Optimal. Leaf size=30 \[ -\sin (x)+\log (\sin (x)) \sin (x)+\frac {\sin ^3(x)}{9}-\frac {1}{3} \log (\sin (x)) \sin ^3(x) \]
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Rubi [A]
time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2713, 2634, 12,
4441} \begin {gather*} \frac {\sin ^3(x)}{9}-\sin (x)-\frac {1}{3} \sin ^3(x) \log (\sin (x))+\sin (x) \log (\sin (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2634
Rule 2713
Rule 4441
Rubi steps
\begin {align*} \int \cos ^3(x) \log (\sin (x)) \, dx &=\log (\sin (x)) \sin (x)-\frac {1}{3} \log (\sin (x)) \sin ^3(x)-\int \frac {1}{6} \cos (x) (5+\cos (2 x)) \, dx\\ &=\log (\sin (x)) \sin (x)-\frac {1}{3} \log (\sin (x)) \sin ^3(x)-\frac {1}{6} \int \cos (x) (5+\cos (2 x)) \, dx\\ &=\log (\sin (x)) \sin (x)-\frac {1}{3} \log (\sin (x)) \sin ^3(x)-\frac {1}{6} \text {Subst}\left (\int \left (6-2 x^2\right ) \, dx,x,\sin (x)\right )\\ &=-\sin (x)+\log (\sin (x)) \sin (x)+\frac {\sin ^3(x)}{9}-\frac {1}{3} \log (\sin (x)) \sin ^3(x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 30, normalized size = 1.00 \begin {gather*} -\sin (x)+\log (\sin (x)) \sin (x)+\frac {\sin ^3(x)}{9}-\frac {1}{3} \log (\sin (x)) \sin ^3(x) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.18, size = 197, normalized size = 6.57
method | result | size |
default | \(-\frac {i \left (\frac {{\mathrm e}^{3 i x} \ln \left (i \left (-{\mathrm e}^{2 i x}+1\right ) {\mathrm e}^{-i x}\right )}{3}-\frac {{\mathrm e}^{3 i x}}{9}-\frac {11 \,{\mathrm e}^{i x}}{3}+3 \,{\mathrm e}^{i x} \ln \left (i \left (-{\mathrm e}^{2 i x}+1\right ) {\mathrm e}^{-i x}\right )-3 \,{\mathrm e}^{-i x} \ln \left (i \left (-{\mathrm e}^{2 i x}+1\right ) {\mathrm e}^{-i x}\right )+\frac {11 \,{\mathrm e}^{-i x}}{3}-\frac {{\mathrm e}^{-3 i x} \ln \left (i \left (-{\mathrm e}^{2 i x}+1\right ) {\mathrm e}^{-i x}\right )}{3}+\frac {{\mathrm e}^{-3 i x}}{9}-\frac {\ln \left (2\right ) {\mathrm e}^{3 i x}}{3}-3 \ln \left (2\right ) {\mathrm e}^{i x}+3 \,{\mathrm e}^{-i x} \ln \left (2\right )+\frac {\ln \left (2\right ) {\mathrm e}^{-3 i x}}{3}\right )}{8}\) | \(197\) |
risch | \(\text {Expression too large to display}\) | \(577\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.31, size = 25, normalized size = 0.83 \begin {gather*} \frac {1}{9} \, \sin \left (x\right )^{3} - \frac {1}{3} \, {\left (\sin \left (x\right )^{3} - 3 \, \sin \left (x\right )\right )} \log \left (\sin \left (x\right )\right ) - \sin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.52, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{3} \, {\left (\cos \left (x\right )^{2} + 2\right )} \log \left (\sin \left (x\right )\right ) \sin \left (x\right ) - \frac {1}{9} \, {\left (\cos \left (x\right )^{2} + 8\right )} \sin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.84, size = 42, normalized size = 1.40 \begin {gather*} \frac {2 \log {\left (\sin {\left (x \right )} \right )} \sin ^{3}{\left (x \right )}}{3} + \log {\left (\sin {\left (x \right )} \right )} \sin {\left (x \right )} \cos ^{2}{\left (x \right )} - \frac {8 \sin ^{3}{\left (x \right )}}{9} - \sin {\left (x \right )} \cos ^{2}{\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.39, size = 26, normalized size = 0.87 \begin {gather*} -\frac {1}{3} \, \log \left (\sin \left (x\right )\right ) \sin \left (x\right )^{3} + \frac {1}{9} \, \sin \left (x\right )^{3} + \log \left (\sin \left (x\right )\right ) \sin \left (x\right ) - \sin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \ln \left (\sin \left (x\right )\right )\,{\cos \left (x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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