Optimal. Leaf size=52 \[ -\frac{2 (x+1)}{3 \left (x^2-x+1\right )}+\frac{1}{2} \log \left (x^2-x+1\right )-\frac{11 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.0515259, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {1660, 634, 618, 204, 628} \[ -\frac{2 (x+1)}{3 \left (x^2-x+1\right )}+\frac{1}{2} \log \left (x^2-x+1\right )-\frac{11 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x \left (1+x+x^2\right )}{\left (1-x+x^2\right )^2} \, dx &=-\frac{2 (1+x)}{3 \left (1-x+x^2\right )}+\frac{1}{3} \int \frac{4+3 x}{1-x+x^2} \, dx\\ &=-\frac{2 (1+x)}{3 \left (1-x+x^2\right )}+\frac{1}{2} \int \frac{-1+2 x}{1-x+x^2} \, dx+\frac{11}{6} \int \frac{1}{1-x+x^2} \, dx\\ &=-\frac{2 (1+x)}{3 \left (1-x+x^2\right )}+\frac{1}{2} \log \left (1-x+x^2\right )-\frac{11}{3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )\\ &=-\frac{2 (1+x)}{3 \left (1-x+x^2\right )}-\frac{11 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{1}{2} \log \left (1-x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0195145, size = 52, normalized size = 1. \[ -\frac{2 (x+1)}{3 \left (x^2-x+1\right )}+\frac{1}{2} \log \left (x^2-x+1\right )+\frac{11 \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 45, normalized size = 0.9 \begin{align*}{\frac{1}{{x}^{2}-x+1} \left ( -{\frac{2\,x}{3}}-{\frac{2}{3}} \right ) }+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{2}}+{\frac{11\,\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 = 1.49714, size = 58, normalized size = 1.12 \begin{align*} \frac{11}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{2 \,{\left (x + 1\right )}}{3 \,{\left (x^{2} - x + 1\right )}} + \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 = 2.34933, size = 167, normalized size = 3.21 \begin{align*} \frac{22 \, \sqrt{3}{\left (x^{2} - x + 1\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + 9 \,{\left (x^{2} - x + 1\right )} \log \left (x^{2} - x + 1\right ) - 12 \, 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 = 0.13767, size = 51, normalized size = 0.98 \begin{align*} - \frac{2 x + 2}{3 x^{2} - 3 x + 3} + \frac{\log{\left (x^{2} - x + 1 \right )}}{2} + \frac{11 \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 = 1.33756, size = 58, normalized size = 1.12 \begin{align*} \frac{11}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{2 \,{\left (x + 1\right )}}{3 \,{\left (x^{2} - x + 1\right )}} + \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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