Optimal. Leaf size=14 \[ -\sin (x)+\tanh ^{-1}(\sin (x))+\sin (x) \log (\cos (x)) \]
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Rubi [A] time = 0.0195638, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 5, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {2637, 2554, 2592, 321, 206} \[ -\sin (x)+\tanh ^{-1}(\sin (x))+\sin (x) \log (\cos (x)) \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2554
Rule 2592
Rule 321
Rule 206
Rubi steps
\begin{align*} \int \cos (x) \log (\cos (x)) \, dx &=\log (\cos (x)) \sin (x)+\int \sin (x) \tan (x) \, dx\\ &=\log (\cos (x)) \sin (x)+\operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,\sin (x)\right )\\ &=-\sin (x)+\log (\cos (x)) \sin (x)+\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sin (x)\right )\\ &=\tanh ^{-1}(\sin (x))-\sin (x)+\log (\cos (x)) \sin (x)\\ \end{align*}
Mathematica [B] time = 0.0114056, size = 43, normalized size = 3.07 \[ -\sin (x)-\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )+\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )+\sin (x) \log (\cos (x)) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.017, size = 73, normalized size = 5.2 \begin{align*} -{\frac{i}{2}}\ln \left ( 2 \right ){{\rm e}^{-ix}}+{\frac{i}{2}}\ln \left ( 2 \right ){{\rm e}^{ix}}+{\frac{i}{2}}{{\rm e}^{-ix}}\ln \left ( 2\,\cos \left ( x \right ) \right ) -{\frac{i}{2}}\ln \left ( 2\,\cos \left ( x \right ) \right ){{\rm e}^{ix}}-{\frac{i}{2}}{{\rm e}^{-ix}}+{\frac{i}{2}}{{\rm e}^{ix}}-2\,i\arctan \left ({{\rm e}^{ix}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00651, size = 146, normalized size = 10.43 \begin{align*} \frac{2 \, \log \left (-\frac{\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1}{\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1}\right ) \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 )}} - \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 )}} + \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right ) - \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.39617, size = 100, normalized size = 7.14 \begin{align*} \log \left (\cos \left (x\right )\right ) \sin \left (x\right ) + \frac{1}{2} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac{1}{2} \, \log \left (-\sin \left (x\right ) + 1\right ) - \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.28682, size = 223, normalized size = 15.93 \begin{align*} - \frac{\log{\left (- \frac{\tan ^{2}{\left (\frac{x}{2} \right )}}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1} + \frac{1}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1} \right )} \tan ^{2}{\left (\frac{x}{2} \right )}}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1} + \frac{2 \log{\left (- \frac{\tan ^{2}{\left (\frac{x}{2} \right )}}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1} + \frac{1}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1} \right )} \tan{\left (\frac{x}{2} \right )}}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1} - \frac{\log{\left (- \frac{\tan ^{2}{\left (\frac{x}{2} \right )}}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1} + \frac{1}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1} \right )}}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1} + \frac{2 \log{\left (\tan{\left (\frac{x}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac{x}{2} \right )}}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1} + \frac{2 \log{\left (\tan{\left (\frac{x}{2} \right )} + 1 \right )}}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1} - \frac{\log{\left (\tan ^{2}{\left (\frac{x}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac{x}{2} \right )}}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1} - \frac{\log{\left (\tan ^{2}{\left (\frac{x}{2} \right )} + 1 \right )}}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1} - \frac{2 \tan{\left (\frac{x}{2} \right )}}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19677, size = 36, normalized size = 2.57 \begin{align*} \log \left (\cos \left (x\right )\right ) \sin \left (x\right ) + \frac{1}{2} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac{1}{2} \, \log \left (-\sin \left (x\right ) + 1\right ) - \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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