Optimal. Leaf size=20 \[ \frac {1}{3} \sin ^3(x) \log (\sin (x))-\frac {\sin ^3(x)}{9} \]
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Rubi [A] time = 0.03, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2564, 30, 2554, 12} \[ \frac {1}{3} \sin ^3(x) \log (\sin (x))-\frac {\sin ^3(x)}{9} \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2554
Rule 2564
Rubi steps
\begin {align*} \int \cos (x) \log (\sin (x)) \sin ^2(x) \, dx &=\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} \operatorname {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*}
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Mathematica [A] time = 0.02, size = 15, normalized size = 0.75 \[ \frac {1}{9} \sin ^3(x) (3 \log (\sin (x))-1) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 24, normalized size = 1.20 \[ -\frac {1}{3} \, {\left (\cos \relax (x)^{2} - 1\right )} \log \left (\sin \relax (x)\right ) \sin \relax (x) + \frac {1}{9} \, {\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 16, normalized size = 0.80 \[ \frac {1}{3} \, \log \left (\sin \relax (x)\right ) \sin \relax (x)^{3} - \frac {1}{9} \, \sin \relax (x)^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 17, normalized size = 0.85 \[ \frac {\ln \left (\sin \relax (x )\right ) \left (\sin ^{3}\relax (x )\right )}{3}-\frac {\left (\sin ^{3}\relax (x )\right )}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 16, normalized size = 0.80 \[ \frac {1}{3} \, \log \left (\sin \relax (x)\right ) \sin \relax (x)^{3} - \frac {1}{9} \, \sin \relax (x)^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 11, normalized size = 0.55 \[ \frac {{\sin \relax (x)}^3\,\left (\ln \left (\sin \relax (x)\right )-\frac {1}{3}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.39, size = 17, normalized size = 0.85 \[ \frac {\log {\left (\sin {\relax (x )} \right )} \sin ^{3}{\relax (x )}}{3} - \frac {\sin ^{3}{\relax (x )}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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