Optimal. Leaf size=10 \[ \cos (x)+\log (1-\cos (x)) \]
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Rubi [A] time = 0.03, antiderivative size = 10, normalized size of antiderivative = 1.00, 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 43
Rule 2833
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*}
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Mathematica [A] time = 0.02, size = 12, normalized size = 1.20 \[ \cos (x)+2 \log \left (\sin \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.38, size = 10, normalized size = 1.00 \[ \cos \relax (x) + \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 10, normalized size = 1.00 \[ \cos \relax (x) + \log \left (-\cos \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 9, normalized size = 0.90 \[ \cos \relax (x )+\ln \left (-1+\cos \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 8, normalized size = 0.80 \[ \cos \relax (x) + \log \left (\cos \relax (x) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.92, size = 8, normalized size = 0.80 \[ \ln \left (\cos \relax (x)-1\right )+\cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 8, normalized size = 0.80 \[ \log {\left (\cos {\relax (x )} - 1 \right )} + \cos {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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