Integrand size = 8, antiderivative size = 30 \[ \int \cos ^3(x) \log (\sin (x)) \, dx=-\sin (x)+\log (\sin (x)) \sin (x)+\frac {\sin ^3(x)}{9}-\frac {1}{3} \log (\sin (x)) \sin ^3(x) \]
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Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2713, 2634, 12, 4441} \[ \int \cos ^3(x) \log (\sin (x)) \, dx=\frac {\sin ^3(x)}{9}-\sin (x)-\frac {1}{3} \sin ^3(x) \log (\sin (x))+\sin (x) \log (\sin (x)) \]
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Rule 12
Rule 2634
Rule 2713
Rule 4441
Rubi steps \begin{align*} \text {integral}& = \log (\sin (x)) \sin (x)-\frac {1}{3} \log (\sin (x)) \sin ^3(x)-\int \frac {1}{6} \cos (x) (5+\cos (2 x)) \, dx \\ & = \log (\sin (x)) \sin (x)-\frac {1}{3} \log (\sin (x)) \sin ^3(x)-\frac {1}{6} \int \cos (x) (5+\cos (2 x)) \, dx \\ & = \log (\sin (x)) \sin (x)-\frac {1}{3} \log (\sin (x)) \sin ^3(x)-\frac {1}{6} \text {Subst}\left (\int \left (6-2 x^2\right ) \, dx,x,\sin (x)\right ) \\ & = -\sin (x)+\log (\sin (x)) \sin (x)+\frac {\sin ^3(x)}{9}-\frac {1}{3} \log (\sin (x)) \sin ^3(x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \cos ^3(x) \log (\sin (x)) \, dx=-\sin (x)+\log (\sin (x)) \sin (x)+\frac {\sin ^3(x)}{9}-\frac {1}{3} \log (\sin (x)) \sin ^3(x) \]
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Time = 0.38 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87
method | result | size |
parallelrisch | \(\frac {\left (3 \ln \left (\sin \left (x \right )\right )-1\right ) \sin \left (3 x \right )}{36}+\frac {3 \ln \left (\sin \left (x \right )\right ) \sin \left (x \right )}{4}-\frac {11 \sin \left (x \right )}{12}\) | \(26\) |
derivativedivides | \(-\sin \left (x \right )+\ln \left (\sin \left (x \right )\right ) \sin \left (x \right )+\frac {\left (\sin ^{3}\left (x \right )\right )}{9}-\frac {\ln \left (\sin \left (x \right )\right ) \left (\sin ^{3}\left (x \right )\right )}{3}\) | \(27\) |
default | \(-\sin \left (x \right )+\ln \left (\sin \left (x \right )\right ) \sin \left (x \right )+\frac {\left (\sin ^{3}\left (x \right )\right )}{9}-\frac {\ln \left (\sin \left (x \right )\right ) \left (\sin ^{3}\left (x \right )\right )}{3}\) | \(27\) |
risch | \(\text {Expression too large to display}\) | \(577\) |
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Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \cos ^3(x) \log (\sin (x)) \, dx=\frac {1}{3} \, {\left (\cos \left (x\right )^{2} + 2\right )} \log \left (\sin \left (x\right )\right ) \sin \left (x\right ) - \frac {1}{9} \, {\left (\cos \left (x\right )^{2} + 8\right )} \sin \left (x\right ) \]
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Time = 0.66 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \cos ^3(x) \log (\sin (x)) \, dx=\frac {2 \log {\left (\sin {\left (x \right )} \right )} \sin ^{3}{\left (x \right )}}{3} + \log {\left (\sin {\left (x \right )} \right )} \sin {\left (x \right )} \cos ^{2}{\left (x \right )} - \frac {8 \sin ^{3}{\left (x \right )}}{9} - \sin {\left (x \right )} \cos ^{2}{\left (x \right )} \]
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Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \cos ^3(x) \log (\sin (x)) \, dx=\frac {1}{9} \, \sin \left (x\right )^{3} - \frac {1}{3} \, {\left (\sin \left (x\right )^{3} - 3 \, \sin \left (x\right )\right )} \log \left (\sin \left (x\right )\right ) - \sin \left (x\right ) \]
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Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \cos ^3(x) \log (\sin (x)) \, dx=-\frac {1}{3} \, \log \left (\sin \left (x\right )\right ) \sin \left (x\right )^{3} + \frac {1}{9} \, \sin \left (x\right )^{3} + \log \left (\sin \left (x\right )\right ) \sin \left (x\right ) - \sin \left (x\right ) \]
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Timed out. \[ \int \cos ^3(x) \log (\sin (x)) \, dx=\int \ln \left (\sin \left (x\right )\right )\,{\cos \left (x\right )}^3 \,d x \]
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