Optimal. Leaf size=55 \[ \frac{2 (1-2 x)}{3 \left (x^2-x+1\right )}+\frac{3}{2} \log \left (x^2-x+1\right )+x-\frac{7 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.0662584, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1660, 1657, 634, 618, 204, 628} \[ \frac{2 (1-2 x)}{3 \left (x^2-x+1\right )}+\frac{3}{2} \log \left (x^2-x+1\right )+x-\frac{7 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 1657
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^2 \left (1+x+x^2\right )}{\left (1-x+x^2\right )^2} \, dx &=\frac{2 (1-2 x)}{3 \left (1-x+x^2\right )}+\frac{1}{3} \int \frac{2+6 x+3 x^2}{1-x+x^2} \, dx\\ &=\frac{2 (1-2 x)}{3 \left (1-x+x^2\right )}+\frac{1}{3} \int \left (3-\frac{1-9 x}{1-x+x^2}\right ) \, dx\\ &=x+\frac{2 (1-2 x)}{3 \left (1-x+x^2\right )}-\frac{1}{3} \int \frac{1-9 x}{1-x+x^2} \, dx\\ &=x+\frac{2 (1-2 x)}{3 \left (1-x+x^2\right )}+\frac{7}{6} \int \frac{1}{1-x+x^2} \, dx+\frac{3}{2} \int \frac{-1+2 x}{1-x+x^2} \, dx\\ &=x+\frac{2 (1-2 x)}{3 \left (1-x+x^2\right )}+\frac{3}{2} \log \left (1-x+x^2\right )-\frac{7}{3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )\\ &=x+\frac{2 (1-2 x)}{3 \left (1-x+x^2\right )}-\frac{7 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{3}{2} \log \left (1-x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.026913, size = 55, normalized size = 1. \[ -\frac{2 (2 x-1)}{3 \left (x^2-x+1\right )}+\frac{3}{2} \log \left (x^2-x+1\right )+x+\frac{7 \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 46, normalized size = 0.8 \begin{align*} x+{\frac{1}{{x}^{2}-x+1} \left ( -{\frac{4\,x}{3}}+{\frac{2}{3}} \right ) }+{\frac{3\,\ln \left ({x}^{2}-x+1 \right ) }{2}}+{\frac{7\,\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52309, size = 62, normalized size = 1.13 \begin{align*} \frac{7}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + x - \frac{2 \,{\left (2 \, x - 1\right )}}{3 \,{\left (x^{2} - x + 1\right )}} + \frac{3}{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 = 2.32918, size = 192, normalized size = 3.49 \begin{align*} \frac{18 \, x^{3} + 14 \, \sqrt{3}{\left (x^{2} - x + 1\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 18 \, x^{2} + 27 \,{\left (x^{2} - x + 1\right )} \log \left (x^{2} - x + 1\right ) - 6 \, x + 12}{18 \,{\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 = 0.13327, size = 54, normalized size = 0.98 \begin{align*} x - \frac{4 x - 2}{3 x^{2} - 3 x + 3} + \frac{3 \log{\left (x^{2} - x + 1 \right )}}{2} + \frac{7 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17382, size = 62, normalized size = 1.13 \begin{align*} \frac{7}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + x - \frac{2 \,{\left (2 \, x - 1\right )}}{3 \,{\left (x^{2} - x + 1\right )}} + \frac{3}{2} \, \log \left (x^{2} - x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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