Optimal. Leaf size=7 \[ -2 \log (\cos (x)+1) \]
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Rubi [A] time = 0.0443849, antiderivative size = 7, normalized size of antiderivative = 1., 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 = {12, 31} \[ -2 \log (\cos (x)+1) \]
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rubi steps
\begin{align*} \int \frac{\sec (x) \sin (2 x)}{1+\cos (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{2}{1+x} \, dx,x,\cos (x)\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\cos (x)\right )\right )\\ &=-2 \log (1+\cos (x))\\ \end{align*}
Mathematica [A] time = 0.0055193, size = 9, normalized size = 1.29 \[ -4 \log \left (\cos \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 8, normalized size = 1.1 \begin{align*} -2\,\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 [A] time = 0.979069, size = 9, normalized size = 1.29 \begin{align*} -2 \, \log \left (\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 = 2.08175, size = 35, normalized size = 5. \begin{align*} -2 \, \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 [A] time = 7.66422, size = 8, normalized size = 1.14 \begin{align*} - 2 \log{\left (\cos{\left (x \right )} + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10496, size = 23, normalized size = 3.29 \begin{align*} 2 \, \log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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