Optimal. Leaf size=14 \[ -\frac {1}{2} \cot ^2(x)-\log (\sin (x)) \]
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Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3473, 3475} \[ -\frac {1}{2} \cot ^2(x)-\log (\sin (x)) \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rubi steps
\begin {align*} \int \cot ^3(x) \, dx &=-\frac {1}{2} \cot ^2(x)-\int \cot (x) \, dx\\ &=-\frac {1}{2} \cot ^2(x)-\log (\sin (x))\\ \end {align*}
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Mathematica [A] time = 0.00, size = 14, normalized size = 1.00 \[ -\frac {1}{2} \csc ^2(x)-\log (\sin (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 31, normalized size = 2.21 \[ -\frac {\log \left (\frac {\tan \relax (x)^{2}}{\tan \relax (x)^{2} + 1}\right ) \tan \relax (x)^{2} + \tan \relax (x)^{2} + 1}{2 \, \tan \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.19, size = 29, normalized size = 2.07 \[ \frac {\tan \relax (x)^{2} - 1}{2 \, \tan \relax (x)^{2}} + \frac {1}{2} \, \log \left (\tan \relax (x)^{2} + 1\right ) - \frac {1}{2} \, \log \left (\tan \relax (x)^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 22, normalized size = 1.57 \[ \frac {\ln \left (\tan ^{2}\relax (x )+1\right )}{2}-\ln \left (\tan \relax (x )\right )-\frac {1}{2 \tan \relax (x )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 14, normalized size = 1.00 \[ -\frac {1}{2 \, \sin \relax (x)^{2}} - \frac {1}{2} \, \log \left (\sin \relax (x)^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 21, normalized size = 1.50 \[ \frac {\ln \left ({\mathrm {tan}\relax (x)}^2+1\right )}{2}-\ln \left (\mathrm {tan}\relax (x)\right )-\frac {1}{2\,{\mathrm {tan}\relax (x)}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 14, normalized size = 1.00 \[ - \log {\left (\sin {\relax (x )} \right )} - \frac {1}{2 \sin ^{2}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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