Integrand size = 13, antiderivative size = 42 \[ \int \frac {\sin ^3(x)}{a+a \sin (x)} \, dx=\frac {3 x}{2 a}+\frac {2 \cos (x)}{a}-\frac {3 \cos (x) \sin (x)}{2 a}+\frac {\cos (x) \sin ^2(x)}{a+a \sin (x)} \]
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Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2846, 2813} \[ \int \frac {\sin ^3(x)}{a+a \sin (x)} \, dx=\frac {3 x}{2 a}+\frac {2 \cos (x)}{a}+\frac {\sin ^2(x) \cos (x)}{a \sin (x)+a}-\frac {3 \sin (x) \cos (x)}{2 a} \]
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Rule 2813
Rule 2846
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (x) \sin ^2(x)}{a+a \sin (x)}-\frac {\int \sin (x) (2 a-3 a \sin (x)) \, dx}{a^2} \\ & = \frac {3 x}{2 a}+\frac {2 \cos (x)}{a}-\frac {3 \cos (x) \sin (x)}{2 a}+\frac {\cos (x) \sin ^2(x)}{a+a \sin (x)} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(87\) vs. \(2(42)=84\).
Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.07 \[ \int \frac {\sin ^3(x)}{a+a \sin (x)} \, dx=\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (4 (1+3 x) \cos \left (\frac {x}{2}\right )+3 \cos \left (\frac {3 x}{2}\right )+\cos \left (\frac {5 x}{2}\right )-20 \sin \left (\frac {x}{2}\right )+12 x \sin \left (\frac {x}{2}\right )+3 \sin \left (\frac {3 x}{2}\right )-\sin \left (\frac {5 x}{2}\right )\right )}{8 a (1+\sin (x))} \]
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Time = 0.36 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.69
method | result | size |
parallelrisch | \(\frac {6 x +8-\sin \left (2 x \right )+4 \cos \left (x \right )-4 \tan \left (x \right )+4 \sec \left (x \right )}{4 a}\) | \(29\) |
risch | \(\frac {3 x}{2 a}+\frac {{\mathrm e}^{i x}}{2 a}+\frac {{\mathrm e}^{-i x}}{2 a}+\frac {2}{\left ({\mathrm e}^{i x}+i\right ) a}-\frac {\sin \left (2 x \right )}{4 a}\) | \(52\) |
default | \(\frac {\frac {16}{8 \tan \left (\frac {x}{2}\right )+8}+\frac {2 \left (\frac {\left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{2}+\tan ^{2}\left (\frac {x}{2}\right )-\frac {\tan \left (\frac {x}{2}\right )}{2}+1\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{2}}+3 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a}\) | \(58\) |
norman | \(\frac {-\frac {\tan ^{5}\left (\frac {x}{2}\right )}{a}+\frac {4 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}+\frac {5 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}+\frac {3 x}{2 a}+\frac {8}{3 a}+\frac {3 x \tan \left (\frac {x}{2}\right )}{2 a}+\frac {9 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {9 x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {9 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {9 x \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {3 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {3 x \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{2 a}-\frac {4 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {\tan \left (\frac {x}{2}\right )}{3 a}+\frac {5 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{3 a}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )}\) | \(178\) |
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Time = 0.32 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.26 \[ \int \frac {\sin ^3(x)}{a+a \sin (x)} \, dx=\frac {\cos \left (x\right )^{3} + 3 \, {\left (x + 1\right )} \cos \left (x\right ) + 2 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{2} - 3 \, x - \cos \left (x\right ) + 2\right )} \sin \left (x\right ) + 3 \, x + 2}{2 \, {\left (a \cos \left (x\right ) + a \sin \left (x\right ) + a\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 665 vs. \(2 (39) = 78\).
Time = 0.88 (sec) , antiderivative size = 665, normalized size of antiderivative = 15.83 \[ \int \frac {\sin ^3(x)}{a+a \sin (x)} \, dx=\frac {3 x \tan ^{5}{\left (\frac {x}{2} \right )}}{2 a \tan ^{5}{\left (\frac {x}{2} \right )} + 2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{3}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a \tan {\left (\frac {x}{2} \right )} + 2 a} + \frac {3 x \tan ^{4}{\left (\frac {x}{2} \right )}}{2 a \tan ^{5}{\left (\frac {x}{2} \right )} + 2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{3}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a \tan {\left (\frac {x}{2} \right )} + 2 a} + \frac {6 x \tan ^{3}{\left (\frac {x}{2} \right )}}{2 a \tan ^{5}{\left (\frac {x}{2} \right )} + 2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{3}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a \tan {\left (\frac {x}{2} \right )} + 2 a} + \frac {6 x \tan ^{2}{\left (\frac {x}{2} \right )}}{2 a \tan ^{5}{\left (\frac {x}{2} \right )} + 2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{3}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a \tan {\left (\frac {x}{2} \right )} + 2 a} + \frac {3 x \tan {\left (\frac {x}{2} \right )}}{2 a \tan ^{5}{\left (\frac {x}{2} \right )} + 2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{3}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a \tan {\left (\frac {x}{2} \right )} + 2 a} + \frac {3 x}{2 a \tan ^{5}{\left (\frac {x}{2} \right )} + 2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{3}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a \tan {\left (\frac {x}{2} \right )} + 2 a} + \frac {6 \tan ^{4}{\left (\frac {x}{2} \right )}}{2 a \tan ^{5}{\left (\frac {x}{2} \right )} + 2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{3}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a \tan {\left (\frac {x}{2} \right )} + 2 a} + \frac {6 \tan ^{3}{\left (\frac {x}{2} \right )}}{2 a \tan ^{5}{\left (\frac {x}{2} \right )} + 2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{3}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a \tan {\left (\frac {x}{2} \right )} + 2 a} + \frac {10 \tan ^{2}{\left (\frac {x}{2} \right )}}{2 a \tan ^{5}{\left (\frac {x}{2} \right )} + 2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{3}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a \tan {\left (\frac {x}{2} \right )} + 2 a} + \frac {2 \tan {\left (\frac {x}{2} \right )}}{2 a \tan ^{5}{\left (\frac {x}{2} \right )} + 2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{3}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a \tan {\left (\frac {x}{2} \right )} + 2 a} + \frac {8}{2 a \tan ^{5}{\left (\frac {x}{2} \right )} + 2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{3}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a \tan {\left (\frac {x}{2} \right )} + 2 a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (38) = 76\).
Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 3.05 \[ \int \frac {\sin ^3(x)}{a+a \sin (x)} \, dx=\frac {\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {5 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 4}{a + \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {2 \, a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {2 \, a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {a \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {a \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}} + \frac {3 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a} \]
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Time = 0.32 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.33 \[ \int \frac {\sin ^3(x)}{a+a \sin (x)} \, dx=\frac {3 \, x}{2 \, a} + \frac {\tan \left (\frac {1}{2} \, x\right )^{3} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} - \tan \left (\frac {1}{2} \, x\right ) + 2}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{2} a} + \frac {2}{a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}} \]
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Time = 5.95 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.40 \[ \int \frac {\sin ^3(x)}{a+a \sin (x)} \, dx=\frac {3\,x}{2\,a}+\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\mathrm {tan}\left (\frac {x}{2}\right )+4}{a\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}^2\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )} \]
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