Optimal. Leaf size=10 \[ \cos (x)+\log (1-\cos (x)) \]
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Rubi [A] time = 0.0684733, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4392, 2833, 43} \[ \cos (x)+\log (1-\cos (x)) \]
Antiderivative was successfully verified.
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Rule 4392
Rule 2833
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos (x)}{-\cot (x)+\csc (x)} \, dx &=\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 = 0.0100583, size = 20, normalized size = 2. \[ 2 \log \left (\sin \left (\frac{x}{2}\right )\right )-2 \sin ^2\left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.078, 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 [B] time = 1.66173, size = 62, normalized size = 6.2 \begin{align*} \frac{2}{\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1} + 2 \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) - \log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.480792, 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{\cos{\left (x \right )}}{\cot{\left (x \right )} - \csc{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09205, 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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