Optimal. Leaf size=31 \[ -\frac {1}{2} \log \left (x^2+x+1\right )+x+\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1657, 634, 618, 204, 628} \[ -\frac {1}{2} \log \left (x^2+x+1\right )+x+\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1657
Rubi steps
\begin {align*} \int \frac {1+x^2}{1+x+x^2} \, dx &=\int \left (1-\frac {x}{1+x+x^2}\right ) \, dx\\ &=x-\int \frac {x}{1+x+x^2} \, dx\\ &=x+\frac {1}{2} \int \frac {1}{1+x+x^2} \, dx-\frac {1}{2} \int \frac {1+2 x}{1+x+x^2} \, dx\\ &=x-\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 )\\ &=x+\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 = 31, normalized size = 1.00 \[ -\frac {1}{2} \log \left (x^2+x+1\right )+x+\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 27, normalized size = 0.87 \[ \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + x - \frac {1}{2} \, \log \left (x^{2} + x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 27, normalized size = 0.87 \[ \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + x - \frac {1}{2} \, \log \left (x^{2} + x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 28, normalized size = 0.90 \[ x +\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{3}-\frac {\ln \left (x^{2}+x +1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 27, normalized size = 0.87 \[ \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + x - \frac {1}{2} \, \log \left (x^{2} + x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 29, normalized size = 0.94 \[ x-\frac {\ln \left (x^2+x+1\right )}{2}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x}{3}+\frac {\sqrt {3}}{3}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 36, normalized size = 1.16 \[ x - \frac {\log {\left (x^{2} + x + 1 \right )}}{2} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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