Optimal. Leaf size=22 \[ \frac{x}{2}-\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \]
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Rubi [A] time = 0.0306626, antiderivative size = 30, normalized size of antiderivative = 1.36, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3137, 3124, 31} \[ \frac{x}{2}-\frac{1}{2} \log \left (\tan \left (\frac{x}{2}\right )+1\right )-\frac{1}{2} \log (\sin (x)+\cos (x)+1) \]
Antiderivative was successfully verified.
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Rule 3137
Rule 3124
Rule 31
Rubi steps
\begin{align*} \int \frac{\sin (x)}{1+\cos (x)+\sin (x)} \, dx &=\frac{x}{2}-\frac{1}{2} \log (1+\cos (x)+\sin (x))-\frac{1}{2} \int \frac{1}{1+\cos (x)+\sin (x)} \, dx\\ &=\frac{x}{2}-\frac{1}{2} \log (1+\cos (x)+\sin (x))-\operatorname{Subst}\left (\int \frac{1}{2+2 x} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=\frac{x}{2}-\frac{1}{2} \log (1+\cos (x)+\sin (x))-\frac{1}{2} \log \left (1+\tan \left (\frac{x}{2}\right )\right )\\ \end{align*}
Mathematica [A] time = 0.047012, size = 22, normalized size = 1. \[ \frac{x}{2}-\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 25, normalized size = 1.1 \begin{align*} -\ln \left ( 1+\tan \left ({\frac{x}{2}} \right ) \right ) +{\frac{1}{2}\ln \left ( 1+ \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2} \right ) }+{\frac{x}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.48185, size = 55, normalized size = 2.5 \begin{align*} \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) - \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right ) + \frac{1}{2} \, \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 = 1.86841, size = 39, normalized size = 1.77 \begin{align*} \frac{1}{2} \, x - \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 = 0.31174, size = 22, normalized size = 1. \begin{align*} \frac{x}{2} - \log{\left (\tan{\left (\frac{x}{2} \right )} + 1 \right )} + \frac{\log{\left (\tan ^{2}{\left (\frac{x}{2} \right )} + 1 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1655, size = 34, normalized size = 1.55 \begin{align*} \frac{1}{2} \, x + \frac{1}{2} \, \log \left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right ) - \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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