Optimal. Leaf size=19 \[ \frac{\cos ^2(x)}{2 a}-\frac{\cos (x)}{a} \]
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Rubi [A] time = 0.0399363, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2667} \[ \frac{\cos ^2(x)}{2 a}-\frac{\cos (x)}{a} \]
Antiderivative was successfully verified.
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Rule 2667
Rubi steps
\begin{align*} \int \frac{\sin ^3(x)}{a+a \cos (x)} \, dx &=-\frac{\operatorname{Subst}(\int (a-x) \, dx,x,a \cos (x))}{a^3}\\ &=-\frac{\cos (x)}{a}+\frac{\cos ^2(x)}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0120588, size = 13, normalized size = 0.68 \[ \frac{2 \sin ^4\left (\frac{x}{2}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 16, normalized size = 0.8 \begin{align*}{\frac{1}{a} \left ({\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}}{2}}-\cos \left ( x \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0833, size = 19, normalized size = 1. \begin{align*} \frac{\cos \left (x\right )^{2} - 2 \, \cos \left (x\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54458, size = 39, normalized size = 2.05 \begin{align*} \frac{\cos \left (x\right )^{2} - 2 \, \cos \left (x\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.976852, size = 88, normalized size = 4.63 \begin{align*} \frac{4 \tan ^{4}{\left (\frac{x}{2} \right )}}{3 a \tan ^{4}{\left (\frac{x}{2} \right )} + 6 a \tan ^{2}{\left (\frac{x}{2} \right )} + 3 a} - \frac{4 \tan ^{2}{\left (\frac{x}{2} \right )}}{3 a \tan ^{4}{\left (\frac{x}{2} \right )} + 6 a \tan ^{2}{\left (\frac{x}{2} \right )} + 3 a} - \frac{2}{3 a \tan ^{4}{\left (\frac{x}{2} \right )} + 6 a \tan ^{2}{\left (\frac{x}{2} \right )} + 3 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14154, size = 19, normalized size = 1. \begin{align*} \frac{\cos \left (x\right )^{2} - 2 \, \cos \left (x\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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