Integrand size = 13, antiderivative size = 65 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^2} \, dx=\frac {5 \text {arctanh}(\cos (x))}{a^2}-\frac {4 \cot (x)}{a^2}-\frac {\cot ^3(x)}{3 a^2}+\frac {\cot (x) \csc (x)}{a^2}-\frac {\cos (x)}{3 a^2 (1+\sin (x))^2}-\frac {13 \cos (x)}{3 a^2 (1+\sin (x))} \]
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Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2845, 3057, 2827, 3852, 3853, 3855} \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^2} \, dx=\frac {5 \text {arctanh}(\cos (x))}{a^2}-\frac {4 \cot ^3(x)}{a^2}-\frac {12 \cot (x)}{a^2}+\frac {5 \cot (x) \csc (x)}{a^2}+\frac {10 \cot (x) \csc ^2(x)}{3 a^2 (\sin (x)+1)}+\frac {\cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2} \]
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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)}{3 (a+a \sin (x))^2}+\frac {\int \frac {\csc ^4(x) (6 a-4 a \sin (x))}{a+a \sin (x)} \, dx}{3 a^2} \\ & = \frac {10 \cot (x) \csc ^2(x)}{3 a^2 (1+\sin (x))}+\frac {\cot (x) \csc ^2(x)}{3 (a+a \sin (x))^2}+\frac {\int \csc ^4(x) \left (36 a^2-30 a^2 \sin (x)\right ) \, dx}{3 a^4} \\ & = \frac {10 \cot (x) \csc ^2(x)}{3 a^2 (1+\sin (x))}+\frac {\cot (x) \csc ^2(x)}{3 (a+a \sin (x))^2}-\frac {10 \int \csc ^3(x) \, dx}{a^2}+\frac {12 \int \csc ^4(x) \, dx}{a^2} \\ & = \frac {5 \cot (x) \csc (x)}{a^2}+\frac {10 \cot (x) \csc ^2(x)}{3 a^2 (1+\sin (x))}+\frac {\cot (x) \csc ^2(x)}{3 (a+a \sin (x))^2}-\frac {5 \int \csc (x) \, dx}{a^2}-\frac {12 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (x)\right )}{a^2} \\ & = \frac {5 \text {arctanh}(\cos (x))}{a^2}-\frac {12 \cot (x)}{a^2}-\frac {4 \cot ^3(x)}{a^2}+\frac {5 \cot (x) \csc (x)}{a^2}+\frac {10 \cot (x) \csc ^2(x)}{3 a^2 (1+\sin (x))}+\frac {\cot (x) \csc ^2(x)}{3 (a+a \sin (x))^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(238\) vs. \(2(65)=130\).
Time = 3.18 (sec) , antiderivative size = 238, normalized size of antiderivative = 3.66 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^2} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (-\cos \left (\frac {x}{2}\right ) \left (1+\cot \left (\frac {x}{2}\right )\right )^3+16 \sin \left (\frac {x}{2}\right )+6 \left (1+\cot \left (\frac {x}{2}\right )\right )^3 \sin \left (\frac {x}{2}\right )-8 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+208 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2-44 \cot \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3+120 \log \left (\cos \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3-120 \log \left (\sin \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3+44 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3 \tan \left (\frac {x}{2}\right )-6 \cos \left (\frac {x}{2}\right ) \left (1+\tan \left (\frac {x}{2}\right )\right )^3+\sin \left (\frac {x}{2}\right ) \left (1+\tan \left (\frac {x}{2}\right )\right )^3\right )}{24 a^2 (1+\sin (x))^2} \]
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Time = 0.57 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.38
method | result | size |
default | \(\frac {\frac {\left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3}-2 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+15 \tan \left (\frac {x}{2}\right )-\frac {1}{3 \tan \left (\frac {x}{2}\right )^{3}}+\frac {2}{\tan \left (\frac {x}{2}\right )^{2}}-\frac {15}{\tan \left (\frac {x}{2}\right )}-40 \ln \left (\tan \left (\frac {x}{2}\right )\right )-\frac {32}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {16}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {80}{\tan \left (\frac {x}{2}\right )+1}}{8 a^{2}}\) | \(90\) |
parallelrisch | \(\frac {\left (-60 \cos \left (4 x \right )+240 \cos \left (2 x \right )-180\right ) \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+\left (-192 \tan \left (x \right )+460\right ) \cos \left (2 x \right )-115 \cos \left (4 x \right )-120 \cos \left (3 x \right )-192 \sin \left (2 x \right )+96 \sin \left (4 x \right )+96 \left (\tan ^{3}\left (x \right )\right )-64 \left (\tan ^{2}\left (x \right )\right ) \sec \left (x \right )+\left (-32 \left (\sec ^{2}\left (x \right )\right )-320 \sin \left (x \right )+192\right ) \tan \left (x \right )+120 \cos \left (x \right )-345}{12 a^{2} \left (3+\cos \left (4 x \right )-4 \cos \left (2 x \right )\right )}\) | \(113\) |
risch | \(-\frac {2 \left (-85 \,{\mathrm e}^{6 i x}+45 i {\mathrm e}^{7 i x}+153 \,{\mathrm e}^{4 i x}-135 i {\mathrm e}^{5 i x}+15 \,{\mathrm e}^{8 i x}-99 \,{\mathrm e}^{2 i x}+155 i {\mathrm e}^{3 i x}+24-57 i {\mathrm e}^{i x}\right )}{3 \left ({\mathrm e}^{2 i x}-1\right )^{3} \left ({\mathrm e}^{i x}+i\right )^{3} a^{2}}-\frac {5 \ln \left ({\mathrm e}^{i x}-1\right )}{a^{2}}+\frac {5 \ln \left ({\mathrm e}^{i x}+1\right )}{a^{2}}\) | \(114\) |
norman | \(\frac {-\frac {1}{24 a}+\frac {\tan \left (\frac {x}{2}\right )}{8 a}-\frac {5 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{4 a}+\frac {5 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{4 a}-\frac {\tan ^{8}\left (\frac {x}{2}\right )}{8 a}+\frac {\tan ^{9}\left (\frac {x}{2}\right )}{24 a}-\frac {115 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{6 a}-\frac {145 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{4 a}-\frac {85 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{4 a}}{\tan \left (\frac {x}{2}\right )^{3} a \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {5 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{2}}\) | \(122\) |
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Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (59) = 118\).
