Optimal. Leaf size=42 \[ \frac {3 x}{2 a}+\frac {2 \cos (x)}{a}+\frac {\sin ^2(x) \cos (x)}{a \sin (x)+a}-\frac {3 \sin (x) \cos (x)}{2 a} \]
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Rubi [A] time = 0.05, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2767, 2734} \[ \frac {3 x}{2 a}+\frac {2 \cos (x)}{a}+\frac {\sin ^2(x) \cos (x)}{a \sin (x)+a}-\frac {3 \sin (x) \cos (x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 2767
Rubi steps
\begin {align*} \int \frac {\sin ^3(x)}{a+a \sin (x)} \, dx &=\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*}
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Mathematica [B] time = 0.09, size = 87, normalized size = 2.07 \[ \frac {\left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \left (12 x \sin \left (\frac {x}{2}\right )-20 \sin \left (\frac {x}{2}\right )+3 \sin \left (\frac {3 x}{2}\right )-\sin \left (\frac {5 x}{2}\right )+4 (3 x+1) \cos \left (\frac {x}{2}\right )+3 \cos \left (\frac {3 x}{2}\right )+\cos \left (\frac {5 x}{2}\right )\right )}{8 a (\sin (x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 53, normalized size = 1.26 \[ \frac {\cos \relax (x)^{3} + 3 \, {\left (x + 1\right )} \cos \relax (x) + 2 \, \cos \relax (x)^{2} - {\left (\cos \relax (x)^{2} - 3 \, x - \cos \relax (x) + 2\right )} \sin \relax (x) + 3 \, x + 2}{2 \, {\left (a \cos \relax (x) + a \sin \relax (x) + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.95, size = 56, normalized size = 1.33 \[ \frac {3 \, x}{2 \, a} + \frac {\tan \left (\frac {1}{2} \, x\right )^{3} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} - \tan \left (\frac {1}{2} \, x\right ) + 2}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{2} a} + \frac {2}{a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 100, normalized size = 2.38 \[ \frac {\tan ^{3}\left (\frac {x}{2}\right )}{a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+\frac {2 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}-\frac {\tan \left (\frac {x}{2}\right )}{a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+\frac {2}{a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+\frac {3 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a}+\frac {2}{a \left (\tan \left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.65, size = 128, normalized size = 3.05 \[ \frac {\frac {\sin \relax (x)}{\cos \relax (x) + 1} + \frac {5 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {3 \, \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {3 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + 4}{a + \frac {a \sin \relax (x)}{\cos \relax (x) + 1} + \frac {2 \, a \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {2 \, a \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {a \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {a \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}}} + \frac {3 \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.91, size = 59, normalized size = 1.40 \[ \frac {3\,x}{2\,a}+\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\mathrm {tan}\left (\frac {x}{2}\right )+4}{a\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}^2\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.79, size = 665, normalized size = 15.83 \[ \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} \]
Verification of antiderivative is not currently implemented for this CAS.
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