Optimal. Leaf size=15 \[ -x-\cot (x)-\cot (x) \log (\sin (x)) \]
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Rubi [A] time = 0.0205751, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3767, 8, 2554, 3473} \[ -x-\cot (x)-\cot (x) \log (\sin (x)) \]
Antiderivative was successfully verified.
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Rule 3767
Rule 8
Rule 2554
Rule 3473
Rubi steps
\begin{align*} \int \csc ^2(x) \log (\sin (x)) \, dx &=-\cot (x) \log (\sin (x))+\int \cot ^2(x) \, dx\\ &=-\cot (x)-\cot (x) \log (\sin (x))-\int 1 \, dx\\ &=-x-\cot (x)-\cot (x) \log (\sin (x))\\ \end{align*}
Mathematica [A] time = 0.0143132, size = 15, normalized size = 1. \[ -x-\cot (x)-\cot (x) \log (\sin (x)) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.055, size = 72, normalized size = 4.8 \begin{align*}{\frac{-2\,i\ln \left ( 2\,\sin \left ( x \right ) \right ){{\rm e}^{2\,ix}}}{{{\rm e}^{2\,ix}}-1}}-{\frac{2\,i}{{{\rm e}^{2\,ix}}-1}}+i\ln \left ({{\rm e}^{ix}}-1 \right ) +i\ln \left ({{\rm e}^{ix}}+1 \right ) +{\frac{2\,i\ln \left ( 2 \right ) }{{{\rm e}^{2\,ix}}-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.50098, size = 109, normalized size = 7.27 \begin{align*} -\frac{1}{2} \,{\left (\frac{\cos \left (x\right ) + 1}{\sin \left (x\right )} - \frac{\sin \left (x\right )}{\cos \left (x\right ) + 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 ) - \frac{\cos \left (x\right ) + 1}{2 \, \sin \left (x\right )} + \frac{\sin \left (x\right )}{2 \,{\left (\cos \left (x\right ) + 1\right )}} - 2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.22429, size = 68, normalized size = 4.53 \begin{align*} -\frac{\cos \left (x\right ) \log \left (\sin \left (x\right )\right ) + x \sin \left (x\right ) + \cos \left (x\right )}{\sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 106.402, size = 17, normalized size = 1.13 \begin{align*} - x - \log{\left (\sin{\left (x \right )} \right )} \cot{\left (x \right )} - \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30822, size = 26, normalized size = 1.73 \begin{align*} -x - \frac{\log \left (\sin \left (x\right )\right )}{\tan \left (x\right )} - \frac{1}{\tan \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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