Integrand size = 6, antiderivative size = 15 \[ \int \log (\sin (x)) \sin (x) \, dx=-\text {arctanh}(\cos (x))+\cos (x)-\cos (x) \log (\sin (x)) \]
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Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {2718, 2634, 2672, 327, 212} \[ \int \log (\sin (x)) \sin (x) \, dx=-\text {arctanh}(\cos (x))+\cos (x)-\cos (x) \log (\sin (x)) \]
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Rule 212
Rule 327
Rule 2634
Rule 2672
Rule 2718
Rubi steps \begin{align*} \text {integral}& = -\cos (x) \log (\sin (x))+\int \cos (x) \cot (x) \, dx \\ & = -\cos (x) \log (\sin (x))-\text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cos (x)\right ) \\ & = \cos (x)-\cos (x) \log (\sin (x))-\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (x)\right ) \\ & = -\text {arctanh}(\cos (x))+\cos (x)-\cos (x) \log (\sin (x)) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.73 \[ \int \log (\sin (x)) \sin (x) \, dx=\cos (x)-\log \left (\cos \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )\right )-\cos (x) \log (\sin (x)) \]
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Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33
method | result | size |
parallelrisch | \(-\cos \left (x \right ) \ln \left (\sin \left (x \right )\right )+\cos \left (x \right )+\ln \left (-\cot \left (x \right )+\csc \left (x \right )\right )+1\) | \(20\) |
norman | \(\frac {2 \tan \left (\frac {x}{2}\right )^{2} \ln \left (\frac {2 \tan \left (\frac {x}{2}\right )}{1+\tan \left (\frac {x}{2}\right )^{2}}\right )+2}{1+\tan \left (\frac {x}{2}\right )^{2}}+\ln \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )\) | \(49\) |
default | \(-\frac {{\mathrm e}^{i x} \ln \left (i \left (1-{\mathrm e}^{2 i x}\right ) {\mathrm e}^{-i x}\right )}{2}+\frac {{\mathrm e}^{i x}}{2}+\ln \left (-1+{\mathrm e}^{i x}\right )-\ln \left ({\mathrm e}^{i x}+1\right )-\frac {{\mathrm e}^{-i x} \ln \left (i \left (1-{\mathrm e}^{2 i x}\right ) {\mathrm e}^{-i x}\right )}{2}+\frac {{\mathrm e}^{-i x}}{2}+\frac {\ln \left (2\right ) \left ({\mathrm e}^{i x}+{\mathrm e}^{-i x}\right )}{2}\) | \(111\) |
risch | \(\ln \left ({\mathrm e}^{i x}\right ) \cos \left (x \right )+\frac {{\mathrm e}^{-i x} \ln \left (2\right )}{2}-\frac {{\mathrm e}^{-i x} \ln \left ({\mathrm e}^{2 i x}-1\right )}{2}-\frac {{\mathrm e}^{i x} \ln \left ({\mathrm e}^{2 i x}-1\right )}{2}+\frac {{\mathrm e}^{i x} \ln \left (2\right )}{2}+\frac {{\mathrm e}^{i x}}{2}-\frac {i {\mathrm e}^{-i x} \operatorname {csgn}\left (\sin \left (x \right )\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \pi }{4}-\frac {i {\mathrm e}^{-i x} \operatorname {csgn}\left (i \sin \left (x \right )\right ) \operatorname {csgn}\left (\sin \left (x \right )\right ) \pi }{4}+\frac {i {\mathrm e}^{i x} \pi \,\operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {csgn}\left (i \sin \left (x \right )\right )^{2}}{4}+\frac {{\mathrm e}^{-i x}}{2}-\ln \left ({\mathrm e}^{i x}+1\right )+\ln \left (-1+{\mathrm e}^{i x}\right )+\frac {i {\mathrm e}^{i x} \pi }{4}+\frac {i {\mathrm e}^{-i x} \pi }{4}-\frac {i {\mathrm e}^{i x} \pi \operatorname {csgn}\left (\sin \left (x \right )\right )^{3}}{4}-\frac {i {\mathrm e}^{-i x} \pi \operatorname {csgn}\left (\sin \left (x \right )\right )^{3}}{4}+\frac {i {\mathrm e}^{-i x} \pi \operatorname {csgn}\left (i \sin \left (x \right )\right )^{3}}{4}+\frac {i {\mathrm e}^{i x} \pi \operatorname {csgn}\left (i \sin \left (x \right )\right )^{3}}{4}-\frac {i {\mathrm e}^{i x} \pi \operatorname {csgn}\left (i \sin \left (x \right )\right )^{2}}{4}-\frac {i {\mathrm e}^{-i x} \operatorname {csgn}\left (i \sin \left (x \right )\right )^{2} \pi }{4}-\frac {i {\mathrm e}^{i x} \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )^{2}}{4}-\frac {i {\mathrm e}^{i x} \pi \,\operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {csgn}\left (i \sin \left (x \right )\right )}{4}+\frac {i {\mathrm e}^{-i x} \operatorname {csgn}\left (i \sin \left (x \right )\right )^{2} \operatorname {csgn}\left (\sin \left (x \right )\right ) \pi }{4}-\frac {i {\mathrm e}^{i x} \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )^{2}}{4}-\frac {i {\mathrm e}^{-i x} \operatorname {csgn}\left (\sin \left (x \right )\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \pi }{4}-\frac {i {\mathrm e}^{-i x} \operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \pi }{4}-\frac {i {\mathrm e}^{i x} \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 i x}-i\right ) \operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )}{4}\) | \(445\) |
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Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.87 \[ \int \log (\sin (x)) \sin (x) \, dx=-\cos \left (x\right ) \log \left (\sin \left (x\right )\right ) + \cos \left (x\right ) - \frac {1}{2} \, \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + \frac {1}{2} \, \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (15) = 30\).
Time = 0.57 (sec) , antiderivative size = 105, normalized size of antiderivative = 7.00 \[ \int \log (\sin (x)) \sin (x) \, dx=\frac {2 \log {\left (\frac {\tan {\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} + 1} \right )} \tan ^{2}{\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} + 1} + \frac {\log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} + 1} + \frac {\log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} + 1} + \frac {2 \log {\left (2 \right )} \tan ^{2}{\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} + 1} + \frac {2}{\tan ^{2}{\left (\frac {x}{2} \right )} + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (15) = 30\).
Time = 0.19 (sec) , antiderivative size = 89, normalized size of antiderivative = 5.93 \[ \int \log (\sin (x)) \sin (x) \, dx=-\frac {2 \, \log \left (\frac {2 \, \sin \left (x\right )}{{\left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )} {\left (\cos \left (x\right ) + 1\right )}}\right )}{\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1} + \frac {2}{\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1} - \log \left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right ) + \log \left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.73 \[ \int \log (\sin (x)) \sin (x) \, dx=-\cos \left (x\right ) \log \left (\sin \left (x\right )\right ) + \cos \left (x\right ) - \frac {1}{2} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac {1}{2} \, \log \left (-\cos \left (x\right ) + 1\right ) \]
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Timed out. \[ \int \log (\sin (x)) \sin (x) \, dx=\int \ln \left (\sin \left (x\right )\right )\,\sin \left (x\right ) \,d x \]
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