Integrand size = 21, antiderivative size = 114 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {13 \text {arctanh}(\cos (c+d x))}{8 a^3 d}-\frac {4 \cot (c+d x)}{a^3 d}-\frac {5 \cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {13 \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d} \]
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Time = 0.16 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2787, 2836, 3853, 3855, 3852} \[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {13 \text {arctanh}(\cos (c+d x))}{8 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}-\frac {5 \cot ^3(c+d x)}{3 a^3 d}-\frac {4 \cot (c+d x)}{a^3 d}+\frac {3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}+\frac {13 \cot (c+d x) \csc (c+d x)}{8 a^3 d} \]
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Rule 2787
Rule 2836
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \csc ^6(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6} \\ & = \frac {\int \left (-a^3 \csc ^3(c+d x)+3 a^3 \csc ^4(c+d x)-3 a^3 \csc ^5(c+d x)+a^3 \csc ^6(c+d x)\right ) \, dx}{a^6} \\ & = -\frac {\int \csc ^3(c+d x) \, dx}{a^3}+\frac {\int \csc ^6(c+d x) \, dx}{a^3}+\frac {3 \int \csc ^4(c+d x) \, dx}{a^3}-\frac {3 \int \csc ^5(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 {9 \int \csc ^3(c+d x) \, dx}{4 a^3}-\frac {\text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^3 d} \\ & = \frac {\text {arctanh}(\cos (c+d x))}{2 a^3 d}-\frac {4 \cot (c+d x)}{a^3 d}-\frac {5 \cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {13 \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 {9 \int \csc (c+d x) \, dx}{8 a^3} \\ & = \frac {13 \text {arctanh}(\cos (c+d x))}{8 a^3 d}-\frac {4 \cot (c+d x)}{a^3 d}-\frac {5 \cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {13 \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 = 1.71 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.66 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\csc ^5(c+d x) \left (-1600 \cos (c+d x)+1520 \cos (3 (c+d x))-304 \cos (5 (c+d x))+1950 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-1950 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+1500 \sin (2 (c+d x))-975 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+975 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-390 \sin (4 (c+d x))+195 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-195 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))\right )}{1920 a^3 d} \]
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Result contains complex when optimal does not.
Time = 0.49 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.28
method | result | size |
risch | \(-\frac {195 \,{\mathrm e}^{9 i \left (d x +c \right )}-720 i {\mathrm e}^{6 i \left (d x +c \right )}-750 \,{\mathrm e}^{7 i \left (d x +c \right )}+2320 i {\mathrm e}^{4 i \left (d x +c \right )}-1520 i {\mathrm e}^{2 i \left (d x +c \right )}+750 \,{\mathrm e}^{3 i \left (d x +c \right )}+304 i-195 \,{\mathrm e}^{i \left (d x +c \right )}}{60 a^{3} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {13 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d \,a^{3}}-\frac {13 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d \,a^{3}}\) | \(146\) |
parallelrisch | \(\frac {-6 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+45 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-45 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-170 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+170 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+480 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-480 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1380 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-1560 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1380 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{960 a^{3} d}\) | \(148\) |
derivativedivides | \(\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {17 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-16 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+46 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-52 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {46}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {17}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {16}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {3}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}}{32 d \,a^{3}}\) | \(150\) |
default | \(\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {17 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-16 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+46 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-52 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {46}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {17}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {16}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {3}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}}{32 d \,a^{3}}\) | \(150\) |
norman | \(\frac {-\frac {1}{160 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d a}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{192 d a}+\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{48 d a}-\frac {13 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d a}+\frac {13 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d a}-\frac {\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )}{48 d a}+\frac {\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )}{192 d a}-\frac {\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d a}+\frac {\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )}{160 d a}-\frac {128 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {79 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}-\frac {511 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}-\frac {3581 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{480 d a}-\frac {2633 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {13 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{3}}\) | \(321\) |
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Time = 0.27 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.57 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {608 \, \cos \left (d x + c\right )^{5} - 1520 \, \cos \left (d x + c\right )^{3} - 195 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 195 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 30 \, {\left (13 \, \cos \left (d x + c\right )^{3} - 19 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 960 \, \cos \left (d x + c\right )}{240 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (104) = 208\).
Time = 0.23 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.05 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {\frac {1380 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {480 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {170 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {45 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {6 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {1560 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac {{\left (\frac {45 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {170 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {480 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {1380 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 6\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a^{3} \sin \left (d x + c\right )^{5}}}{960 \, d} \]
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Time = 0.37 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.64 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {1560 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {3562 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1380 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 170 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} - \frac {6 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 170 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 480 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1380 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{960 \, d} \]
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Time = 10.40 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.55 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {6\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+45\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-170\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+480\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-1380\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+1380\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-480\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+170\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1560\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{960\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]
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