Integrand size = 29, antiderivative size = 153 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {5 a^2 x}{8}+\frac {5 a^2 \text {arctanh}(\cos (c+d x))}{d}-\frac {4 a^2 \cos (c+d x)}{d}-\frac {2 a^2 \cos ^3(c+d x)}{3 d}+\frac {a^2 \cot (c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{d}-\frac {5 a^2 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 \cos (c+d x) \sin ^3(c+d x)}{4 d} \]
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Time = 0.18 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2951, 3855, 3852, 8, 3853, 2718, 2715, 2713} \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {5 a^2 \text {arctanh}(\cos (c+d x))}{d}-\frac {2 a^2 \cos ^3(c+d x)}{3 d}-\frac {4 a^2 \cos (c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot (c+d x)}{d}+\frac {a^2 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {5 a^2 \sin (c+d x) \cos (c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{d}+\frac {5 a^2 x}{8} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2718
Rule 2951
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (-6 a^8 \csc (c+d x)-2 a^8 \csc ^2(c+d x)+2 a^8 \csc ^3(c+d x)+a^8 \csc ^4(c+d x)+6 a^8 \sin (c+d x)+2 a^8 \sin ^2(c+d x)-2 a^8 \sin ^3(c+d x)-a^8 \sin ^4(c+d x)\right ) \, dx}{a^6} \\ & = a^2 \int \csc ^4(c+d x) \, dx-a^2 \int \sin ^4(c+d x) \, dx-\left (2 a^2\right ) \int \csc ^2(c+d x) \, dx+\left (2 a^2\right ) \int \csc ^3(c+d x) \, dx+\left (2 a^2\right ) \int \sin ^2(c+d x) \, dx-\left (2 a^2\right ) \int \sin ^3(c+d x) \, dx-\left (6 a^2\right ) \int \csc (c+d x) \, dx+\left (6 a^2\right ) \int \sin (c+d x) \, dx \\ & = \frac {6 a^2 \text {arctanh}(\cos (c+d x))}{d}-\frac {6 a^2 \cos (c+d x)}{d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{d}-\frac {a^2 \cos (c+d x) \sin (c+d x)}{d}+\frac {a^2 \cos (c+d x) \sin ^3(c+d x)}{4 d}-\frac {1}{4} \left (3 a^2\right ) \int \sin ^2(c+d x) \, dx+a^2 \int 1 \, dx+a^2 \int \csc (c+d x) \, dx-\frac {a^2 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}+\frac {\left (2 a^2\right ) \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} \\ & = a^2 x+\frac {5 a^2 \text {arctanh}(\cos (c+d x))}{d}-\frac {4 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 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{d}-\frac {5 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 {5 a^2 x}{8}+\frac {5 a^2 \text {arctanh}(\cos (c+d x))}{d}-\frac {4 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 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot (c+d x) \csc (c+d x)}{d}-\frac {5 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 = 8.41 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.37 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 (1+\sin (c+d x))^2 \left (60 (c+d x)-432 \cos (c+d x)-16 \cos (3 (c+d x))+64 \cot \left (\frac {1}{2} (c+d x)\right )-24 \csc ^2\left (\frac {1}{2} (c+d x)\right )+480 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-480 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 \sec ^2\left (\frac {1}{2} (c+d x)\right )+32 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-2 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-24 \sin (2 (c+d x))-3 \sin (4 (c+d x))-64 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{96 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4} \]
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Time = 0.45 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.16
method | result | size |
parallelrisch | \(\frac {a^{2} \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (960 \left (-3 \sin \left (d x +c \right )+\sin \left (3 d x +3 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 d x \sin \left (3 d x +3 c \right )+360 d x \sin \left (d x +c \right )-1200 \sin \left (2 d x +2 c \right )+1088 \sin \left (3 d x +3 c \right )+384 \sin \left (4 d x +4 c \right )+16 \sin \left (6 d x +6 c \right )-45 \cos \left (d x +c \right )-193 \cos \left (3 d x +3 c \right )-15 \cos \left (5 d x +5 c \right )-3 \cos \left (7 d x +7 c \right )-3264 \sin \left (d x +c \right )\right )}{6144 d}\) | \(178\) |
derivativedivides | \(\frac {a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+2 a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{2}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )}{d}\) | \(224\) |
default | \(\frac {a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+2 a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{2}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )}{d}\) | \(224\) |
risch | \(\frac {5 a^{2} x}{8}+\frac {i a^{2} {\mathrm e}^{4 i \left (d x +c \right )}}{64 d}+\frac {i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {9 a^{2} {\mathrm e}^{i \left (d x +c \right )}}{4 d}-\frac {9 a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{4 d}-\frac {i a^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {i a^{2} {\mathrm e}^{-4 i \left (d x +c \right )}}{64 d}+\frac {2 a^{2} \left (6 i {\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{5 i \left (d x +c \right )}-6 i {\mathrm e}^{2 i \left (d x +c \right )}+4 i-3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {5 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}-\frac {5 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {a^{2} \cos \left (3 d x +3 c \right )}{6 d}\) | \(241\) |
norman | \(\frac {-\frac {a^{2}}{24 d}-\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {11 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {3 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {15 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {15 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {3 a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {11 a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {a^{2} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {a^{2} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {5 a^{2} x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {5 a^{2} x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {15 a^{2} x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {5 a^{2} x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {5 a^{2} x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {34 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {34 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {59 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {367 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {5 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(386\) |
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Time = 0.29 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.43 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {6 \, a^{2} \cos \left (d x + c\right )^{7} - 3 \, a^{2} \cos \left (d x + c\right )^{5} + 20 \, a^{2} \cos \left (d x + c\right )^{3} - 15 \, a^{2} \cos \left (d x + c\right ) + 60 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 60 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - {\left (16 \, a^{2} \cos \left (d x + c\right )^{5} - 15 \, a^{2} d x \cos \left (d x + c\right )^{2} + 80 \, a^{2} \cos \left (d x + c\right )^{3} + 15 \, a^{2} d x - 120 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \]
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Time = 0.32 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.24 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {4 \, {\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{2} + 3 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{2} - 4 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a^{2}}{24 \, d} \]
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Time = 0.40 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.79 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, {\left (d x + c\right )} a^{2} - 120 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {220 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} + \frac {2 \, {\left (15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 144 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 9 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 336 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 9 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 304 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 112 \, a^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
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Time = 10.39 (sec) , antiderivative size = 384, normalized size of antiderivative = 2.51 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {5\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {5\,a^2\,\mathrm {atan}\left (\frac {25\,a^4}{16\,\left (\frac {25\,a^4}{2}+\frac {25\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}-\frac {25\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\left (\frac {25\,a^4}{2}+\frac {25\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}\right )}{4\,d}-\frac {5\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {-15\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+98\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-\frac {41\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+232\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-\frac {104\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {644\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}-8\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {248\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-\frac {11\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+2\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^2}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )} \]
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