Optimal. Leaf size=32 \[ -\frac{1}{2} \log \left (x^2+x+1\right )+x-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0192937, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {703, 634, 618, 204, 628} \[ -\frac{1}{2} \log \left (x^2+x+1\right )+x-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 703
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^2}{1+x+x^2} \, dx &=x+\int \frac{-1-x}{1+x+x^2} \, dx\\ &=x-\frac{1}{2} \int \frac{1}{1+x+x^2} \, dx-\frac{1}{2} \int \frac{1+2 x}{1+x+x^2} \, dx\\ &=x-\frac{1}{2} \log \left (1+x+x^2\right )+\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=x-\frac{\tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{1}{2} \log \left (1+x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0078181, size = 32, normalized size = 1. \[ -\frac{1}{2} \log \left (x^2+x+1\right )+x-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 28, normalized size = 0.9 \begin{align*} x-{\frac{\ln \left ({x}^{2}+x+1 \right ) }{2}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50221, size = 36, normalized size = 1.12 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + x - \frac{1}{2} \, \log \left (x^{2} + x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.23919, size = 96, normalized size = 3. \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + x - \frac{1}{2} \, \log \left (x^{2} + x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.105458, size = 36, normalized size = 1.12 \begin{align*} x - \frac{\log{\left (x^{2} + x + 1 \right )}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0799, size = 36, normalized size = 1.12 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + x - \frac{1}{2} \, \log \left (x^{2} + x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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