Optimal. Leaf size=15 \[ \sin (x)-x \cos (x)+x \log \left (e^{\cos (x)}\right ) \]
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Rubi [A] time = 0.0098554, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {2548, 3296, 2637} \[ \sin (x)-x \cos (x)+x \log \left (e^{\cos (x)}\right ) \]
Antiderivative was successfully verified.
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Rule 2548
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int \log \left (e^{\cos (x)}\right ) \, dx &=x \log \left (e^{\cos (x)}\right )+\int x \sin (x) \, dx\\ &=-x \cos (x)+x \log \left (e^{\cos (x)}\right )+\int \cos (x) \, dx\\ &=-x \cos (x)+x \log \left (e^{\cos (x)}\right )+\sin (x)\\ \end{align*}
Mathematica [A] time = 0.009336, size = 15, normalized size = 1. \[ \sin (x)+x \left (\log \left (e^{\cos (x)}\right )-\cos (x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 15, normalized size = 1. \begin{align*} -x\cos \left ( x \right ) +x\ln \left ({{\rm e}^{\cos \left ( x \right ) }} \right ) +\sin \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.92759, size = 3, normalized size = 0.2 \begin{align*} \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.07367, size = 11, normalized size = 0.73 \begin{align*} \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.183113, size = 2, normalized size = 0.13 \begin{align*} \sin{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12051, size = 3, normalized size = 0.2 \begin{align*} \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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