Optimal. Leaf size=11 \[ -\log \left (\cot \left (\frac{x}{2}\right )+1\right ) \]
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Rubi [A] time = 0.0107476, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3121, 31} \[ -\log \left (\cot \left (\frac{x}{2}\right )+1\right ) \]
Antiderivative was successfully verified.
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Rule 3121
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{1-\cos (x)+\sin (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\cot \left (\frac{x}{2}\right )\right )\\ &=-\log \left (1+\cot \left (\frac{x}{2}\right )\right )\\ \end{align*}
Mathematica [B] time = 0.0143862, size = 24, normalized size = 2.18 \[ \log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 16, normalized size = 1.5 \begin{align*} \ln \left ( \tan \left ({\frac{x}{2}} \right ) \right ) -\ln \left ( 1+\tan \left ({\frac{x}{2}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.9437, size = 34, normalized size = 3.09 \begin{align*} -\log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right ) + \log \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 = 2.25173, size = 68, normalized size = 6.18 \begin{align*} \frac{1}{2} \, \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - \frac{1}{2} \, \log \left (\sin \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.237113, size = 14, normalized size = 1.27 \begin{align*} - \log{\left (\tan{\left (\frac{x}{2} \right )} + 1 \right )} + \log{\left (\tan{\left (\frac{x}{2} \right )} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09904, size = 23, normalized size = 2.09 \begin{align*} -\log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) + 1 \right |}\right ) + \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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