Optimal. Leaf size=41 \[ -\frac{2 (2-x)}{3 \left (x^2-x+1\right )}-\frac{10 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.0329209, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {1660, 12, 618, 204} \[ -\frac{2 (2-x)}{3 \left (x^2-x+1\right )}-\frac{10 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 12
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{1+x+x^2}{\left (1-x+x^2\right )^2} \, dx &=-\frac{2 (2-x)}{3 \left (1-x+x^2\right )}+\frac{1}{3} \int \frac{5}{1-x+x^2} \, dx\\ &=-\frac{2 (2-x)}{3 \left (1-x+x^2\right )}+\frac{5}{3} \int \frac{1}{1-x+x^2} \, dx\\ &=-\frac{2 (2-x)}{3 \left (1-x+x^2\right )}-\frac{10}{3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )\\ &=-\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}}\\ \end{align*}
Mathematica [A] time = 0.021879, size = 39, normalized size = 0.95 \[ \frac{2 (x-2)}{3 \left (x^2-x+1\right )}+\frac{10 \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 34, normalized size = 0.8 \begin{align*}{\frac{1}{{x}^{2}-x+1} \left ({\frac{2\,x}{3}}-{\frac{4}{3}} \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 = 1.51796, size = 43, normalized size = 1.05 \begin{align*} \frac{10}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{2 \,{\left (x - 2\right )}}{3 \,{\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 = 1.90323, size = 115, normalized size = 2.8 \begin{align*} \frac{2 \,{\left (5 \, \sqrt{3}{\left (x^{2} - x + 1\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + 3 \, x - 6\right )}}{9 \,{\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 = 0.125305, size = 41, normalized size = 1. \begin{align*} \frac{2 x - 4}{3 x^{2} - 3 x + 3} + \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 = 1.28502, size = 43, normalized size = 1.05 \begin{align*} \frac{10}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{2 \,{\left (x - 2\right )}}{3 \,{\left (x^{2} - x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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