Optimal. Leaf size=15 \[ \frac {1}{2} \log \left (x^2+1\right )+x \cot ^{-1}(x) \]
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Rubi [A] time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4847, 260} \[ \frac {1}{2} \log \left (x^2+1\right )+x \cot ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 260
Rule 4847
Rubi steps
\begin {align*} \int \cot ^{-1}(x) \, dx &=x \cot ^{-1}(x)+\int \frac {x}{1+x^2} \, dx\\ &=x \cot ^{-1}(x)+\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 15, normalized size = 1.00 \[ \frac {1}{2} \log \left (x^2+1\right )+x \cot ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 13, normalized size = 0.87 \[ x \operatorname {arccot}\relax (x) + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 21, normalized size = 1.40 \[ x \arctan \left (\frac {1}{x}\right ) + \frac {1}{2} \, \log \left (\frac {1}{x^{2}} + 1\right ) - \frac {1}{2} \, \log \left (\frac {1}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 14, normalized size = 0.93 \[ x \,\mathrm {arccot}\relax (x )+\frac {\ln \left (x^{2}+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 13, normalized size = 0.87 \[ x \operatorname {arccot}\relax (x) + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 13, normalized size = 0.87 \[ \frac {\ln \left (x^2+1\right )}{2}+x\,\mathrm {acot}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 12, normalized size = 0.80 \[ x \operatorname {acot}{\relax (x )} + \frac {\log {\left (x^{2} + 1 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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