Optimal. Leaf size=21 \[ \frac {1}{2} x^2 \cot ^{-1}(x)+\frac {x}{2}-\frac {1}{2} \tan ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {4853, 321, 203} \[ \frac {1}{2} x^2 \cot ^{-1}(x)+\frac {x}{2}-\frac {1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 321
Rule 4853
Rubi steps
\begin {align*} \int x \cot ^{-1}(x) \, dx &=\frac {1}{2} x^2 \cot ^{-1}(x)+\frac {1}{2} \int \frac {x^2}{1+x^2} \, dx\\ &=\frac {x}{2}+\frac {1}{2} x^2 \cot ^{-1}(x)-\frac {1}{2} \int \frac {1}{1+x^2} \, dx\\ &=\frac {x}{2}+\frac {1}{2} x^2 \cot ^{-1}(x)-\frac {1}{2} \tan ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 21, normalized size = 1.00 \[ \frac {1}{2} x^2 \cot ^{-1}(x)+\frac {x}{2}-\frac {1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 13, normalized size = 0.62 \[ \frac {1}{2} \, {\left (x^{2} + 1\right )} \operatorname {arccot}\relax (x) + \frac {1}{2} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.42, size = 19, normalized size = 0.90 \[ \frac {1}{2} \, x^{2} \arctan \left (\frac {1}{x}\right ) + \frac {1}{2} \, x + \frac {1}{2} \, \arctan \left (\frac {1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 16, normalized size = 0.76 \[ \frac {x^{2} \mathrm {arccot}\relax (x )}{2}+\frac {x}{2}-\frac {\arctan \relax (x )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 15, normalized size = 0.71 \[ \frac {1}{2} \, x^{2} \operatorname {arccot}\relax (x) + \frac {1}{2} \, x - \frac {1}{2} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 15, normalized size = 0.71 \[ \frac {x}{2}-\frac {\mathrm {atan}\relax (x)}{2}+\frac {x^2\,\mathrm {acot}\relax (x)}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 15, normalized size = 0.71 \[ \frac {x^{2} \operatorname {acot}{\relax (x )}}{2} + \frac {x}{2} + \frac {\operatorname {acot}{\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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