Optimal. Leaf size=16 \[ -\frac{\sin ^3(x)}{3 (1-\cos (x))^3} \]
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Rubi [A] time = 0.0308387, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2671} \[ -\frac{\sin ^3(x)}{3 (1-\cos (x))^3} \]
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 = 0.0293909, size = 12, normalized size = 0.75 \[ -\frac{1}{3} \cot ^3\left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, 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 = 1.3989, size = 16, normalized size = 1. \begin{align*} -\frac{{\left (\cos \left (x\right ) + 1\right )}^{3}}{3 \, \sin \left (x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56474, size = 72, normalized size = 4.5 \begin{align*} \frac{\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1}{3 \,{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.27375, size = 10, normalized size = 0.62 \begin{align*} - \frac{1}{3 \tan ^{3}{\left (\frac{x}{2} \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15603, size = 11, normalized size = 0.69 \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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