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 = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, 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 12
Rule 2554
Rule 2637
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*}
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Mathematica [A] time = 0.00, 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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fricas [A] time = 0.49, size = 17, normalized size = 0.81 \[ \log \left (-\cos \relax (x)^{2} + 1\right ) \sin \relax (x) - 2 \, \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 13, normalized size = 0.62 \[ \log \left (\sin \relax (x)^{2}\right ) \sin \relax (x) - 2 \, \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.60, size = 72, normalized size = 3.43 \[ \frac {i {\mathrm e}^{-i x} \ln \left (-2 \cos \left (2 x \right )+2\right )}{2}-\frac {i {\mathrm e}^{i x} \ln \left (-2 \cos \left (2 x \right )+2\right )}{2}-i \ln \relax (2) {\mathrm e}^{-i x}-i {\mathrm e}^{-i x}+i \ln \relax (2) {\mathrm e}^{i x}+i {\mathrm e}^{i x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 17, normalized size = 0.81 \[ \log \left (-\frac {1}{2} \, \cos \left (2 \, x\right ) + \frac {1}{2}\right ) \sin \relax (x) - 2 \, \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \ln \left (\frac {1}{2}-\frac {\cos \left (2\,x\right )}{2}\right )\,\cos \relax (x) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \log {\left (\frac {1}{2} - \frac {\cos {\left (2 x \right )}}{2} \right )} \cos {\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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