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 = 0.07, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 12
Rule 206
Rule 321
Rule 459
Rule 2554
Rule 2633
Rule 4366
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 0.50, size = 43, normalized size = 1.08 \[ -\frac {1}{9} \, \cos \relax (x)^{3} + \frac {1}{3} \, {\left (\cos \relax (x)^{3} - 3 \, \cos \relax (x)\right )} \log \left (\sin \relax (x)\right ) + \frac {2}{3} \, \cos \relax (x) - \frac {1}{3} \, \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + \frac {1}{3} \, \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 41, normalized size = 1.02 \[ -\frac {1}{9} \, \cos \relax (x)^{3} + \frac {1}{3} \, {\left (\cos \relax (x)^{3} - 3 \, \cos \relax (x)\right )} \log \left (\sin \relax (x)\right ) + \frac {2}{3} \, \cos \relax (x) - \frac {1}{3} \, \log \left (\cos \relax (x) + 1\right ) + \frac {1}{3} \, \log \left (-\cos \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.34, size = 134, normalized size = 3.35 \[ \frac {{\mathrm e}^{-3 i x} \ln \left (2 \sin \relax (x )\right )}{24}-\frac {3 \,{\mathrm e}^{-i x} \ln \left (2 \sin \relax (x )\right )}{8}-\frac {3 \,{\mathrm e}^{i x} \ln \left (2 \sin \relax (x )\right )}{8}+\frac {{\mathrm e}^{3 i x} \ln \left (2 \sin \relax (x )\right )}{24}-\frac {{\mathrm e}^{-3 i x}}{72}-\frac {\ln \relax (2) {\mathrm e}^{-3 i x}}{24}+\frac {7 \,{\mathrm e}^{-i x}}{24}+\frac {3 \ln \relax (2) {\mathrm e}^{-i x}}{8}+\frac {7 \,{\mathrm e}^{i x}}{24}+\frac {3 \ln \relax (2) {\mathrm e}^{i x}}{8}-\frac {{\mathrm e}^{3 i x}}{72}-\frac {\ln \relax (2) {\mathrm e}^{3 i x}}{24}-\frac {2 \ln \left ({\mathrm e}^{i x}+1\right )}{3}+\frac {2 \ln \left ({\mathrm e}^{i x}-1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 179, normalized size = 4.48 \[ -\frac {4 \, {\left (\frac {3 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right )} \log \left (\frac {2 \, \sin \relax (x)}{{\left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right )} {\left (\cos \relax (x) + 1\right )}}\right )}{3 \, {\left (\frac {3 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {3 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {\sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} + 1\right )}} + \frac {2 \, {\left (\frac {12 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {3 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + 5\right )}}{9 \, {\left (\frac {3 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {3 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {\sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} + 1\right )}} - \frac {2}{3} \, \log \left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right ) + \frac {2}{3} \, \log \left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \ln \left (\sin \relax (x)\right )\,{\sin \relax (x)}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 7.74, size = 439, normalized size = 10.98 \[ - \frac {6 \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )} \tan ^{6}{\left (\frac {x}{2} \right )}}{9 \tan ^{6}{\left (\frac {x}{2} \right )} + 27 \tan ^{4}{\left (\frac {x}{2} \right )} + 27 \tan ^{2}{\left (\frac {x}{2} \right )} + 9} - \frac {18 \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )} \tan ^{4}{\left (\frac {x}{2} \right )}}{9 \tan ^{6}{\left (\frac {x}{2} \right )} + 27 \tan ^{4}{\left (\frac {x}{2} \right )} + 27 \tan ^{2}{\left (\frac {x}{2} \right )} + 9} + \frac {18 \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac {x}{2} \right )}}{9 \tan ^{6}{\left (\frac {x}{2} \right )} + 27 \tan ^{4}{\left (\frac {x}{2} \right )} + 27 \tan ^{2}{\left (\frac {x}{2} \right )} + 9} + \frac {6 \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )}}{9 \tan ^{6}{\left (\frac {x}{2} \right )} + 27 \tan ^{4}{\left (\frac {x}{2} \right )} + 27 \tan ^{2}{\left (\frac {x}{2} \right )} + 9} + \frac {12 \log {\left (\tan {\left (\frac {x}{2} \right )} \right )} \tan ^{6}{\left (\frac {x}{2} \right )}}{9 \tan ^{6}{\left (\frac {x}{2} \right )} + 27 \tan ^{4}{\left (\frac {x}{2} \right )} + 27 \tan ^{2}{\left (\frac {x}{2} \right )} + 9} + \frac {36 \log {\left (\tan {\left (\frac {x}{2} \right )} \right )} \tan ^{4}{\left (\frac {x}{2} \right )}}{9 \tan ^{6}{\left (\frac {x}{2} \right )} + 27 \tan ^{4}{\left (\frac {x}{2} \right )} + 27 \tan ^{2}{\left (\frac {x}{2} \right )} + 9} + \frac {12 \log {\relax (2 )} \tan ^{6}{\left (\frac {x}{2} \right )}}{9 \tan ^{6}{\left (\frac {x}{2} \right )} + 27 \tan ^{4}{\left (\frac {x}{2} \right )} + 27 \tan ^{2}{\left (\frac {x}{2} \right )} + 9} + \frac {6 \tan ^{4}{\left (\frac {x}{2} \right )}}{9 \tan ^{6}{\left (\frac {x}{2} \right )} + 27 \tan ^{4}{\left (\frac {x}{2} \right )} + 27 \tan ^{2}{\left (\frac {x}{2} \right )} + 9} + \frac {36 \log {\relax (2 )} \tan ^{4}{\left (\frac {x}{2} \right )}}{9 \tan ^{6}{\left (\frac {x}{2} \right )} + 27 \tan ^{4}{\left (\frac {x}{2} \right )} + 27 \tan ^{2}{\left (\frac {x}{2} \right )} + 9} + \frac {24 \tan ^{2}{\left (\frac {x}{2} \right )}}{9 \tan ^{6}{\left (\frac {x}{2} \right )} + 27 \tan ^{4}{\left (\frac {x}{2} \right )} + 27 \tan ^{2}{\left (\frac {x}{2} \right )} + 9} + \frac {10}{9 \tan ^{6}{\left (\frac {x}{2} \right )} + 27 \tan ^{4}{\left (\frac {x}{2} \right )} + 27 \tan ^{2}{\left (\frac {x}{2} \right )} + 9} \]
Verification of antiderivative is not currently implemented for this CAS.
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