Integrand size = 29, antiderivative size = 100 \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \text {arctanh}(\cos (c+d x))}{4 d}-\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {a^2 \cot (c+d x) \csc (c+d x)}{4 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{2 d} \]
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Time = 0.16 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2952, 2687, 30, 2691, 3853, 3855, 14} \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \text {arctanh}(\cos (c+d x))}{4 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{2 d}+\frac {a^2 \cot (c+d x) \csc (c+d x)}{4 d} \]
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Rule 14
Rule 30
Rule 2687
Rule 2691
Rule 2952
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 \cot ^2(c+d x) \csc ^2(c+d x)+2 a^2 \cot ^2(c+d x) \csc ^3(c+d x)+a^2 \cot ^2(c+d x) \csc ^4(c+d x)\right ) \, dx \\ & = a^2 \int \cot ^2(c+d x) \csc ^2(c+d x) \, dx+a^2 \int \cot ^2(c+d x) \csc ^4(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx \\ & = -\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{2 d}-\frac {1}{2} a^2 \int \csc ^3(c+d x) \, dx+\frac {a^2 \text {Subst}\left (\int x^2 \, dx,x,-\cot (c+d x)\right )}{d}+\frac {a^2 \text {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = -\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot (c+d x) \csc (c+d x)}{4 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{2 d}-\frac {1}{4} a^2 \int \csc (c+d x) \, dx+\frac {a^2 \text {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,-\cot (c+d x)\right )}{d} \\ & = \frac {a^2 \text {arctanh}(\cos (c+d x))}{4 d}-\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {a^2 \cot (c+d x) \csc (c+d x)}{4 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{2 d} \\ \end{align*}
Time = 6.21 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.89 \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \csc ^5(c+d x) \left (200 \cos (c+d x)+20 \cos (3 (c+d x))-28 \cos (5 (c+d x))-150 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+150 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+180 \sin (2 (c+d x))+75 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-75 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+30 \sin (4 (c+d x))-15 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+15 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))\right )}{960 d} \]
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Time = 0.25 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.20
method | result | size |
parallelrisch | \(-\frac {\left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )-\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {25 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {25 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-30 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+30 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+40 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a^{2}}{160 d}\) | \(120\) |
derivativedivides | \(\frac {-\frac {a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )^{3}}+2 a^{2} \left (-\frac {\cos ^{3}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{3}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{8}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+a^{2} \left (-\frac {\cos ^{3}\left (d x +c \right )}{5 \sin \left (d x +c \right )^{5}}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15 \sin \left (d x +c \right )^{3}}\right )}{d}\) | \(136\) |
default | \(\frac {-\frac {a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )^{3}}+2 a^{2} \left (-\frac {\cos ^{3}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{3}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{8}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+a^{2} \left (-\frac {\cos ^{3}\left (d x +c \right )}{5 \sin \left (d x +c \right )^{5}}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15 \sin \left (d x +c \right )^{3}}\right )}{d}\) | \(136\) |
risch | \(-\frac {a^{2} \left (-60 i {\mathrm e}^{8 i \left (d x +c \right )}+15 \,{\mathrm e}^{9 i \left (d x +c \right )}+240 i {\mathrm e}^{6 i \left (d x +c \right )}+90 \,{\mathrm e}^{7 i \left (d x +c \right )}-40 i {\mathrm e}^{4 i \left (d x +c \right )}+80 i {\mathrm e}^{2 i \left (d x +c \right )}-90 \,{\mathrm e}^{3 i \left (d x +c \right )}-28 i-15 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{30 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{4 d}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{4 d}\) | \(158\) |
norman | \(\frac {-\frac {a^{2}}{160 d}-\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 d}-\frac {31 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d}-\frac {a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {37 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d}+\frac {13 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}-\frac {13 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}-\frac {37 a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d}+\frac {a^{2} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {31 a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d}+\frac {a^{2} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {a^{2} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d}-\frac {a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}\) | \(301\) |
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Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (90) = 180\).
Time = 0.27 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.89 \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {56 \, a^{2} \cos \left (d x + c\right )^{5} - 80 \, a^{2} \cos \left (d x + c\right )^{3} + 15 \, {\left (a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 15 \, {\left (a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 30 \, {\left (a^{2} \cos \left (d x + c\right )^{3} + a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.09 \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {15 \, a^{2} {\left (\frac {2 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {40 \, a^{2}}{\tan \left (d x + c\right )^{3}} + \frac {8 \, {\left (5 \, \tan \left (d x + c\right )^{2} + 3\right )} a^{2}}{\tan \left (d x + c\right )^{5}}}{120 \, d} \]
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Time = 0.36 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.64 \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 25 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 90 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {274 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 90 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 25 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \]
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Time = 9.56 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.60 \[ \int \cot ^2(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {5\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}-\frac {a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (-6\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {5\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^2}{5}\right )}{32\,d}-\frac {3\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d} \]
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