Optimal. Leaf size=16 \[ -x-\frac{\sin (x)}{1-\cos (x)} \]
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Rubi [A] time = 0.0263574, antiderivative size = 16, 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 = {2735, 2648} \[ -x-\frac{\sin (x)}{1-\cos (x)} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2648
Rubi steps
\begin{align*} \int \frac{\cos (x)}{1-\cos (x)} \, dx &=-x+\int \frac{1}{1-\cos (x)} \, dx\\ &=-x-\frac{\sin (x)}{1-\cos (x)}\\ \end{align*}
Mathematica [A] time = 0.0231004, size = 21, normalized size = 1.31 \[ \frac{2 x \sin ^2\left (\frac{x}{2}\right )+\sin (x)}{\cos (x)-1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 13, normalized size = 0.8 \begin{align*} - \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}-x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46981, size = 31, normalized size = 1.94 \begin{align*} -\frac{\cos \left (x\right ) + 1}{\sin \left (x\right )} - 2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80118, size = 45, normalized size = 2.81 \begin{align*} -\frac{x \sin \left (x\right ) + \cos \left (x\right ) + 1}{\sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.480344, size = 8, normalized size = 0.5 \begin{align*} - x - \frac{1}{\tan{\left (\frac{x}{2} \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05088, size = 16, normalized size = 1. \begin{align*} -x - \frac{1}{\tan \left (\frac{1}{2} \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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