Optimal. Leaf size=40 \[ -\frac{\cos ^3(x)}{9}+\frac{2 \cos (x)}{3}-\frac{2}{3} \tanh ^{-1}(\cos (x))+\frac{1}{3} \cos ^3(x) \log (\sin (x))-\cos (x) \log (\sin (x)) \]
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Rubi [A] time = 0.0671791, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.875, Rules used = {2633, 2554, 12, 4366, 459, 321, 206} \[ -\frac{\cos ^3(x)}{9}+\frac{2 \cos (x)}{3}-\frac{2}{3} \tanh ^{-1}(\cos (x))+\frac{1}{3} \cos ^3(x) \log (\sin (x))-\cos (x) \log (\sin (x)) \]
Antiderivative was successfully verified.
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Rule 2633
Rule 2554
Rule 12
Rule 4366
Rule 459
Rule 321
Rule 206
Rubi steps
\begin{align*} \int \log (\sin (x)) \sin ^3(x) \, dx &=-\cos (x) \log (\sin (x))+\frac{1}{3} \cos ^3(x) \log (\sin (x))-\int \frac{1}{6} \cos (x) (-5+\cos (2 x)) \cot (x) \, dx\\ &=-\cos (x) \log (\sin (x))+\frac{1}{3} \cos ^3(x) \log (\sin (x))-\frac{1}{6} \int \cos (x) (-5+\cos (2 x)) \cot (x) \, dx\\ &=-\cos (x) \log (\sin (x))+\frac{1}{3} \cos ^3(x) \log (\sin (x))+\frac{1}{6} \operatorname{Subst}\left (\int \frac{2 x^2 \left (-3+x^2\right )}{1-x^2} \, dx,x,\cos (x)\right )\\ &=-\cos (x) \log (\sin (x))+\frac{1}{3} \cos ^3(x) \log (\sin (x))+\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2 \left (-3+x^2\right )}{1-x^2} \, dx,x,\cos (x)\right )\\ &=-\frac{1}{9} \cos ^3(x)-\cos (x) \log (\sin (x))+\frac{1}{3} \cos ^3(x) \log (\sin (x))-\frac{2}{3} \operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,\cos (x)\right )\\ &=\frac{2 \cos (x)}{3}-\frac{\cos ^3(x)}{9}-\cos (x) \log (\sin (x))+\frac{1}{3} \cos ^3(x) \log (\sin (x))-\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (x)\right )\\ &=-\frac{2}{3} \tanh ^{-1}(\cos (x))+\frac{2 \cos (x)}{3}-\frac{\cos ^3(x)}{9}-\cos (x) \log (\sin (x))+\frac{1}{3} \cos ^3(x) \log (\sin (x))\\ \end{align*}
Mathematica [A] time = 0.0402819, size = 47, normalized size = 1.18 \[ \frac{1}{36} \left (24 \left (\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right )\right )+\cos (3 x) (3 \log (\sin (x))-1)-3 \cos (x) (9 \log (\sin (x))-7)\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.032, size = 134, normalized size = 3.4 \begin{align*}{\frac{{{\rm e}^{3\,ix}}\ln \left ( 2\,\sin \left ( x \right ) \right ) }{24}}-{\frac{{{\rm e}^{3\,ix}}}{72}}+{\frac{7\,{{\rm e}^{ix}}}{24}}+{\frac{2\,\ln \left ({{\rm e}^{ix}}-1 \right ) }{3}}-{\frac{2\,\ln \left ({{\rm e}^{ix}}+1 \right ) }{3}}-{\frac{3\,{{\rm e}^{ix}}\ln \left ( 2\,\sin \left ( x \right ) \right ) }{8}}-{\frac{3\,{{\rm e}^{-ix}}\ln \left ( 2\,\sin \left ( x \right ) \right ) }{8}}+{\frac{7\,{{\rm e}^{-ix}}}{24}}+{\frac{{{\rm e}^{-3\,ix}}\ln \left ( 2\,\sin \left ( x \right ) \right ) }{24}}-{\frac{{{\rm e}^{-3\,ix}}}{72}}-{\frac{\ln \left ( 2 \right ){{\rm e}^{3\,ix}}}{24}}+{\frac{3\,\ln \left ( 2 \right ){{\rm e}^{ix}}}{8}}+{\frac{3\,\ln \left ( 2 \right ){{\rm e}^{-ix}}}{8}}-{\frac{\ln \left ( 2 \right ){{\rm e}^{-3\,ix}}}{24}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00537, size = 242, normalized size = 6.05 \begin{align*} -\frac{4 \,{\left (\frac{3 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )} \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 )}{3 \,{\left (\frac{3 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{3 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{\sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + 1\right )}} + \frac{2 \,{\left (\frac{12 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{3 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 5\right )}}{9 \,{\left (\frac{3 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{3 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{\sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + 1\right )}} - \frac{2}{3} \, \log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right ) + \frac{2}{3} \, \log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.24953, size = 169, normalized size = 4.22 \begin{align*} -\frac{1}{9} \, \cos \left (x\right )^{3} + \frac{1}{3} \,{\left (\cos \left (x\right )^{3} - 3 \, \cos \left (x\right )\right )} \log \left (\sin \left (x\right )\right ) + \frac{2}{3} \, \cos \left (x\right ) - \frac{1}{3} \, \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + \frac{1}{3} \, \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 15.1769, size = 445, normalized size = 11.12 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29915, size = 55, normalized size = 1.38 \begin{align*} -\frac{1}{9} \, \cos \left (x\right )^{3} + \frac{1}{3} \,{\left (\cos \left (x\right )^{3} - 3 \, \cos \left (x\right )\right )} \log \left (\sin \left (x\right )\right ) + \frac{2}{3} \, \cos \left (x\right ) - \frac{1}{3} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac{1}{3} \, \log \left (-\cos \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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