Integrand size = 10, antiderivative size = 20 \[ \int \cos (x) \log (\sin (x)) \sin ^2(x) \, dx=-\frac {1}{9} \sin ^3(x)+\frac {1}{3} \log (\sin (x)) \sin ^3(x) \]
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Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2644, 30, 2634, 12} \[ \int \cos (x) \log (\sin (x)) \sin ^2(x) \, dx=\frac {1}{3} \sin ^3(x) \log (\sin (x))-\frac {\sin ^3(x)}{9} \]
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Rule 12
Rule 30
Rule 2634
Rule 2644
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \log (\sin (x)) \sin ^3(x)-\int \frac {1}{3} \cos (x) \sin ^2(x) \, dx \\ & = \frac {1}{3} \log (\sin (x)) \sin ^3(x)-\frac {1}{3} \int \cos (x) \sin ^2(x) \, dx \\ & = \frac {1}{3} \log (\sin (x)) \sin ^3(x)-\frac {1}{3} \text {Subst}\left (\int x^2 \, dx,x,\sin (x)\right ) \\ & = -\frac {1}{9} \sin ^3(x)+\frac {1}{3} \log (\sin (x)) \sin ^3(x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \cos (x) \log (\sin (x)) \sin ^2(x) \, dx=\frac {1}{9} (-1+3 \log (\sin (x))) \sin ^3(x) \]
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Time = 0.42 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(-\frac {\left (\sin ^{3}\left (x \right )\right )}{9}+\frac {\ln \left (\sin \left (x \right )\right ) \left (\sin ^{3}\left (x \right )\right )}{3}\) | \(17\) |
default | \(-\frac {\left (\sin ^{3}\left (x \right )\right )}{9}+\frac {\ln \left (\sin \left (x \right )\right ) \left (\sin ^{3}\left (x \right )\right )}{3}\) | \(17\) |
parallelrisch | \(\frac {\left (3 \ln \left (\sin \left (x \right )\right )-1\right ) \left (-\sin \left (3 x \right )+3 \sin \left (x \right )\right )}{36}\) | \(21\) |
risch | \(\text {Expression too large to display}\) | \(577\) |
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Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \cos (x) \log (\sin (x)) \sin ^2(x) \, dx=-\frac {1}{3} \, {\left (\cos \left (x\right )^{2} - 1\right )} \log \left (\sin \left (x\right )\right ) \sin \left (x\right ) + \frac {1}{9} \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) \]
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Time = 0.64 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \cos (x) \log (\sin (x)) \sin ^2(x) \, dx=\frac {\log {\left (\sin {\left (x \right )} \right )} \sin ^{3}{\left (x \right )}}{3} - \frac {\sin ^{3}{\left (x \right )}}{9} \]
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Time = 0.20 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \cos (x) \log (\sin (x)) \sin ^2(x) \, dx=\frac {1}{3} \, \log \left (\sin \left (x\right )\right ) \sin \left (x\right )^{3} - \frac {1}{9} \, \sin \left (x\right )^{3} \]
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Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \cos (x) \log (\sin (x)) \sin ^2(x) \, dx=\frac {1}{3} \, \log \left (\sin \left (x\right )\right ) \sin \left (x\right )^{3} - \frac {1}{9} \, \sin \left (x\right )^{3} \]
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Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.55 \[ \int \cos (x) \log (\sin (x)) \sin ^2(x) \, dx=\frac {{\sin \left (x\right )}^3\,\left (\ln \left (\sin \left (x\right )\right )-\frac {1}{3}\right )}{3} \]
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