Optimal. Leaf size=23 \[ -\frac{1}{2} \log \left (x^2+1\right )-\frac{1}{2} \tan ^{-1}(x)^2+x \tan ^{-1}(x) \]
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Rubi [A] time = 0.0484445, antiderivative size = 23, normalized size of antiderivative = 1., 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 = {4916, 4846, 260, 4884} \[ -\frac{1}{2} \log \left (x^2+1\right )-\frac{1}{2} \tan ^{-1}(x)^2+x \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 4916
Rule 4846
Rule 260
Rule 4884
Rubi steps
\begin{align*} \int \frac{x^2 \tan ^{-1}(x)}{1+x^2} \, dx &=\int \tan ^{-1}(x) \, dx-\int \frac{\tan ^{-1}(x)}{1+x^2} \, dx\\ &=x \tan ^{-1}(x)-\frac{1}{2} \tan ^{-1}(x)^2-\int \frac{x}{1+x^2} \, dx\\ &=x \tan ^{-1}(x)-\frac{1}{2} \tan ^{-1}(x)^2-\frac{1}{2} \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.015335, size = 23, normalized size = 1. \[ -\frac{1}{2} \log \left (x^2+1\right )-\frac{1}{2} \tan ^{-1}(x)^2+x \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 20, normalized size = 0.9 \begin{align*} x\arctan \left ( x \right ) -{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{2}}{2}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41145, size = 32, normalized size = 1.39 \begin{align*}{\left (x - \arctan \left (x\right )\right )} \arctan \left (x\right ) + \frac{1}{2} \, \arctan \left (x\right )^{2} - \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.40857, size = 68, normalized size = 2.96 \begin{align*} x \arctan \left (x\right ) - \frac{1}{2} \, \arctan \left (x\right )^{2} - \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.385271, size = 19, normalized size = 0.83 \begin{align*} x \operatorname{atan}{\left (x \right )} - \frac{\log{\left (x^{2} + 1 \right )}}{2} - \frac{\operatorname{atan}^{2}{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08159, size = 26, normalized size = 1.13 \begin{align*} x \arctan \left (x\right ) - \frac{1}{2} \, \arctan \left (x\right )^{2} - \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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