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 = 0.0388632, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, 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 2564
Rule 30
Rule 2554
Rule 12
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*}
Mathematica [A] time = 0.0145919, 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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Maple [A] time = 0.007, size = 17, normalized size = 0.9 \begin{align*} -{\frac{ \left ( \sin \left ( x \right ) \right ) ^{3}}{9}}+{\frac{\ln \left ( \sin \left ( x \right ) \right ) \left ( \sin \left ( x \right ) \right ) ^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995932, size = 22, normalized size = 1.1 \begin{align*} \frac{1}{3} \, \log \left (\sin \left (x\right )\right ) \sin \left (x\right )^{3} - \frac{1}{9} \, \sin \left (x\right )^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.2643, size = 92, normalized size = 4.6 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.832, size = 17, normalized size = 0.85 \begin{align*} \frac{\log{\left (\sin{\left (x \right )} \right )} \sin ^{3}{\left (x \right )}}{3} - \frac{\sin ^{3}{\left (x \right )}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17397, size = 22, normalized size = 1.1 \begin{align*} \frac{1}{3} \, \log \left (\sin \left (x\right )\right ) \sin \left (x\right )^{3} - \frac{1}{9} \, \sin \left (x\right )^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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