Optimal. Leaf size=21 \[ \sin (x) \log \left (\frac{1}{2} (1-\cos (2 x))\right )-2 \sin (x) \]
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Rubi [A] time = 0.021194, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2637, 2554, 12} \[ \sin (x) \log \left (\frac{1}{2} (1-\cos (2 x))\right )-2 \sin (x) \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2554
Rule 12
Rubi steps
\begin{align*} \int \cos (x) \log \left (\frac{1}{2} (1-\cos (2 x))\right ) \, dx &=\log \left (\frac{1}{2} (1-\cos (2 x))\right ) \sin (x)-\int 2 \cos (x) \, dx\\ &=\log \left (\frac{1}{2} (1-\cos (2 x))\right ) \sin (x)-2 \int \cos (x) \, dx\\ &=-2 \sin (x)+\log \left (\frac{1}{2} (1-\cos (2 x))\right ) \sin (x)\\ \end{align*}
Mathematica [A] time = 0.0039362, size = 13, normalized size = 0.62 \[ \sin (x) \log \left (\sin ^2(x)\right )-2 \sin (x) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.046, size = 72, normalized size = 3.4 \begin{align*} i\ln \left ( 2 \right ){{\rm e}^{ix}}-i\ln \left ( 2 \right ){{\rm e}^{-ix}}-{\frac{i}{2}}{{\rm e}^{ix}}\ln \left ( 2-2\,\cos \left ( 2\,x \right ) \right ) +{\frac{i}{2}}{{\rm e}^{-ix}}\ln \left ( 2-2\,\cos \left ( 2\,x \right ) \right ) +i{{\rm e}^{ix}}-i{{\rm e}^{-ix}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01556, size = 23, normalized size = 1.1 \begin{align*} \log \left (-\frac{1}{2} \, \cos \left (2 \, x\right ) + \frac{1}{2}\right ) \sin \left (x\right ) - 2 \, \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.24085, size = 51, normalized size = 2.43 \begin{align*} \log \left (-\cos \left (x\right )^{2} + 1\right ) \sin \left (x\right ) - 2 \, \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \log{\left (\frac{1}{2} - \frac{\cos{\left (2 x \right )}}{2} \right )} \cos{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34837, size = 18, normalized size = 0.86 \begin{align*} \log \left (\sin \left (x\right )^{2}\right ) \sin \left (x\right ) - 2 \, \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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