Optimal. Leaf size=32 \[ \frac {1}{2} \log \left (x^2-x+1\right )+\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.02, antiderivative size = 32, 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 = {634, 618, 204, 628} \[ \frac {1}{2} \log \left (x^2-x+1\right )+\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {-1+x}{1-x+x^2} \, dx &=-\left (\frac {1}{2} \int \frac {1}{1-x+x^2} \, dx\right )+\frac {1}{2} \int \frac {-1+2 x}{1-x+x^2} \, dx\\ &=\frac {1}{2} \log \left (1-x+x^2\right )+\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )\\ &=-\frac {\tan ^{-1}\left (\frac {-1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (1-x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 1.03 \[ \frac {1}{2} \log \left (x^2-x+1\right )-\frac {\tan ^{-1}\left (\frac {2 x-1}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 28, normalized size = 0.88 \[ -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{2} \, \log \left (x^{2} - x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 28, normalized size = 0.88 \[ -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{2} \, \log \left (x^{2} - x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 0.91 \[ -\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3}+\frac {\ln \left (x^{2}-x +1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.53, size = 28, normalized size = 0.88 \[ -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{2} \, \log \left (x^{2} - x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 30, normalized size = 0.94 \[ \frac {\ln \left (x^2-x+1\right )}{2}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x}{3}-\frac {\sqrt {3}}{3}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 34, normalized size = 1.06 \[ \frac {\log {\left (x^{2} - x + 1 \right )}}{2} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} - \frac {\sqrt {3}}{3} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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