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 = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 638
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*}
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Mathematica [A] time = 0.02, size = 39, normalized size = 1.00 \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+3 x}{\left (1+x+x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 37, normalized size = 0.95 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 32, normalized size = 0.82 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 33, normalized size = 0.85 \begin {gather*} -\frac {10 \sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{9}+\frac {-5 x -7}{3 x^{2}+3 x +3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 32, normalized size = 0.82 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 34, normalized size = 0.87 \begin {gather*} -\frac {10\,\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x}{3}+\frac {\sqrt {3}}{3}\right )}{9}-\frac {\frac {5\,x}{3}+\frac {7}{3}}{x^2+x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 42, normalized size = 1.08 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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