Optimal. Leaf size=10 \[ \cos (x)+\log (1-\cos (x)) \]
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Rubi [A] time = 0.0330559, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2833, 43} \[ \cos (x)+\log (1-\cos (x)) \]
Antiderivative was successfully verified.
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Rule 2833
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos (x) \sin (x)}{1-\cos (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{x}{1+x} \, dx,x,-\cos (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (1+\frac{1}{-1-x}\right ) \, dx,x,-\cos (x)\right )\\ &=\cos (x)+\log (1-\cos (x))\\ \end{align*}
Mathematica [A] time = 0.0174454, size = 12, normalized size = 1.2 \[ \cos (x)+2 \log \left (\sin \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 9, normalized size = 0.9 \begin{align*} \cos \left ( x \right ) +\ln \left ( -1+\cos \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.9537, size = 11, normalized size = 1.1 \begin{align*} \cos \left (x\right ) + \log \left (\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 = 1.92776, size = 45, normalized size = 4.5 \begin{align*} \cos \left (x\right ) + \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.19146, size = 8, normalized size = 0.8 \begin{align*} \log{\left (\cos{\left (x \right )} - 1 \right )} + \cos{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.109, size = 14, normalized size = 1.4 \begin{align*} \cos \left (x\right ) + \log \left (-\cos \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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