Time = 0.30 (sec) , antiderivative size = 266, normalized size of antiderivative = 4.09 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^2} \, dx=-\frac {48 \, \cos \left (x\right )^{5} - 18 \, \cos \left (x\right )^{4} - 108 \, \cos \left (x\right )^{3} + 22 \, \cos \left (x\right )^{2} - 15 \, {\left (\cos \left (x\right )^{5} + 2 \, \cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{3} - 4 \, \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{4} - \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )^{2} + \cos \left (x\right ) + 2\right )} \sin \left (x\right ) + \cos \left (x\right ) + 2\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (x\right )^{5} + 2 \, \cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{3} - 4 \, \cos \left (x\right )^{2} + {\left (\cos \left (x\right )^{4} - \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )^{2} + \cos \left (x\right ) + 2\right )} \sin \left (x\right ) + \cos \left (x\right ) + 2\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 2 \, {\left (24 \, \cos \left (x\right )^{4} + 33 \, \cos \left (x\right )^{3} - 21 \, \cos \left (x\right )^{2} - 32 \, \cos \left (x\right ) - 1\right )} \sin \left (x\right ) + 62 \, \cos \left (x\right ) - 2}{6 \, {\left (a^{2} \cos \left (x\right )^{5} + 2 \, a^{2} \cos \left (x\right )^{4} - 2 \, a^{2} \cos \left (x\right )^{3} - 4 \, a^{2} \cos \left (x\right )^{2} + a^{2} \cos \left (x\right ) + 2 \, a^{2} + {\left (a^{2} \cos \left (x\right )^{4} - a^{2} \cos \left (x\right )^{3} - 3 \, a^{2} \cos \left (x\right )^{2} + a^{2} \cos \left (x\right ) + 2 \, a^{2}\right )} \sin \left (x\right )\right )}} \]
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\[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^2} \, dx=\frac {\int \frac {\csc ^{4}{\left (x \right )}}{\sin ^{2}{\left (x \right )} + 2 \sin {\left (x \right )} + 1}\, dx}{a^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (59) = 118\).
Time = 0.23 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.74 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^2} \, dx=\frac {\frac {3 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {30 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {342 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {561 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac {285 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} - 1}{24 \, {\left (\frac {a^{2} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {3 \, a^{2} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {3 \, a^{2} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {a^{2} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}}\right )}} + \frac {\frac {45 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {6 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {\sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}}{24 \, a^{2}} - \frac {5 \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{2}} \]
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Time = 0.31 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.75 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^2} \, dx=-\frac {5 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a^{2}} + \frac {110 \, \tan \left (\frac {1}{2} \, x\right )^{6} + 45 \, \tan \left (\frac {1}{2} \, x\right )^{5} - 231 \, \tan \left (\frac {1}{2} \, x\right )^{4} - 232 \, \tan \left (\frac {1}{2} \, x\right )^{3} - 30 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 3 \, \tan \left (\frac {1}{2} \, x\right ) - 1}{24 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right )\right )}^{3} a^{2}} + \frac {a^{4} \tan \left (\frac {1}{2} \, x\right )^{3} - 6 \, a^{4} \tan \left (\frac {1}{2} \, x\right )^{2} + 45 \, a^{4} \tan \left (\frac {1}{2} \, x\right )}{24 \, a^{6}} \]
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Time = 5.89 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.55 \[ \int \frac {\csc ^4(x)}{(a+a \sin (x))^2} \, dx=\frac {15\,\mathrm {tan}\left (\frac {x}{2}\right )}{8\,a^2}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{4\,a^2}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{24\,a^2}-\frac {5\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a^2}-\frac {\frac {95\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5}{8}+\frac {187\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{8}+\frac {57\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{4}+\frac {5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{4}-\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{8}+\frac {1}{24}}{a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^3} \]
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