Optimal. Leaf size=23 \[ \frac {1}{2} \log \left (x^2+1\right )+\frac {1}{2} \cot ^{-1}(x)^2+x \cot ^{-1}(x) \]
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Rubi [A] time = 0.05, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {4917, 4847, 260, 4885} \[ \frac {1}{2} \log \left (x^2+1\right )+\frac {1}{2} \cot ^{-1}(x)^2+x \cot ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 260
Rule 4847
Rule 4885
Rule 4917
Rubi steps
\begin {align*} \int \frac {x^2 \cot ^{-1}(x)}{1+x^2} \, dx &=\int \cot ^{-1}(x) \, dx-\int \frac {\cot ^{-1}(x)}{1+x^2} \, dx\\ &=x \cot ^{-1}(x)+\frac {1}{2} \cot ^{-1}(x)^2+\int \frac {x}{1+x^2} \, dx\\ &=x \cot ^{-1}(x)+\frac {1}{2} \cot ^{-1}(x)^2+\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 23, normalized size = 1.00 \[ \frac {1}{2} \log \left (x^2+1\right )+\frac {1}{2} \cot ^{-1}(x)^2+x \cot ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 19, normalized size = 0.83 \[ x \operatorname {arccot}\relax (x) + \frac {1}{2} \, \operatorname {arccot}\relax (x)^{2} + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \operatorname {arccot}\relax (x)}{x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 26, normalized size = 1.13 \[ -\mathrm {arccot}\relax (x ) \arctan \relax (x )+x \,\mathrm {arccot}\relax (x )+\frac {\ln \left (x^{2}+1\right )}{2}-\frac {\arctan \relax (x )^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 24, normalized size = 1.04 \[ {\left (x - \arctan \relax (x)\right )} \operatorname {arccot}\relax (x) - \frac {1}{2} \, \arctan \relax (x)^{2} + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.64, size = 19, normalized size = 0.83 \[ \frac {{\mathrm {acot}\relax (x)}^2}{2}+x\,\mathrm {acot}\relax (x)+\frac {\ln \left (x^2+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 19, normalized size = 0.83 \[ x \operatorname {acot}{\relax (x )} + \frac {\log {\left (x^{2} + 1 \right )}}{2} + \frac {\operatorname {acot}^{2}{\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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