Optimal. Leaf size=62 \[ \frac{x^2}{2}+\frac{2 (2-x)}{3 \left (x^2-x+1\right )}+2 \log \left (x^2-x+1\right )+3 x+\frac{10 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.0743509, antiderivative size = 62, 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} \[ \frac{x^2}{2}+\frac{2 (2-x)}{3 \left (x^2-x+1\right )}+2 \log \left (x^2-x+1\right )+3 x+\frac{10 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
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^3 \left (1+x+x^2\right )}{\left (1-x+x^2\right )^2} \, dx &=\frac{2 (2-x)}{3 \left (1-x+x^2\right )}+\frac{1}{3} \int \frac{-2+6 x+6 x^2+3 x^3}{1-x+x^2} \, dx\\ &=\frac{2 (2-x)}{3 \left (1-x+x^2\right )}+\frac{1}{3} \int \left (9+3 x-\frac{11-12 x}{1-x+x^2}\right ) \, dx\\ &=3 x+\frac{x^2}{2}+\frac{2 (2-x)}{3 \left (1-x+x^2\right )}-\frac{1}{3} \int \frac{11-12 x}{1-x+x^2} \, dx\\ &=3 x+\frac{x^2}{2}+\frac{2 (2-x)}{3 \left (1-x+x^2\right )}-\frac{5}{3} \int \frac{1}{1-x+x^2} \, dx+2 \int \frac{-1+2 x}{1-x+x^2} \, dx\\ &=3 x+\frac{x^2}{2}+\frac{2 (2-x)}{3 \left (1-x+x^2\right )}+2 \log \left (1-x+x^2\right )+\frac{10}{3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )\\ &=3 x+\frac{x^2}{2}+\frac{2 (2-x)}{3 \left (1-x+x^2\right )}+\frac{10 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}}+2 \log \left (1-x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0332674, size = 60, normalized size = 0.97 \[ \frac{x^2}{2}-\frac{2 (x-2)}{3 \left (x^2-x+1\right )}+2 \log \left (x^2-x+1\right )+3 x-\frac{10 \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 53, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2}}+3\,x+{\frac{1}{{x}^{2}-x+1} \left ( -{\frac{2\,x}{3}}+{\frac{4}{3}} \right ) }+2\,\ln \left ({x}^{2}-x+1 \right ) -{\frac{10\,\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 = 1.43281, size = 69, normalized size = 1.11 \begin{align*} \frac{1}{2} \, x^{2} - \frac{10}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + 3 \, x - \frac{2 \,{\left (x - 2\right )}}{3 \,{\left (x^{2} - x + 1\right )}} + 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 = 2.34005, size = 204, normalized size = 3.29 \begin{align*} \frac{9 \, x^{4} + 45 \, x^{3} - 20 \, \sqrt{3}{\left (x^{2} - x + 1\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 45 \, x^{2} + 36 \,{\left (x^{2} - x + 1\right )} \log \left (x^{2} - x + 1\right ) + 42 \, x + 24}{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 = 0.143439, size = 60, normalized size = 0.97 \begin{align*} \frac{x^{2}}{2} + 3 x - \frac{2 x - 4}{3 x^{2} - 3 x + 3} + 2 \log{\left (x^{2} - x + 1 \right )} - \frac{10 \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 = 1.19867, size = 69, normalized size = 1.11 \begin{align*} \frac{1}{2} \, x^{2} - \frac{10}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + 3 \, x - \frac{2 \,{\left (x - 2\right )}}{3 \,{\left (x^{2} - x + 1\right )}} + 2 \, \log \left (x^{2} - x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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