Integrand size = 27, antiderivative size = 102 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {x}{a}-\frac {3 \text {arctanh}(\cos (c+d x))}{8 a d}-\frac {\cot (c+d x)}{a d}+\frac {\cot ^3(c+d x)}{3 a d}+\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 a d} \]
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Time = 0.11 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2918, 2691, 3855, 3554, 8} \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 \text {arctanh}(\cos (c+d x))}{8 a d}+\frac {\cot ^3(c+d x)}{3 a d}-\frac {\cot (c+d x)}{a d}-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 a d}+\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {x}{a} \]
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Rule 8
Rule 2691
Rule 2918
Rule 3554
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \cot ^4(c+d x) \, dx}{a}+\frac {\int \cot ^4(c+d x) \csc (c+d x) \, dx}{a} \\ & = \frac {\cot ^3(c+d x)}{3 a d}-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 a d}-\frac {3 \int \cot ^2(c+d x) \csc (c+d x) \, dx}{4 a}+\frac {\int \cot ^2(c+d x) \, dx}{a} \\ & = -\frac {\cot (c+d x)}{a d}+\frac {\cot ^3(c+d x)}{3 a d}+\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 a d}+\frac {3 \int \csc (c+d x) \, dx}{8 a}-\frac {\int 1 \, dx}{a} \\ & = -\frac {x}{a}-\frac {3 \text {arctanh}(\cos (c+d x))}{8 a d}-\frac {\cot (c+d x)}{a d}+\frac {\cot ^3(c+d x)}{3 a d}+\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 a d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(232\) vs. \(2(102)=204\).
Time = 1.00 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.27 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc ^4(c+d x) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (72 c+72 d x+18 \cos (c+d x)+30 \cos (3 (c+d x))+24 c \cos (4 (c+d x))+24 d x \cos (4 (c+d x))+27 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+9 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-12 \cos (2 (c+d x)) \left (8 c+8 d x+3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-27 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-9 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+32 \sin (2 (c+d x))-32 \sin (4 (c+d x))\right )}{192 a d (1+\sin (c+d x))} \]
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Time = 0.30 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.24
method | result | size |
parallelrisch | \(\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-192 d x +120 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+72 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{192 d a}\) | \(126\) |
derivativedivides | \(\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {10}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-32 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}\) | \(136\) |
default | \(\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {10}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-32 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}\) | \(136\) |
risch | \(-\frac {x}{a}-\frac {48 i {\mathrm e}^{6 i \left (d x +c \right )}+15 \,{\mathrm e}^{7 i \left (d x +c \right )}-96 i {\mathrm e}^{4 i \left (d x +c \right )}+9 \,{\mathrm e}^{5 i \left (d x +c \right )}+80 i {\mathrm e}^{2 i \left (d x +c \right )}+9 \,{\mathrm e}^{3 i \left (d x +c \right )}-32 i+15 \,{\mathrm e}^{i \left (d x +c \right )}}{12 a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d a}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d a}\) | \(152\) |
norman | \(\frac {-\frac {1}{64 a d}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{192 d a}+\frac {29 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}-\frac {91 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}+\frac {91 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}-\frac {29 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}-\frac {5 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}+\frac {\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d a}-\frac {x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d a}-\frac {11 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}-\frac {7 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d}\) | \(325\) |
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Time = 0.26 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.68 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {48 \, d x \cos \left (d x + c\right )^{4} - 96 \, d x \cos \left (d x + c\right )^{2} + 30 \, \cos \left (d x + c\right )^{3} + 48 \, d x + 9 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 9 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 16 \, {\left (4 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 18 \, \cos \left (d x + c\right )}{48 \, {\left (a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{2} + a d\right )}} \]
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Timed out. \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (94) = 188\).
Time = 0.32 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.13 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {120 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {24 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {8 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{a} - \frac {384 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {72 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {{\left (\frac {8 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {24 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {120 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{4}}{a \sin \left (d x + c\right )^{4}}}{192 \, d} \]
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Time = 0.35 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.64 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {192 \, {\left (d x + c\right )}}{a} - \frac {72 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} - \frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{4}} + \frac {150 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
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Time = 10.44 (sec) , antiderivative size = 317, normalized size of antiderivative = 3.11 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-8\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+8\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+384\,\mathrm {atan}\left (\frac {8\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-3\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+72\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{192\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4} \]
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