Optimal. Leaf size=9 \[ \log (\sin (x))-x \cot (x) \]
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Rubi [A] time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4184, 3475} \[ \log (\sin (x))-x \cot (x) \]
Antiderivative was successfully verified.
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Rule 3475
Rule 4184
Rubi steps
\begin {align*} \int x \csc ^2(x) \, dx &=-x \cot (x)+\int \cot (x) \, dx\\ &=-x \cot (x)+\log (\sin (x))\\ \end {align*}
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Mathematica [A] time = 0.02, size = 9, normalized size = 1.00 \[ \log (\sin (x))-x \cot (x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 20, normalized size = 2.22 \[ -\frac {x \cos \relax (x) - \log \left (\frac {1}{2} \, \sin \relax (x)\right ) \sin \relax (x)}{\sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.96, size = 52, normalized size = 5.78 \[ \frac {x \tan \left (\frac {1}{2} \, x\right )^{2} + \log \left (\frac {16 \, \tan \left (\frac {1}{2} \, x\right )^{2}}{\tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, x\right ) - x}{2 \, \tan \left (\frac {1}{2} \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 10, normalized size = 1.11 \[ -x \cot \relax (x )+\ln \left (\sin \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 104, normalized size = 11.56 \[ \frac {{\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + {\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) - 4 \, x \sin \left (2 \, x\right )}{2 \, {\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 9, normalized size = 1.00 \[ \ln \left (\sin \relax (x)\right )-x\,\mathrm {cot}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \csc ^{2}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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