Integrand size = 16, antiderivative size = 21 \[ \int \cos (x) \log \left (\frac {1}{2} (1-\cos (2 x))\right ) \, dx=-2 \sin (x)+\log \left (\frac {1}{2} (1-\cos (2 x))\right ) \sin (x) \]
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Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2717, 2634, 12} \[ \int \cos (x) \log \left (\frac {1}{2} (1-\cos (2 x))\right ) \, dx=\sin (x) \log \left (\frac {1}{2} (1-\cos (2 x))\right )-2 \sin (x) \]
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Rule 12
Rule 2634
Rule 2717
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \cos (x) \log \left (\frac {1}{2} (1-\cos (2 x))\right ) \, dx=-2 \sin (x)+\log \left (\sin ^2(x)\right ) \sin (x) \]
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Result contains complex when optimal does not.
Time = 6.89 (sec) , antiderivative size = 111, normalized size of antiderivative = 5.29
method | result | size |
default | \(-\frac {i \left ({\mathrm e}^{i x} \ln \left (\left (-{\mathrm e}^{4 i x}+2 \,{\mathrm e}^{2 i x}-1\right ) {\mathrm e}^{-2 i x}\right )-2 \,{\mathrm e}^{i x}-{\mathrm e}^{-i x} \ln \left (\left (-{\mathrm e}^{4 i x}+2 \,{\mathrm e}^{2 i x}-1\right ) {\mathrm e}^{-2 i x}\right )+2 \,{\mathrm e}^{-i x}-2 \ln \left (2\right ) \left ({\mathrm e}^{i x}-{\mathrm e}^{-i x}\right )\right )}{2}\) | \(111\) |
risch | \(\text {Expression too large to display}\) | \(796\) |
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Time = 0.32 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \cos (x) \log \left (\frac {1}{2} (1-\cos (2 x))\right ) \, dx=\log \left (-\cos \left (x\right )^{2} + 1\right ) \sin \left (x\right ) - 2 \, \sin \left (x\right ) \]
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\[ \int \cos (x) \log \left (\frac {1}{2} (1-\cos (2 x))\right ) \, dx=\int \log {\left (\frac {1}{2} - \frac {\cos {\left (2 x \right )}}{2} \right )} \cos {\left (x \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \cos (x) \log \left (\frac {1}{2} (1-\cos (2 x))\right ) \, dx=\log \left (-\frac {1}{2} \, \cos \left (2 \, x\right ) + \frac {1}{2}\right ) \sin \left (x\right ) - 2 \, \sin \left (x\right ) \]
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Time = 0.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \cos (x) \log \left (\frac {1}{2} (1-\cos (2 x))\right ) \, dx=\log \left (\sin \left (x\right )^{2}\right ) \sin \left (x\right ) - 2 \, \sin \left (x\right ) \]
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Timed out. \[ \int \cos (x) \log \left (\frac {1}{2} (1-\cos (2 x))\right ) \, dx=\int \ln \left (\frac {1}{2}-\frac {\cos \left (2\,x\right )}{2}\right )\,\cos \left (x\right ) \,d x \]
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