Optimal. Leaf size=15 \[ -x-\cot (x)-\cot (x) \log (\sin (x)) \]
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Rubi [A] time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3767, 8, 2554, 3473} \[ -x-\cot (x)-\cot (x) \log (\sin (x)) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2554
Rule 3473
Rule 3767
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*}
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Mathematica [A] time = 0.02, size = 15, normalized size = 1.00 \[ -x-\cot (x)-\cot (x) \log (\sin (x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 19, normalized size = 1.27 \[ -\frac {\cos \relax (x) \log \left (\sin \relax (x)\right ) + x \sin \relax (x) + \cos \relax (x)}{\sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 19, normalized size = 1.27 \[ -x - \frac {\log \left (\sin \relax (x)\right )}{\tan \relax (x)} - \frac {1}{\tan \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.72, size = 72, normalized size = 4.80 \[ -\frac {2 i {\mathrm e}^{2 i x} \ln \left (2 \sin \relax (x )\right )}{{\mathrm e}^{2 i x}-1}+i \ln \left ({\mathrm e}^{i x}+1\right )+i \ln \left ({\mathrm e}^{i x}-1\right )-\frac {2 i}{{\mathrm e}^{2 i x}-1}+\frac {2 i \ln \relax (2)}{{\mathrm e}^{2 i x}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.99, size = 81, normalized size = 5.40 \[ -\frac {1}{2} \, {\left (\frac {\cos \relax (x) + 1}{\sin \relax (x)} - \frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )} \log \left (\frac {2 \, \sin \relax (x)}{{\left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right )} {\left (\cos \relax (x) + 1\right )}}\right ) - \frac {\cos \relax (x) + 1}{2 \, \sin \relax (x)} + \frac {\sin \relax (x)}{2 \, {\left (\cos \relax (x) + 1\right )}} - 2 \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.58, size = 57, normalized size = 3.80 \[ -2\,x-\ln \left ({\mathrm {e}}^{x\,2{}\mathrm {i}}-1\right )\,1{}\mathrm {i}-\frac {\ln \left (\frac {{\mathrm {e}}^{-x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )\,2{}\mathrm {i}}{{\mathrm {e}}^{x\,2{}\mathrm {i}}-1}-\frac {2{}\mathrm {i}}{{\mathrm {e}}^{x\,2{}\mathrm {i}}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.68, size = 17, normalized size = 1.13 \[ - x - \log {\left (\sin {\relax (x )} \right )} \cot {\relax (x )} - \frac {\cos {\relax (x )}}{\sin {\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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