Integrand size = 13, antiderivative size = 103 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^3} \, dx=\frac {23 \text {arctanh}(\cos (x))}{2 a^3}-\frac {136 \cot (x)}{5 a^3}-\frac {136 \cot ^3(x)}{15 a^3}+\frac {23 \cot (x) \csc (x)}{2 a^3}+\frac {\cot (x) \csc ^2(x)}{5 (a+a \sin (x))^3}+\frac {13 \cot (x) \csc ^2(x)}{15 a (a+a \sin (x))^2}+\frac {23 \cot (x) \csc ^2(x)}{3 \left (a^3+a^3 \sin (x)\right )} \]
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Time = 0.18 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2845, 3057, 2827, 3852, 3853, 3855} \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^3} \, dx=\frac {23 \text {arctanh}(\cos (x))}{2 a^3}-\frac {136 \cot ^3(x)}{15 a^3}-\frac {136 \cot (x)}{5 a^3}+\frac {23 \cot (x) \csc (x)}{2 a^3}+\frac {23 \cot (x) \csc ^2(x)}{3 \left (a^3 \sin (x)+a^3\right )}+\frac {13 \cot (x) \csc ^2(x)}{15 a (a \sin (x)+a)^2}+\frac {\cot (x) \csc ^2(x)}{5 (a \sin (x)+a)^3} \]
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Rule 2827
Rule 2845
Rule 3057
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\cot (x) \csc ^2(x)}{5 (a+a \sin (x))^3}+\frac {\int \frac {\csc ^4(x) (8 a-5 a \sin (x))}{(a+a \sin (x))^2} \, dx}{5 a^2} \\ & = \frac {\cot (x) \csc ^2(x)}{5 (a+a \sin (x))^3}+\frac {13 \cot (x) \csc ^2(x)}{15 a (a+a \sin (x))^2}+\frac {\int \frac {\csc ^4(x) \left (63 a^2-52 a^2 \sin (x)\right )}{a+a \sin (x)} \, dx}{15 a^4} \\ & = \frac {\cot (x) \csc ^2(x)}{5 (a+a \sin (x))^3}+\frac {13 \cot (x) \csc ^2(x)}{15 a (a+a \sin (x))^2}+\frac {23 \cot (x) \csc ^2(x)}{3 \left (a^3+a^3 \sin (x)\right )}+\frac {\int \csc ^4(x) \left (408 a^3-345 a^3 \sin (x)\right ) \, dx}{15 a^6} \\ & = \frac {\cot (x) \csc ^2(x)}{5 (a+a \sin (x))^3}+\frac {13 \cot (x) \csc ^2(x)}{15 a (a+a \sin (x))^2}+\frac {23 \cot (x) \csc ^2(x)}{3 \left (a^3+a^3 \sin (x)\right )}-\frac {23 \int \csc ^3(x) \, dx}{a^3}+\frac {136 \int \csc ^4(x) \, dx}{5 a^3} \\ & = \frac {23 \cot (x) \csc (x)}{2 a^3}+\frac {\cot (x) \csc ^2(x)}{5 (a+a \sin (x))^3}+\frac {13 \cot (x) \csc ^2(x)}{15 a (a+a \sin (x))^2}+\frac {23 \cot (x) \csc ^2(x)}{3 \left (a^3+a^3 \sin (x)\right )}-\frac {23 \int \csc (x) \, dx}{2 a^3}-\frac {136 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (x)\right )}{5 a^3} \\ & = \frac {23 \text {arctanh}(\cos (x))}{2 a^3}-\frac {136 \cot (x)}{5 a^3}-\frac {136 \cot ^3(x)}{15 a^3}+\frac {23 \cot (x) \csc (x)}{2 a^3}+\frac {\cot (x) \csc ^2(x)}{5 (a+a \sin (x))^3}+\frac {13 \cot (x) \csc ^2(x)}{15 a (a+a \sin (x))^2}+\frac {23 \cot (x) \csc ^2(x)}{3 \left (a^3+a^3 \sin (x)\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(299\) vs. \(2(103)=206\).
