Optimal. Leaf size=11 \[ -\frac {1}{2} \log \left (\cos ^2(x)+1\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 11, normalized size of antiderivative = 1.00, 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 260
Rule 4335
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*}
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Mathematica [A] time = 0.03, size = 11, normalized size = 1.00 \[ -\frac {1}{2} \log (\cos (2 x)+3) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 11, normalized size = 1.00 \[ -\frac {1}{2} \, \log \left (\frac {1}{2} \, \cos \relax (x)^{2} + \frac {1}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 9, normalized size = 0.82 \[ -\frac {1}{2} \, \log \left (\cos \relax (x)^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 10, normalized size = 0.91 \[ -\frac {\ln \left (1+\cos ^{2}\relax (x )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 9, normalized size = 0.82 \[ -\frac {1}{2} \, \log \left (\cos \relax (x)^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.01, size = 17, normalized size = 1.55 \[ -\mathrm {atanh}\left (\frac {16}{3\,\left (12\,{\mathrm {tan}\relax (x)}^2+16\right )}-\frac {1}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 10, normalized size = 0.91 \[ - \frac {\log {\left (\cos ^{2}{\relax (x )} + 1 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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