Optimal. Leaf size=43 \[ \frac {\arctan \left (\frac {\sqrt {3} x}{2-x}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (\frac {1+x}{\sqrt {1-x+x^2}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 41, normalized size of antiderivative = 0.95, number of steps
used = 6, number of rules used = 6, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {206, 31, 648,
632, 210, 642} \begin {gather*} -\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (x^2-x+1\right )+\frac {1}{3} \log (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 210
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\frac {1}{3} \int \frac {1}{1+x} \, dx+\frac {1}{3} \int \frac {2-x}{1-x+x^2} \, dx\\ &=\frac {1}{3} \log (1+x)-\frac {1}{6} \int \frac {-1+2 x}{1-x+x^2} \, dx+\frac {1}{2} \int \frac {1}{1-x+x^2} \, dx\\ &=\frac {1}{3} \log (1+x)-\frac {1}{6} \log \left (1-x+x^2\right )-\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )\\ &=\frac {\arctan \left (\frac {-1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log (1+x)-\frac {1}{6} \log \left (1-x+x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 40, normalized size = 0.93 \begin {gather*} \frac {\arctan \left (\frac {-1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log (1+x)-\frac {1}{6} \log \left (1-x+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 35, normalized size = 0.81
method | result | size |
default | \(\frac {\ln \left (1+x \right )}{3}-\frac {\ln \left (x^{2}-x +1\right )}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3}\) | \(35\) |
risch | \(-\frac {\ln \left (4 x^{2}-4 x +4\right )}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3}+\frac {\ln \left (1+x \right )}{3}\) | \(37\) |
meijerg | \(\frac {x \ln \left (1+\left (x^{3}\right )^{\frac {1}{3}}\right )}{3 \left (x^{3}\right )^{\frac {1}{3}}}-\frac {x \ln \left (1-\left (x^{3}\right )^{\frac {1}{3}}+\left (x^{3}\right )^{\frac {2}{3}}\right )}{6 \left (x^{3}\right )^{\frac {1}{3}}}+\frac {x \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{3}\right )^{\frac {1}{3}}}{2-\left (x^{3}\right )^{\frac {1}{3}}}\right )}{3 \left (x^{3}\right )^{\frac {1}{3}}}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 34, normalized size = 0.79 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{6} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{3} \, \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.64, size = 34, normalized size = 0.79 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{6} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{3} \, \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 41, normalized size = 0.95 \begin {gather*} \frac {\log {\left (x + 1 \right )}}{3} - \frac {\log {\left (x^{2} - x + 1 \right )}}{6} + \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 = 0.45, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{6} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{3} \, \log \left ({\left | x + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 31, normalized size = 0.72 \begin {gather*} \frac {\ln \left (x+1\right )}{3}-\frac {\ln \left ({\left (x-\frac {1}{2}\right )}^2+\frac {3}{4}\right )}{6}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,\left (x-\frac {1}{2}\right )}{3}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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