Time = 1.67 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.90 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^3} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (48 \sin \left (\frac {x}{2}\right )-5 \cos \left (\frac {x}{2}\right ) \left (1+\cot \left (\frac {x}{2}\right )\right )^5 \sin ^2\left (\frac {x}{2}\right )+45 \left (1+\cot \left (\frac {x}{2}\right )\right )^5 \sin ^3\left (\frac {x}{2}\right )-24 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+352 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2-176 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3+2752 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^4-400 \cot \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5+1380 \log \left (\cos \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5-1380 \log \left (\sin \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5+400 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5 \tan \left (\frac {x}{2}\right )-45 \cos ^3\left (\frac {x}{2}\right ) \left (1+\tan \left (\frac {x}{2}\right )\right )^5+5 \cos ^2\left (\frac {x}{2}\right ) \sin \left (\frac {x}{2}\right ) \left (1+\tan \left (\frac {x}{2}\right )\right )^5\right )}{120 a^3 (1+\sin (x))^3} \]
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Time = 0.66 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.07
method | result | size |
default | \(\frac {\frac {\left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3}-3 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+27 \tan \left (\frac {x}{2}\right )-\frac {1}{3 \tan \left (\frac {x}{2}\right )^{3}}+\frac {3}{\tan \left (\frac {x}{2}\right )^{2}}-\frac {27}{\tan \left (\frac {x}{2}\right )}-92 \ln \left (\tan \left (\frac {x}{2}\right )\right )-\frac {64}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {32}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {256}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {96}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {240}{\tan \left (\frac {x}{2}\right )+1}}{8 a^{3}}\) | \(110\) |
parallelrisch | \(\frac {\left (-1380 \cos \left (4 x \right )+5520 \cos \left (2 x \right )-4140\right ) \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+\left (-4352 \tan \left (x \right )+12804\right ) \cos \left (2 x \right )-3201 \cos \left (4 x \right )-2760 \cos \left (3 x \right )-4352 \sin \left (2 x \right )+2176 \sin \left (4 x \right )+\left (1088 \left (\sec ^{2}\left (x \right )\right )+2176\right ) \left (\tan ^{3}\left (x \right )\right )+\left (-768 \left (\sec ^{3}\left (x \right )\right )-1472 \sec \left (x \right )\right ) \left (\tan ^{2}\left (x \right )\right )+\left (-320 \left (\sec ^{4}\left (x \right )\right )-7360 \sin \left (x \right )+4352\right ) \tan \left (x \right )+2760 \cos \left (x \right )-9603}{120 a^{3} \left (3+\cos \left (4 x \right )-4 \cos \left (2 x \right )\right )}\) | \(128\) |
risch | \(-\frac {-544-12622 \,{\mathrm e}^{4 i x}+5347 \,{\mathrm e}^{2 i x}-4370 \,{\mathrm e}^{8 i x}+345 \,{\mathrm e}^{10 i x}+1725 i {\mathrm e}^{9 i x}+11684 \,{\mathrm e}^{6 i x}-8050 i {\mathrm e}^{7 i x}+13340 i {\mathrm e}^{5 i x}-9230 i {\mathrm e}^{3 i x}+2375 i {\mathrm e}^{i x}}{15 \left ({\mathrm e}^{2 i x}-1\right )^{3} \left ({\mathrm e}^{i x}+i\right )^{5} a^{3}}+\frac {23 \ln \left ({\mathrm e}^{i x}+1\right )}{2 a^{3}}-\frac {23 \ln \left ({\mathrm e}^{i x}-1\right )}{2 a^{3}}\) | \(129\) |
norman | \(\frac {-\frac {1}{24 a}+\frac {\tan \left (\frac {x}{2}\right )}{6 a}-\frac {23 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{12 a}+\frac {23 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{12 a}-\frac {\tan ^{10}\left (\frac {x}{2}\right )}{6 a}+\frac {\tan ^{11}\left (\frac {x}{2}\right )}{24 a}-\frac {228 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}-\frac {1067 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{20 a}-\frac {611 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{2 a}-\frac {1567 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{8 a}-\frac {567 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{8 a}}{\tan \left (\frac {x}{2}\right )^{3} a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {23 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{2 a^{3}}\) | \(144\) |
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Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (89) = 178\).
