Optimal. Leaf size=54 \[ -\frac{x+2}{6 \left (x^2+x+1\right )^2}-\frac{2 x+1}{6 \left (x^2+x+1\right )}-\frac{2 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.0205593, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {638, 614, 618, 204} \[ -\frac{x+2}{6 \left (x^2+x+1\right )^2}-\frac{2 x+1}{6 \left (x^2+x+1\right )}-\frac{2 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 638
Rule 614
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{x}{\left (1+x+x^2\right )^3} \, dx &=-\frac{2+x}{6 \left (1+x+x^2\right )^2}-\frac{1}{2} \int \frac{1}{\left (1+x+x^2\right )^2} \, dx\\ &=-\frac{2+x}{6 \left (1+x+x^2\right )^2}-\frac{1+2 x}{6 \left (1+x+x^2\right )}-\frac{1}{3} \int \frac{1}{1+x+x^2} \, dx\\ &=-\frac{2+x}{6 \left (1+x+x^2\right )^2}-\frac{1+2 x}{6 \left (1+x+x^2\right )}+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=-\frac{2+x}{6 \left (1+x+x^2\right )^2}-\frac{1+2 x}{6 \left (1+x+x^2\right )}-\frac{2 \tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )}{3 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0253443, size = 49, normalized size = 0.91 \[ \frac{1}{18} \left (-\frac{3 \left (2 x^3+3 x^2+4 x+3\right )}{\left (x^2+x+1\right )^2}-4 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 48, normalized size = 0.9 \begin{align*}{\frac{-2-x}{6\, \left ({x}^{2}+x+1 \right ) ^{2}}}-{\frac{1+2\,x}{6\,{x}^{2}+6\,x+6}}-{\frac{2\,\sqrt{3}}{9}\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 = 1.44099, size = 73, normalized size = 1.35 \begin{align*} -\frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{2 \, x^{3} + 3 \, x^{2} + 4 \, x + 3}{6 \,{\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.87343, size = 189, normalized size = 3.5 \begin{align*} -\frac{6 \, x^{3} + 4 \, \sqrt{3}{\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 9 \, x^{2} + 12 \, x + 9}{18 \,{\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.135304, size = 63, normalized size = 1.17 \begin{align*} - \frac{2 x^{3} + 3 x^{2} + 4 x + 3}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} - \frac{2 \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 = 1.1185, size = 57, normalized size = 1.06 \begin{align*} -\frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{2 \, x^{3} + 3 \, x^{2} + 4 \, x + 3}{6 \,{\left (x^{2} + x + 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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