Optimal. Leaf size=12 \[ -\frac{1}{2 (1-\cos (x))^2} \]
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Rubi [A] time = 0.0188377, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2667, 32} \[ -\frac{1}{2 (1-\cos (x))^2} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 32
Rubi steps
\begin{align*} \int \frac{\sin (x)}{(1-\cos (x))^3} \, dx &=\operatorname{Subst}\left (\int \frac{1}{(1+x)^3} \, dx,x,-\cos (x)\right )\\ &=-\frac{1}{2 (1-\cos (x))^2}\\ \end{align*}
Mathematica [A] time = 0.0101411, size = 12, normalized size = 1. \[ -\frac{1}{8} \csc ^4\left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 11, normalized size = 0.9 \begin{align*} -{\frac{1}{2\, \left ( 1-\cos \left ( x \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4021, size = 11, normalized size = 0.92 \begin{align*} -\frac{1}{2 \,{\left (\cos \left (x\right ) - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5605, size = 43, normalized size = 3.58 \begin{align*} -\frac{1}{2 \,{\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 = 0.816951, size = 15, normalized size = 1.25 \begin{align*} - \frac{1}{2 \cos ^{2}{\left (x \right )} - 4 \cos{\left (x \right )} + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1406, size = 11, normalized size = 0.92 \begin{align*} -\frac{1}{2 \,{\left (\cos \left (x\right ) - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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