Optimal. Leaf size=39 \[ -\frac{5 x+7}{3 \left (x^2+x+1\right )}-\frac{10 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.0131921, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {638, 618, 204} \[ -\frac{5 x+7}{3 \left (x^2+x+1\right )}-\frac{10 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 638
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{-1+3 x}{\left (1+x+x^2\right )^2} \, dx &=-\frac{7+5 x}{3 \left (1+x+x^2\right )}-\frac{5}{3} \int \frac{1}{1+x+x^2} \, dx\\ &=-\frac{7+5 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{7+5 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 = 0.0225667, size = 39, normalized size = 1. \[ \frac{-5 x-7}{3 \left (x^2+x+1\right )}-\frac{10 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 33, normalized size = 0.9 \begin{align*}{\frac{-7-5\,x}{3\,{x}^{2}+3\,x+3}}-{\frac{10\,\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 = 1.58553, size = 43, normalized size = 1.1 \begin{align*} -\frac{10}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{5 \, x + 7}{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 = 1.56255, size = 120, normalized size = 3.08 \begin{align*} -\frac{10 \, \sqrt{3}{\left (x^{2} + x + 1\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 15 \, x + 21}{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 = 0.152542, size = 42, normalized size = 1.08 \begin{align*} - \frac{5 x + 7}{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 = 1.28852, size = 43, normalized size = 1.1 \begin{align*} -\frac{10}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{5 \, x + 7}{3 \,{\left (x^{2} + x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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