Integrand size = 13, antiderivative size = 35 \[ \int \frac {\sin ^2(x)}{(a+a \sin (x))^2} \, dx=\frac {x}{a^2}+\frac {5 \cos (x)}{3 a^2 (1+\sin (x))}-\frac {\cos (x)}{3 (a+a \sin (x))^2} \]
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Time = 0.06 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2837, 2814, 2727} \[ \int \frac {\sin ^2(x)}{(a+a \sin (x))^2} \, dx=\frac {x}{a^2}+\frac {5 \cos (x)}{3 a^2 (\sin (x)+1)}-\frac {\cos (x)}{3 (a \sin (x)+a)^2} \]
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Rule 2727
Rule 2814
Rule 2837
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (x)}{3 (a+a \sin (x))^2}+\frac {\int \frac {-2 a+3 a \sin (x)}{a+a \sin (x)} \, dx}{3 a^2} \\ & = \frac {x}{a^2}-\frac {\cos (x)}{3 (a+a \sin (x))^2}-\frac {5 \int \frac {1}{a+a \sin (x)} \, dx}{3 a} \\ & = \frac {x}{a^2}-\frac {\cos (x)}{3 (a+a \sin (x))^2}+\frac {5 \cos (x)}{3 \left (a^2+a^2 \sin (x)\right )} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.97 \[ \int \frac {\sin ^2(x)}{(a+a \sin (x))^2} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (3 (-4+3 x) \cos \left (\frac {x}{2}\right )+(10-3 x) \cos \left (\frac {3 x}{2}\right )+6 (-3+2 x+x \cos (x)) \sin \left (\frac {x}{2}\right )\right )}{6 a^2 (1+\sin (x))^2} \]
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Time = 0.34 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94
method | result | size |
parallelrisch | \(\frac {3 x +4-5 \tan \left (x \right )+6 \sec \left (x \right )+2 \tan \left (x \right ) \left (\sec ^{2}\left (x \right )\right )-2 \left (\sec ^{3}\left (x \right )\right )}{3 a^{2}}\) | \(33\) |
risch | \(\frac {x}{a^{2}}+\frac {4 \,{\mathrm e}^{2 i x}-\frac {10}{3}+6 i {\mathrm e}^{i x}}{\left ({\mathrm e}^{i x}+i\right )^{3} a^{2}}\) | \(39\) |
default | \(\frac {-\frac {4}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {2}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {8}{4 \tan \left (\frac {x}{2}\right )+4}+2 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a^{2}}\) | \(44\) |
norman | \(\frac {\frac {x}{a}+\frac {x \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}+\frac {2 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}+\frac {6 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}+\frac {12 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}+\frac {8}{3 a}+\frac {3 x \tan \left (\frac {x}{2}\right )}{a}+\frac {5 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}+\frac {7 x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}+\frac {7 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}+\frac {5 x \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}+\frac {3 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}+\frac {6 \tan \left (\frac {x}{2}\right )}{a}+\frac {22 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {20 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{3 a}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{2} a \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}\) | \(179\) |
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Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (31) = 62\).
Time = 0.30 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.34 \[ \int \frac {\sin ^2(x)}{(a+a \sin (x))^2} \, dx=\frac {{\left (3 \, x - 5\right )} \cos \left (x\right )^{2} - {\left (3 \, x + 4\right )} \cos \left (x\right ) - {\left ({\left (3 \, x + 5\right )} \cos \left (x\right ) + 6 \, x + 1\right )} \sin \left (x\right ) - 6 \, x + 1}{3 \, {\left (a^{2} \cos \left (x\right )^{2} - a^{2} \cos \left (x\right ) - 2 \, a^{2} - {\left (a^{2} \cos \left (x\right ) + 2 \, a^{2}\right )} \sin \left (x\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (32) = 64\).
Time = 1.02 (sec) , antiderivative size = 321, normalized size of antiderivative = 9.17 \[ \int \frac {\sin ^2(x)}{(a+a \sin (x))^2} \, dx=\frac {3 x \tan ^{3}{\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} + \frac {9 x \tan ^{2}{\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} + \frac {9 x \tan {\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} + \frac {3 x}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} + \frac {6 \tan ^{2}{\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} + \frac {18 \tan {\left (\frac {x}{2} \right )}}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} + \frac {8}{3 a^{2} \tan ^{3}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 9 a^{2} \tan {\left (\frac {x}{2} \right )} + 3 a^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (31) = 62\).
Time = 0.30 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.57 \[ \int \frac {\sin ^2(x)}{(a+a \sin (x))^2} \, dx=\frac {2 \, {\left (\frac {9 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {3 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 4\right )}}{3 \, {\left (a^{2} + \frac {3 \, a^{2} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {3 \, a^{2} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}\right )}} + \frac {2 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^2(x)}{(a+a \sin (x))^2} \, dx=\frac {x}{a^{2}} + \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, x\right ) + 4\right )}}{3 \, a^{2} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{3}} \]
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Time = 5.64 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \frac {\sin ^2(x)}{(a+a \sin (x))^2} \, dx=\frac {x}{a^2}+\frac {2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+6\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {8}{3}}{a^2\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^3} \]
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