Optimal. Leaf size=47 \[ -\frac{2 x}{a^2}-\frac{4 \cos (x)}{3 a^2}-\frac{2 \cos (x)}{a^2 (\sin (x)+1)}+\frac{\sin ^2(x) \cos (x)}{3 (a \sin (x)+a)^2} \]
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Rubi [A] time = 0.143152, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {2765, 2968, 3023, 12, 2735, 2648} \[ -\frac{2 x}{a^2}-\frac{4 \cos (x)}{3 a^2}-\frac{2 \cos (x)}{a^2 (\sin (x)+1)}+\frac{\sin ^2(x) \cos (x)}{3 (a \sin (x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2765
Rule 2968
Rule 3023
Rule 12
Rule 2735
Rule 2648
Rubi steps
\begin{align*} \int \frac{\sin ^3(x)}{(a+a \sin (x))^2} \, dx &=\frac{\cos (x) \sin ^2(x)}{3 (a+a \sin (x))^2}-\frac{\int \frac{\sin (x) (2 a-4 a \sin (x))}{a+a \sin (x)} \, dx}{3 a^2}\\ &=\frac{\cos (x) \sin ^2(x)}{3 (a+a \sin (x))^2}-\frac{\int \frac{2 a \sin (x)-4 a \sin ^2(x)}{a+a \sin (x)} \, dx}{3 a^2}\\ &=-\frac{4 \cos (x)}{3 a^2}+\frac{\cos (x) \sin ^2(x)}{3 (a+a \sin (x))^2}-\frac{\int \frac{6 a^2 \sin (x)}{a+a \sin (x)} \, dx}{3 a^3}\\ &=-\frac{4 \cos (x)}{3 a^2}+\frac{\cos (x) \sin ^2(x)}{3 (a+a \sin (x))^2}-\frac{2 \int \frac{\sin (x)}{a+a \sin (x)} \, dx}{a}\\ &=-\frac{2 x}{a^2}-\frac{4 \cos (x)}{3 a^2}+\frac{\cos (x) \sin ^2(x)}{3 (a+a \sin (x))^2}+\frac{2 \int \frac{1}{a+a \sin (x)} \, dx}{a}\\ &=-\frac{2 x}{a^2}-\frac{4 \cos (x)}{3 a^2}+\frac{\cos (x) \sin ^2(x)}{3 (a+a \sin (x))^2}-\frac{2 \cos (x)}{a^2+a^2 \sin (x)}\\ \end{align*}
Mathematica [A] time = 0.224515, size = 84, normalized size = 1.79 \[ -\frac{\left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \left (6 (6 x-5) \cos \left (\frac{x}{2}\right )+(41-12 x) \cos \left (\frac{3 x}{2}\right )-3 \cos \left (\frac{5 x}{2}\right )+6 \sin \left (\frac{x}{2}\right ) (8 x+4 (x+1) \cos (x)+\cos (2 x)-9)\right )}{12 a^2 (\sin (x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 66, normalized size = 1.4 \begin{align*} -2\,{\frac{1}{{a}^{2} \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) }}-4\,{\frac{\arctan \left ( \tan \left ( x/2 \right ) \right ) }{{a}^{2}}}+{\frac{4}{3\,{a}^{2}} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}-2\,{\frac{1}{{a}^{2} \left ( \tan \left ( x/2 \right ) +1 \right ) ^{2}}}-4\,{\frac{1}{{a}^{2} \left ( \tan \left ( x/2 \right ) +1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.49702, size = 194, normalized size = 4.13 \begin{align*} -\frac{4 \,{\left (\frac{12 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{11 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{9 \, \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}} + 5\right )}}{3 \,{\left (a^{2} + \frac{3 \, a^{2} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{4 \, a^{2} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{4 \, 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{a^{2} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}\right )}} - \frac{4 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.45906, size = 259, normalized size = 5.51 \begin{align*} -\frac{{\left (6 \, x - 11\right )} \cos \left (x\right )^{2} + 3 \, \cos \left (x\right )^{3} -{\left (6 \, x + 13\right )} \cos \left (x\right ) -{\left (2 \,{\left (3 \, x + 7\right )} \cos \left (x\right ) + 3 \, \cos \left (x\right )^{2} + 12 \, x + 1\right )} \sin \left (x\right ) - 12 \, 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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 105.718, size = 848, normalized size = 18.04 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34885, size = 69, normalized size = 1.47 \begin{align*} -\frac{2 \, x}{a^{2}} - \frac{2}{{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )} a^{2}} - \frac{2 \,{\left (6 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 15 \, \tan \left (\frac{1}{2} \, x\right ) + 7\right )}}{3 \, a^{2}{\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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