Optimal. Leaf size=10 \[ -\frac{1}{1-\cos (x)} \]
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Rubi [A] time = 0.019053, antiderivative size = 10, 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} \[ -\frac{1}{1-\cos (x)} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 32
Rubi steps
\begin{align*} \int \frac{\sin (x)}{(1-\cos (x))^2} \, dx &=\operatorname{Subst}\left (\int \frac{1}{(1+x)^2} \, dx,x,-\cos (x)\right )\\ &=-\frac{1}{1-\cos (x)}\\ \end{align*}
Mathematica [A] time = 0.0093907, size = 12, normalized size = 1.2 \[ -\frac{1}{2} \csc ^2\left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 11, normalized size = 1.1 \begin{align*} - \left ( 1-\cos \left ( x \right ) \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13927, size = 8, normalized size = 0.8 \begin{align*} \frac{1}{\cos \left (x\right ) - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5355, size = 22, normalized size = 2.2 \begin{align*} \frac{1}{\cos \left (x\right ) - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.467631, size = 7, normalized size = 0.7 \begin{align*} \frac{\cos{\left (x \right )}}{\cos{\left (x \right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13099, size = 8, normalized size = 0.8 \begin{align*} \frac{1}{\cos \left (x\right ) - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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