Optimal. Leaf size=7 \[ -\frac{\cos (x)}{a} \]
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Rubi [A] time = 0.0413498, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3175, 2638} \[ -\frac{\cos (x)}{a} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 2638
Rubi steps
\begin{align*} \int \frac{\sin ^3(x)}{a-a \cos ^2(x)} \, dx &=\frac{\int \sin (x) \, dx}{a}\\ &=-\frac{\cos (x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0020658, size = 7, normalized size = 1. \[ -\frac{\cos (x)}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 8, normalized size = 1.1 \begin{align*} -{\frac{\cos \left ( x \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.945899, size = 9, normalized size = 1.29 \begin{align*} -\frac{\cos \left (x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85319, size = 15, normalized size = 2.14 \begin{align*} -\frac{\cos \left (x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.00868, size = 36, normalized size = 5.14 \begin{align*} \frac{2 \tan ^{2}{\left (\frac{x}{2} \right )}}{3 a \tan ^{2}{\left (\frac{x}{2} \right )} + 3 a} - \frac{4}{3 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 = 1.12301, size = 9, normalized size = 1.29 \begin{align*} -\frac{\cos \left (x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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