Optimal. Leaf size=30 \[ -\frac{x^2}{2 \left (x^2+1\right )}+\frac{1}{2} \log \left (x^2+1\right )+x-\tan ^{-1}(x) \]
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Rubi [A] time = 0.0395859, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {1804, 774, 635, 203, 260} \[ -\frac{x^2}{2 \left (x^2+1\right )}+\frac{1}{2} \log \left (x^2+1\right )+x-\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 1804
Rule 774
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{x^2 \left (1+x+x^2\right )}{\left (1+x^2\right )^2} \, dx &=-\frac{x^2}{2 \left (1+x^2\right )}-\frac{1}{2} \int \frac{(-2-2 x) x}{1+x^2} \, dx\\ &=x-\frac{x^2}{2 \left (1+x^2\right )}-\frac{1}{2} \int \frac{2-2 x}{1+x^2} \, dx\\ &=x-\frac{x^2}{2 \left (1+x^2\right )}-\int \frac{1}{1+x^2} \, dx+\int \frac{x}{1+x^2} \, dx\\ &=x-\frac{x^2}{2 \left (1+x^2\right )}-\tan ^{-1}(x)+\frac{1}{2} \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0101956, size = 27, normalized size = 0.9 \[ \frac{1}{2 \left (x^2+1\right )}+\frac{1}{2} \log \left (x^2+1\right )+x-\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 24, normalized size = 0.8 \begin{align*} x+{\frac{1}{2\,{x}^{2}+2}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{2}}-\arctan \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52833, size = 31, normalized size = 1.03 \begin{align*} x + \frac{1}{2 \,{\left (x^{2} + 1\right )}} - \arctan \left (x\right ) + \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 = 0.984418, size = 111, normalized size = 3.7 \begin{align*} \frac{2 \, x^{3} - 2 \,{\left (x^{2} + 1\right )} \arctan \left (x\right ) +{\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) + 2 \, x + 1}{2 \,{\left (x^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.110712, size = 20, normalized size = 0.67 \begin{align*} x + \frac{\log{\left (x^{2} + 1 \right )}}{2} - \operatorname{atan}{\left (x \right )} + \frac{1}{2 x^{2} + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15406, size = 31, normalized size = 1.03 \begin{align*} x + \frac{1}{2 \,{\left (x^{2} + 1\right )}} - \arctan \left (x\right ) + \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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