Time = 0.29 (sec) , antiderivative size = 333, normalized size of antiderivative = 3.23 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^3} \, dx=\frac {1088 \, \cos \left (x\right )^{6} + 2574 \, \cos \left (x\right )^{5} - 2428 \, \cos \left (x\right )^{4} - 5338 \, \cos \left (x\right )^{3} + 1372 \, \cos \left (x\right )^{2} + 345 \, {\left (\cos \left (x\right )^{6} - 2 \, \cos \left (x\right )^{5} - 6 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{3} + 9 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{5} + 3 \, \cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{3} - 7 \, \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 345 \, {\left (\cos \left (x\right )^{6} - 2 \, \cos \left (x\right )^{5} - 6 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{3} + 9 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{5} + 3 \, \cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{3} - 7 \, \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 2 \, {\left (544 \, \cos \left (x\right )^{5} - 743 \, \cos \left (x\right )^{4} - 1957 \, \cos \left (x\right )^{3} + 712 \, \cos \left (x\right )^{2} + 1398 \, \cos \left (x\right ) + 6\right )} \sin \left (x\right ) + 2784 \, \cos \left (x\right ) - 12}{60 \, {\left (a^{3} \cos \left (x\right )^{6} - 2 \, a^{3} \cos \left (x\right )^{5} - 6 \, a^{3} \cos \left (x\right )^{4} + 4 \, a^{3} \cos \left (x\right )^{3} + 9 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3} - {\left (a^{3} \cos \left (x\right )^{5} + 3 \, a^{3} \cos \left (x\right )^{4} - 3 \, a^{3} \cos \left (x\right )^{3} - 7 \, a^{3} \cos \left (x\right )^{2} + 2 \, a^{3} \cos \left (x\right ) + 4 \, a^{3}\right )} \sin \left (x\right )\right )}} \]
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\[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^3} \, dx=\frac {\int \frac {\csc ^{4}{\left (x \right )}}{\sin ^{3}{\left (x \right )} + 3 \sin ^{2}{\left (x \right )} + 3 \sin {\left (x \right )} + 1}\, dx}{a^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (89) = 178\).
Time = 0.20 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.25 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^3} \, dx=\frac {\frac {20 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {230 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {4777 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {15785 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac {22390 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} - \frac {14940 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac {4005 \, \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} - 5}{120 \, {\left (\frac {a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {10 \, a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {10 \, a^{3} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {5 \, a^{3} \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} + \frac {a^{3} \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}}\right )}} + \frac {\frac {81 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {9 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {\sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}}{24 \, a^{3}} - \frac {23 \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{2 \, a^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.24 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^3} \, dx=-\frac {23 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{2 \, a^{3}} + \frac {506 \, \tan \left (\frac {1}{2} \, x\right )^{3} - 81 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, x\right ) - 1}{24 \, a^{3} \tan \left (\frac {1}{2} \, x\right )^{3}} - \frac {2 \, {\left (225 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 810 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 1160 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 760 \, \tan \left (\frac {1}{2} \, x\right ) + 197\right )}}{15 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} + \frac {a^{6} \tan \left (\frac {1}{2} \, x\right )^{3} - 9 \, a^{6} \tan \left (\frac {1}{2} \, x\right )^{2} + 81 \, a^{6} \tan \left (\frac {1}{2} \, x\right )}{24 \, a^{9}} \]
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Time = 6.14 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.14 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^3} \, dx=\frac {27\,\mathrm {tan}\left (\frac {x}{2}\right )}{8\,a^3}-\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8\,a^3}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{24\,a^3}-\frac {23\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{2\,a^3}-\frac {\frac {267\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7}{8}+\frac {249\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6}{2}+\frac {2239\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5}{12}+\frac {3157\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{24}+\frac {4777\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{120}+\frac {23\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{12}-\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{6}+\frac {1}{24}}{a^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \]
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