Optimal. Leaf size=14 \[ \frac{\sin ^3(x)}{3 (\cos (x)+1)^3} \]
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Rubi [A] time = 0.0308714, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2671} \[ \frac{\sin ^3(x)}{3 (\cos (x)+1)^3} \]
Antiderivative was successfully verified.
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Rule 2671
Rubi steps
\begin{align*} \int \frac{\sin ^2(x)}{(1+\cos (x))^3} \, dx &=\frac{\sin ^3(x)}{3 (1+\cos (x))^3}\\ \end{align*}
Mathematica [A] time = 0.0280643, size = 12, normalized size = 0.86 \[ \frac{1}{3} \tan ^3\left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 9, normalized size = 0.6 \begin{align*}{\frac{1}{3} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.60098, size = 16, normalized size = 1.14 \begin{align*} \frac{\sin \left (x\right )^{3}}{3 \,{\left (\cos \left (x\right ) + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53684, size = 70, normalized size = 5. \begin{align*} -\frac{{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right )}{3 \,{\left (\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.60673, size = 7, normalized size = 0.5 \begin{align*} \frac{\tan ^{3}{\left (\frac{x}{2} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16325, size = 11, normalized size = 0.79 \begin{align*} \frac{1}{3} \, \tan \left (\frac{1}{2} \, x\right )^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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