Integrand size = 27, antiderivative size = 99 \[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {13 x}{8 a^3}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\cos (c+d x)}{a^3 d}-\frac {\cos ^3(c+d x)}{a^3 d}-\frac {13 \cos (c+d x) \sin (c+d x)}{8 a^3 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a^3 d} \]
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Time = 0.19 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {2954, 2952, 2715, 8, 2672, 327, 212, 2645, 30, 2648} \[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {\cos ^3(c+d x)}{a^3 d}+\frac {\cos (c+d x)}{a^3 d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a^3 d}-\frac {13 \sin (c+d x) \cos (c+d x)}{8 a^3 d}-\frac {13 x}{8 a^3} \]
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Rule 8
Rule 30
Rule 212
Rule 327
Rule 2645
Rule 2648
Rule 2672
Rule 2715
Rule 2952
Rule 2954
Rubi steps \begin{align*} \text {integral}& = \frac {\int \cos (c+d x) \cot (c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6} \\ & = \frac {\int \left (-3 a^3 \cos ^2(c+d x)+a^3 \cos (c+d x) \cot (c+d x)+3 a^3 \cos ^2(c+d x) \sin (c+d x)-a^3 \cos ^2(c+d x) \sin ^2(c+d x)\right ) \, dx}{a^6} \\ & = \frac {\int \cos (c+d x) \cot (c+d x) \, dx}{a^3}-\frac {\int \cos ^2(c+d x) \sin ^2(c+d x) \, dx}{a^3}-\frac {3 \int \cos ^2(c+d x) \, dx}{a^3}+\frac {3 \int \cos ^2(c+d x) \sin (c+d x) \, dx}{a^3} \\ & = -\frac {3 \cos (c+d x) \sin (c+d x)}{2 a^3 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a^3 d}-\frac {\int \cos ^2(c+d x) \, dx}{4 a^3}-\frac {3 \int 1 \, dx}{2 a^3}-\frac {\text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int x^2 \, dx,x,\cos (c+d x)\right )}{a^3 d} \\ & = -\frac {3 x}{2 a^3}+\frac {\cos (c+d x)}{a^3 d}-\frac {\cos ^3(c+d x)}{a^3 d}-\frac {13 \cos (c+d x) \sin (c+d x)}{8 a^3 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a^3 d}-\frac {\int 1 \, dx}{8 a^3}-\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{a^3 d} \\ & = -\frac {13 x}{8 a^3}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\cos (c+d x)}{a^3 d}-\frac {\cos ^3(c+d x)}{a^3 d}-\frac {13 \cos (c+d x) \sin (c+d x)}{8 a^3 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a^3 d} \\ \end{align*}
Time = 0.97 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.81 \[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-52 c-52 d x+8 \cos (c+d x)-8 \cos (3 (c+d x))-32 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+32 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-24 \sin (2 (c+d x))+\sin (4 (c+d x))}{32 a^3 d} \]
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Time = 0.35 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.66
method | result | size |
parallelrisch | \(\frac {-52 d x +32 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \cos \left (3 d x +3 c \right )+8 \cos \left (d x +c \right )+\sin \left (4 d x +4 c \right )-24 \sin \left (2 d x +2 c \right )}{32 d \,a^{3}}\) | \(65\) |
derivativedivides | \(\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2 \left (-\frac {11 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {19 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {19 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {13 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d \,a^{3}}\) | \(125\) |
default | \(\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2 \left (-\frac {11 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {19 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {19 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {13 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d \,a^{3}}\) | \(125\) |
risch | \(-\frac {13 x}{8 a^{3}}+\frac {{\mathrm e}^{i \left (d x +c \right )}}{8 d \,a^{3}}+\frac {{\mathrm e}^{-i \left (d x +c \right )}}{8 d \,a^{3}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{3}}+\frac {\sin \left (4 d x +4 c \right )}{32 d \,a^{3}}-\frac {\cos \left (3 d x +3 c \right )}{4 d \,a^{3}}-\frac {3 \sin \left (2 d x +2 c \right )}{4 d \,a^{3}}\) | \(132\) |
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Time = 0.27 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.85 \[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {8 \, \cos \left (d x + c\right )^{3} + 13 \, d x - {\left (2 \, \cos \left (d x + c\right )^{3} - 13 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 8 \, \cos \left (d x + c\right ) + 4 \, \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 4 \, \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{8 \, a^{3} d} \]
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Timed out. \[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (93) = 186\).
Time = 0.31 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.72 \[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {\frac {11 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {16 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {19 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {19 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {16 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {11 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{3} + \frac {4 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} + \frac {13 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {4 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{4 \, d} \]
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Time = 0.33 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.30 \[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {13 \, {\left (d x + c\right )}}{a^{3}} - \frac {8 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {2 \, {\left (11 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 16 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 19 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 19 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 16 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 11 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a^{3}}}{8 \, d} \]
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Time = 11.37 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.24 \[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {13\,\mathrm {atan}\left (\frac {169}{16\,\left (\frac {169\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}+\frac {13}{2}\right )}-\frac {13\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\left (\frac {169\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}+\frac {13}{2}\right )}\right )}{4\,a^3\,d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}-\frac {-\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {19\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+\frac {19\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {11\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )} \]
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