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 = 0.0082851, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, 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 4853
Rule 321
Rule 203
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*}
Mathematica [A] time = 0.0023428, size = 21, normalized size = 1. \[ \frac{1}{2} x^2 \cot ^{-1}(x)+\frac{x}{2}-\frac{1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 16, normalized size = 0.8 \begin{align*}{\frac{x}{2}}+{\frac{{x}^{2}{\rm arccot} \left (x\right )}{2}}-{\frac{\arctan \left ( x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44582, size = 20, normalized size = 0.95 \begin{align*} \frac{1}{2} \, x^{2} \operatorname{arccot}\left (x\right ) + \frac{1}{2} \, x - \frac{1}{2} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64884, size = 45, normalized size = 2.14 \begin{align*} \frac{1}{2} \,{\left (x^{2} + 1\right )} \operatorname{arccot}\left (x\right ) + \frac{1}{2} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.247543, size = 15, normalized size = 0.71 \begin{align*} \frac{x^{2} \operatorname{acot}{\left (x \right )}}{2} + \frac{x}{2} + \frac{\operatorname{acot}{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08124, size = 23, normalized size = 1.1 \begin{align*} \frac{1}{2} \, x^{2} \arctan \left (\frac{1}{x}\right ) + \frac{1}{2} \, x - \frac{1}{2} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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