Optimal. Leaf size=11 \[ -\frac{1}{2} \log \left (\cos ^2(x)+1\right ) \]
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Rubi [A] time = 0.0317343, antiderivative size = 11, normalized size of antiderivative = 1., 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 = {4335, 260} \[ -\frac{1}{2} \log \left (\cos ^2(x)+1\right ) \]
Antiderivative was successfully verified.
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Rule 4335
Rule 260
Rubi steps
\begin{align*} \int \frac{\cos (x) \sin (x)}{1+\cos ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,\cos (x)\right )\\ &=-\frac{1}{2} \log \left (1+\cos ^2(x)\right )\\ \end{align*}
Mathematica [A] time = 0.028184, size = 11, normalized size = 1. \[ -\frac{1}{2} \log (\cos (2 x)+3) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 10, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ( 1+ \left ( \cos \left ( x \right ) \right ) ^{2} \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.947922, size = 12, normalized size = 1.09 \begin{align*} -\frac{1}{2} \, \log \left (\cos \left (x\right )^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05358, size = 41, normalized size = 3.73 \begin{align*} -\frac{1}{2} \, \log \left (\frac{1}{2} \, \cos \left (x\right )^{2} + \frac{1}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.194233, size = 10, normalized size = 0.91 \begin{align*} - \frac{\log{\left (\cos ^{2}{\left (x \right )} + 1 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07364, size = 12, normalized size = 1.09 \begin{align*} -\frac{1}{2} \, \log \left (\cos \left (x\right )^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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