Optimal. Leaf size=5 \[ -\log (\cos (x)) \]
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Rubi [B] time = 0.0210279, antiderivative size = 11, normalized size of antiderivative = 2.2, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2667, 31} \[ -\frac{1}{2} \log (\cos (2 x)+1) \]
Antiderivative was successfully verified.
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Rule 2667
Rule 31
Rubi steps
\begin{align*} \int \frac{\sin (2 x)}{1+\cos (2 x)} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\cos (2 x)\right )\right )\\ &=-\frac{1}{2} \log (1+\cos (2 x))\\ \end{align*}
Mathematica [A] time = 0.0027969, size = 5, normalized size = 1. \[ -\log (\cos (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 10, normalized size = 2. \begin{align*} -{\frac{\ln \left ( 1+\cos \left ( 2\,x \right ) \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14241, size = 12, normalized size = 2.4 \begin{align*} -\frac{1}{2} \, \log \left (\cos \left (2 \, x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.58993, size = 41, normalized size = 8.2 \begin{align*} -\frac{1}{2} \, \log \left (\frac{1}{2} \, \cos \left (2 \, 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 = 0.127385, size = 10, normalized size = 2. \begin{align*} - \frac{\log{\left (\cos{\left (2 x \right )} + 1 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1576, size = 12, normalized size = 2.4 \begin{align*} -\frac{1}{2} \, \log \left (\cos \left (2 \, x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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