9 a^{2} x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {8 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {16 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {40 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {16 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(308\) |
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Time = 0.30 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.16 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {6 \, a^{2} \cos \left (d x + c\right )^{5} - 9 \, a^{2} \cos \left (d x + c\right )^{3} + 24 \, a^{2} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 24 \, a^{2} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 27 \, a^{2} \cos \left (d x + c\right ) - {\left (16 \, a^{2} \cos \left (d x + c\right )^{3} - 27 \, a^{2} d x + 48 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d \sin \left (d x + c\right )} \]
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Timed out. \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.10 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {32 \, {\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right ) - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{2} + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} - 48 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} + 2}{\tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{2}}{96 \, d} \]
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Time = 0.34 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.81 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {27 \, {\left (d x + c\right )} a^{2} - 48 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 12 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {12 \, {\left (4 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{2}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \frac {2 \, {\left (3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 21 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 192 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 21 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 160 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 64 \, a^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
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Time = 9.31 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.67 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {2\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {9\,a^2\,\mathrm {atan}\left (\frac {81\,a^4}{16\,\left (9\,a^4+\frac {81\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}-\frac {9\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{9\,a^4+\frac {81\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}}\right )}{4\,d}-\frac {\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2}-16\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2}-32\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {19\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}-\frac {80\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {7\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}-\frac {32\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+a^2}{d\,\left (2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}+\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d} \]
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