Integrand size = 27, antiderivative size = 134 \[ \int \frac {\cot ^8(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {5 \text {arctanh}(\cos (c+d x))}{128 a d}+\frac {\cot ^7(c+d x)}{7 a d}+\frac {5 \cot (c+d x) \csc (c+d x)}{128 a d}-\frac {5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}+\frac {5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d} \]
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Time = 0.17 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2918, 2691, 3853, 3855, 2687, 30} \[ \int \frac {\cot ^8(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {5 \text {arctanh}(\cos (c+d x))}{128 a d}+\frac {\cot ^7(c+d x)}{7 a d}-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}+\frac {5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}-\frac {5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}+\frac {5 \cot (c+d x) \csc (c+d x)}{128 a d} \]
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Rule 30
Rule 2687
Rule 2691
Rule 2918
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \cot ^6(c+d x) \csc ^2(c+d x) \, dx}{a}+\frac {\int \cot ^6(c+d x) \csc ^3(c+d x) \, dx}{a} \\ & = -\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}-\frac {5 \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx}{8 a}-\frac {\text {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{a d} \\ & = \frac {\cot ^7(c+d x)}{7 a d}+\frac {5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}+\frac {5 \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx}{16 a} \\ & = \frac {\cot ^7(c+d x)}{7 a d}-\frac {5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}+\frac {5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}-\frac {5 \int \csc ^3(c+d x) \, dx}{64 a} \\ & = \frac {\cot ^7(c+d x)}{7 a d}+\frac {5 \cot (c+d x) \csc (c+d x)}{128 a d}-\frac {5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}+\frac {5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}-\frac {5 \int \csc (c+d x) \, dx}{128 a} \\ & = \frac {5 \text {arctanh}(\cos (c+d x))}{128 a d}+\frac {\cot ^7(c+d x)}{7 a d}+\frac {5 \cot (c+d x) \csc (c+d x)}{128 a d}-\frac {5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}+\frac {5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(291\) vs. \(2(134)=268\).
Time = 1.91 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.17 \[ \int \frac {\cot ^8(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^8(c+d x) \left (-24710 \cos (c+d x)-12530 \cos (3 (c+d x))-5558 \cos (5 (c+d x))-210 \cos (7 (c+d x))+3675 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-5880 \cos (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+2940 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-840 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+105 \cos (8 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-3675 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+5880 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-2940 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+840 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-105 \cos (8 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+5376 \sin (2 (c+d x))+5376 \sin (4 (c+d x))+2304 \sin (6 (c+d x))+384 \sin (8 (c+d x))\right )}{344064 a d} \]
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Time = 0.42 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.69
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {2 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-6 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-10 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {10}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}+\frac {2}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}+\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{256 d a}\) | \(226\) |
default | \(\frac {\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {2 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-6 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-10 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {10}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}+\frac {2}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}+\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{256 d a}\) | \(226\) |
parallelrisch | \(\frac {21 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-21 \left (\cot ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-48 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+48 \left (\cot ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-112 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+112 \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+336 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-336 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+168 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-168 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1008 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1008 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+336 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-336 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1680 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1680 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1680 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{43008 d a}\) | \(226\) |
risch | \(-\frac {13440 i {\mathrm e}^{10 i \left (d x +c \right )}+105 \,{\mathrm e}^{15 i \left (d x +c \right )}+2688 i {\mathrm e}^{14 i \left (d x +c \right )}+2779 \,{\mathrm e}^{13 i \left (d x +c \right )}-8064 i {\mathrm e}^{4 i \left (d x +c \right )}+6265 \,{\mathrm e}^{11 i \left (d x +c \right )}-13440 i {\mathrm e}^{8 i \left (d x +c \right )}+12355 \,{\mathrm e}^{9 i \left (d x +c \right )}+384 i {\mathrm e}^{2 i \left (d x +c \right )}+12355 \,{\mathrm e}^{7 i \left (d x +c \right )}+8064 i {\mathrm e}^{6 i \left (d x +c \right )}+6265 \,{\mathrm e}^{5 i \left (d x +c \right )}-2688 i {\mathrm e}^{12 i \left (d x +c \right )}+2779 \,{\mathrm e}^{3 i \left (d x +c \right )}-384 i+105 \,{\mathrm e}^{i \left (d x +c \right )}}{1344 a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d a}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d a}\) | \(238\) |
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Time = 0.26 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.61 \[ \int \frac {\cot ^8(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {768 \, \cos \left (d x + c\right )^{7} \sin \left (d x + c\right ) - 210 \, \cos \left (d x + c\right )^{7} - 1022 \, \cos \left (d x + c\right )^{5} + 770 \, \cos \left (d x + c\right )^{3} + 105 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 105 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 210 \, \cos \left (d x + c\right )}{5376 \, {\left (a d \cos \left (d x + c\right )^{8} - 4 \, a d \cos \left (d x + c\right )^{6} + 6 \, a d \cos \left (d x + c\right )^{4} - 4 \, a d \cos \left (d x + c\right )^{2} + a d\right )}} \]
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Timed out. \[ \int \frac {\cot ^8(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (122) = 244\).
Time = 0.22 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.64 \[ \int \frac {\cot ^8(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {1680 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {336 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1008 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {168 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {336 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {112 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {48 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {21 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}}{a} - \frac {1680 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {{\left (\frac {48 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {112 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {336 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {168 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1008 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {336 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {1680 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 21\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{8}}{a \sin \left (d x + c\right )^{8}}}{43008 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (122) = 244\).
Time = 0.38 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.04 \[ \int \frac {\cot ^8(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {1680 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} - \frac {21 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 48 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 112 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 336 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 168 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1008 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 336 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1680 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{8}} - \frac {4566 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1680 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 336 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1008 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 168 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 336 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 112 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 21}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}}}{43008 \, d} \]
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Time = 12.89 (sec) , antiderivative size = 435, normalized size of antiderivative = 3.25 \[ \int \frac {\cot ^8(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-21\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+48\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}-48\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+112\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-336\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-168\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+1008\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-336\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-1680\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+1680\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+336\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-1008\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+168\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+336\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-112\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1680\,\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 )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{43008\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8} \]
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