Optimal. Leaf size=14 \[ \frac{2}{\cos (x)+1}+\log (\cos (x)+1) \]
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Rubi [A] time = 0.0377091, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2667, 43} \[ \frac{2}{\cos (x)+1}+\log (\cos (x)+1) \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{\sin ^3(x)}{(1+\cos (x))^3} \, dx &=-\operatorname{Subst}\left (\int \frac{1-x}{(1+x)^2} \, dx,x,\cos (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{1}{-1-x}+\frac{2}{(1+x)^2}\right ) \, dx,x,\cos (x)\right )\\ &=\frac{2}{1+\cos (x)}+\log (1+\cos (x))\\ \end{align*}
Mathematica [A] time = 0.0093198, size = 18, normalized size = 1.29 \[ \tan ^2\left (\frac{x}{2}\right )+2 \log \left (\cos \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 15, normalized size = 1.1 \begin{align*} 2\, \left ( \cos \left ( x \right ) +1 \right ) ^{-1}+\ln \left ( \cos \left ( x \right ) +1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.32383, size = 19, normalized size = 1.36 \begin{align*} \frac{2}{\cos \left (x\right ) + 1} + \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 = 1.63419, size = 74, normalized size = 5.29 \begin{align*} \frac{{\left (\cos \left (x\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 2}{\cos \left (x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.785423, size = 126, normalized size = 9. \begin{align*} \frac{2 \log{\left (\cos{\left (x \right )} + 1 \right )} \cos ^{2}{\left (x \right )}}{2 \cos ^{2}{\left (x \right )} + 4 \cos{\left (x \right )} + 2} + \frac{4 \log{\left (\cos{\left (x \right )} + 1 \right )} \cos{\left (x \right )}}{2 \cos ^{2}{\left (x \right )} + 4 \cos{\left (x \right )} + 2} + \frac{2 \log{\left (\cos{\left (x \right )} + 1 \right )}}{2 \cos ^{2}{\left (x \right )} + 4 \cos{\left (x \right )} + 2} + \frac{\sin ^{2}{\left (x \right )}}{2 \cos ^{2}{\left (x \right )} + 4 \cos{\left (x \right )} + 2} + \frac{2 \cos{\left (x \right )}}{2 \cos ^{2}{\left (x \right )} + 4 \cos{\left (x \right )} + 2} + \frac{2}{2 \cos ^{2}{\left (x \right )} + 4 \cos{\left (x \right )} + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17238, size = 19, normalized size = 1.36 \begin{align*} \frac{2}{\cos \left (x\right ) + 1} + \log \left (\cos \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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