Integrand size = 21, antiderivative size = 82 \[ \int \frac {\cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {\cot (c+d x)}{a^3 d}+\frac {2 \cot (c+d x)}{3 a^3 d (1+\csc (c+d x))^2}-\frac {13 \cot (c+d x)}{3 a^3 d (1+\csc (c+d x))} \]
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Time = 0.17 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2788, 3855, 3852, 8, 3862, 4004, 3879} \[ \int \frac {\cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {3 \text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {\cot (c+d x)}{a^3 d}-\frac {13 \cot (c+d x)}{3 a^3 d (\csc (c+d x)+1)}+\frac {2 \cot (c+d x)}{3 a^3 d (\csc (c+d x)+1)^2} \]
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Rule 8
Rule 2788
Rule 3852
Rule 3855
Rule 3862
Rule 3879
Rule 4004
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (\frac {5}{a}-\frac {3 \csc (c+d x)}{a}+\frac {\csc ^2(c+d x)}{a}+\frac {2}{a (1+\csc (c+d x))^2}-\frac {7}{a (1+\csc (c+d x))}\right ) \, dx}{a^2} \\ & = \frac {5 x}{a^3}+\frac {\int \csc ^2(c+d x) \, dx}{a^3}+\frac {2 \int \frac {1}{(1+\csc (c+d x))^2} \, dx}{a^3}-\frac {3 \int \csc (c+d x) \, dx}{a^3}-\frac {7 \int \frac {1}{1+\csc (c+d x)} \, dx}{a^3} \\ & = \frac {5 x}{a^3}+\frac {3 \text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {2 \cot (c+d x)}{3 a^3 d (1+\csc (c+d x))^2}-\frac {7 \cot (c+d x)}{a^3 d (1+\csc (c+d x))}-\frac {2 \int \frac {-3+\csc (c+d x)}{1+\csc (c+d x)} \, dx}{3 a^3}+\frac {7 \int -1 \, dx}{a^3}-\frac {\text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^3 d} \\ & = \frac {3 \text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {\cot (c+d x)}{a^3 d}+\frac {2 \cot (c+d x)}{3 a^3 d (1+\csc (c+d x))^2}-\frac {7 \cot (c+d x)}{a^3 d (1+\csc (c+d x))}-\frac {8 \int \frac {\csc (c+d x)}{1+\csc (c+d x)} \, dx}{3 a^3} \\ & = \frac {3 \text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {\cot (c+d x)}{a^3 d}+\frac {2 \cot (c+d x)}{3 a^3 d (1+\csc (c+d x))^2}-\frac {13 \cot (c+d x)}{3 a^3 d (1+\csc (c+d x))} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(255\) vs. \(2(82)=164\).
Time = 1.52 (sec) , antiderivative size = 255, normalized size of antiderivative = 3.11 \[ \int \frac {\cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (8 \sin \left (\frac {1}{2} (c+d x)\right )-4 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+44 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2-3 \cot \left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3+18 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3-18 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3+3 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{6 d (a+a \sin (c+d x))^3} \]
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Time = 0.44 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {16}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {20}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{2 d \,a^{3}}\) | \(89\) |
default | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {16}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {20}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{2 d \,a^{3}}\) | \(89\) |
parallelrisch | \(\frac {-18 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-81 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-129 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-70}{6 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}\) | \(97\) |
risch | \(-\frac {2 \left (-29 \,{\mathrm e}^{2 i \left (d x +c \right )}+27 i {\mathrm e}^{3 i \left (d x +c \right )}+14-33 i {\mathrm e}^{i \left (d x +c \right )}+9 \,{\mathrm e}^{4 i \left (d x +c \right )}\right )}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{3} d \,a^{3}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{3}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{3}}\) | \(125\) |
norman | \(\frac {-\frac {18 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {1}{2 a d}+\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}-\frac {41 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d a}-\frac {151 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {117 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}-\frac {469 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}\) | \(169\) |
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Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (78) = 156\).
Time = 0.28 (sec) , antiderivative size = 279, normalized size of antiderivative = 3.40 \[ \int \frac {\cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {28 \, \cos \left (d x + c\right )^{3} - 10 \, \cos \left (d x + c\right )^{2} - 9 \, {\left (\cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 9 \, {\left (\cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, {\left (14 \, \cos \left (d x + c\right )^{2} + 19 \, \cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - 34 \, \cos \left (d x + c\right ) + 4}{6 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 2 \, a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d \cos \left (d x + c\right ) - 2 \, a^{3} d + {\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d \cos \left (d x + c\right ) - 2 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
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\[ \int \frac {\cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\cos ^{2}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (78) = 156\).
Time = 0.26 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.46 \[ \int \frac {\cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {\frac {61 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {105 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {63 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + 3}{\frac {a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {18 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {3 \, \sin \left (d x + c\right )}{a^{3} {\left (\cos \left (d x + c\right ) + 1\right )}}}{6 \, d} \]
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Time = 0.42 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.33 \[ \int \frac {\cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {18 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{3}} - \frac {3 \, {\left (6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \frac {4 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 13\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{3}}}{6 \, d} \]
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Time = 9.94 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.77 \[ \int \frac {\cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^3\,d}-\frac {21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {61\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+1}{d\,\left (2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+6\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d} \]
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