Optimal. Leaf size=58 \[ -\frac {1-2 x}{2 \left (x^2-x+1\right )}-\frac {1-x}{2 \left (x^2-x+1\right )^2}-\frac {2 \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {638, 614, 618, 204} \[ -\frac {1-2 x}{2 \left (x^2-x+1\right )}-\frac {1-x}{2 \left (x^2-x+1\right )^2}-\frac {2 \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 614
Rule 618
Rule 638
Rubi steps
\begin {align*} \int \frac {1+x}{\left (1-x+x^2\right )^3} \, dx &=-\frac {1-x}{2 \left (1-x+x^2\right )^2}+\frac {3}{2} \int \frac {1}{\left (1-x+x^2\right )^2} \, dx\\ &=-\frac {1-x}{2 \left (1-x+x^2\right )^2}-\frac {1-2 x}{2 \left (1-x+x^2\right )}+\int \frac {1}{1-x+x^2} \, dx\\ &=-\frac {1-x}{2 \left (1-x+x^2\right )^2}-\frac {1-2 x}{2 \left (1-x+x^2\right )}-2 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )\\ &=-\frac {1-x}{2 \left (1-x+x^2\right )^2}-\frac {1-2 x}{2 \left (1-x+x^2\right )}-\frac {2 \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 49, normalized size = 0.84 \[ \frac {2 x^3-3 x^2+4 x-2}{2 \left (x^2-x+1\right )^2}+\frac {2 \tan ^{-1}\left (\frac {2 x-1}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 71, normalized size = 1.22 \[ \frac {6 \, x^{3} + 4 \, \sqrt {3} {\left (x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 9 \, x^{2} + 12 \, x - 6}{6 \, {\left (x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 44, normalized size = 0.76 \[ \frac {2}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {2 \, x^{3} - 3 \, x^{2} + 4 \, x - 2}{2 \, {\left (x^{2} - x + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 52, normalized size = 0.90 \[ \frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3}+\frac {3 x -3}{6 \left (x^{2}-x +1\right )^{2}}+\frac {2 x -1}{2 x^{2}-2 x +2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 54, normalized size = 0.93 \[ \frac {2}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {2 \, x^{3} - 3 \, x^{2} + 4 \, x - 2}{2 \, {\left (x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 53, normalized size = 0.91 \[ \frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x}{3}-\frac {\sqrt {3}}{3}\right )}{3}+\frac {x^3-\frac {3\,x^2}{2}+2\,x-1}{x^4-2\,x^3+3\,x^2-2\,x+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 61, normalized size = 1.05 \[ \frac {2 x^{3} - 3 x^{2} + 4 x - 2}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} + \frac {2 \sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} - \frac {\sqrt {3}}{3} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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