Optimal. Leaf size=31 \[ -\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (1+x+x^2\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {648, 632, 210,
642} \begin {gather*} \frac {1}{2} \log \left (x^2+x+1\right )-\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {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 )+\text {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 = 31, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (1+x+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.89, size = 26, normalized size = 0.84 \begin {gather*} -\frac {\sqrt {3} \text {ArcTan}\left [\frac {\sqrt {3} \left (1+2 x\right )}{3}\right ]}{3}+\frac {\text {Log}\left [1+x+x^2\right ]}{2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 27, normalized size = 0.87
method | result | size |
default | \(\frac {\ln \left (x^{2}+x +1\right )}{2}-\frac {\arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}\) | \(27\) |
risch | \(-\frac {\arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}+\frac {\ln \left (4 x^{2}+4 x +4\right )}{2}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 26, normalized size = 0.84 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 26, normalized size = 0.84 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 34, normalized size = 1.10 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 32, normalized size = 1.03 \begin {gather*} \frac {\ln \left (x^{2}+x+1\right )}{2}-\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 28, normalized size = 0.90 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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