Integrand size = 29, antiderivative size = 100 \[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {7 \text {arctanh}(\cos (c+d x))}{8 a^3 d}-\frac {4 \cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {\cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d} \]
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Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2954, 2952, 2691, 3855, 2687, 30, 3853, 14} \[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {7 \text {arctanh}(\cos (c+d x))}{8 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}-\frac {4 \cot ^3(c+d x)}{3 a^3 d}+\frac {3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}+\frac {\cot (c+d x) \csc (c+d x)}{8 a^3 d} \]
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Rule 14
Rule 30
Rule 2687
Rule 2691
Rule 2952
Rule 2954
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \cot ^2(c+d x) \csc ^4(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6} \\ & = \frac {\int \left (-a^3 \cot ^2(c+d x) \csc (c+d x)+3 a^3 \cot ^2(c+d x) \csc ^2(c+d x)-3 a^3 \cot ^2(c+d x) \csc ^3(c+d x)+a^3 \cot ^2(c+d x) \csc ^4(c+d x)\right ) \, dx}{a^6} \\ & = -\frac {\int \cot ^2(c+d x) \csc (c+d x) \, dx}{a^3}+\frac {\int \cot ^2(c+d x) \csc ^4(c+d x) \, dx}{a^3}+\frac {3 \int \cot ^2(c+d x) \csc ^2(c+d x) \, dx}{a^3}-\frac {3 \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx}{a^3} \\ & = \frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}+\frac {\int \csc (c+d x) \, dx}{2 a^3}+\frac {3 \int \csc ^3(c+d x) \, dx}{4 a^3}+\frac {\text {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}+\frac {3 \text {Subst}\left (\int x^2 \, dx,x,-\cot (c+d x)\right )}{a^3 d} \\ & = -\frac {\text {arctanh}(\cos (c+d x))}{2 a^3 d}-\frac {\cot ^3(c+d x)}{a^3 d}+\frac {\cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}+\frac {3 \int \csc (c+d x) \, dx}{8 a^3}+\frac {\text {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d} \\ & = -\frac {7 \text {arctanh}(\cos (c+d x))}{8 a^3 d}-\frac {4 \cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {\cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d} \\ \end{align*}
Time = 2.18 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.89 \[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\csc ^5(c+d x) \left (560 \cos (c+d x)-40 \cos (3 (c+d x))-136 \cos (5 (c+d x))+1050 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-1050 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-780 \sin (2 (c+d x))-525 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+525 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+30 \sin (4 (c+d x))+105 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-105 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))\right )}{1920 a^3 d} \]
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Time = 0.39 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.48
method | result | size |
parallelrisch | \(\frac {6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-45 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+45 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+130 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-130 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+120 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-420 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+840 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+420 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{960 d \,a^{3}}\) | \(148\) |
derivativedivides | \(\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+28 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {13}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {3}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {14}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{32 d \,a^{3}}\) | \(150\) |
default | \(\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+28 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {13}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {3}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {14}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{32 d \,a^{3}}\) | \(150\) |
risch | \(-\frac {-360 i {\mathrm e}^{8 i \left (d x +c \right )}+15 \,{\mathrm e}^{9 i \left (d x +c \right )}+960 i {\mathrm e}^{6 i \left (d x +c \right )}-390 \,{\mathrm e}^{7 i \left (d x +c \right )}-400 i {\mathrm e}^{4 i \left (d x +c \right )}+320 i {\mathrm e}^{2 i \left (d x +c \right )}+390 \,{\mathrm e}^{3 i \left (d x +c \right )}-136 i-15 \,{\mathrm e}^{i \left (d x +c \right )}}{60 a^{3} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}-\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d \,a^{3}}+\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d \,a^{3}}\) | \(158\) |
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Time = 0.28 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.69 \[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {272 \, \cos \left (d x + c\right )^{5} - 320 \, \cos \left (d x + c\right )^{3} - 105 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 105 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 30 \, {\left (\cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (90) = 180\).
Time = 0.23 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.35 \[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {\frac {420 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {120 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {130 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {45 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {6 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {840 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {{\left (\frac {45 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {130 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {120 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {420 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 6\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a^{3} \sin \left (d x + c\right )^{5}}}{960 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (90) = 180\).
Time = 0.40 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.86 \[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {840 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {1918 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 420 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 130 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} + \frac {6 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 130 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 420 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{960 \, d} \]
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Time = 10.46 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.91 \[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-6\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-45\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+130\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-420\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+420\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-130\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+840\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{960\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]
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