Optimal. Leaf size=41 \[ \frac {5}{6} \log \left (x^2-x+1\right )+\frac {1}{3} \log (x+1)-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.04, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1874, 31, 634, 618, 204, 628} \[ \frac {5}{6} \log \left (x^2-x+1\right )+\frac {1}{3} \log (x+1)-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 618
Rule 628
Rule 634
Rule 1874
Rubi steps
\begin {align*} \int \frac {x (1+2 x)}{1+x^3} \, dx &=\frac {1}{3} \int \frac {1}{1+x} \, dx+\frac {1}{3} \int \frac {-1+5 x}{1-x+x^2} \, dx\\ &=\frac {1}{3} \log (1+x)+\frac {1}{2} \int \frac {1}{1-x+x^2} \, dx+\frac {5}{6} \int \frac {-1+2 x}{1-x+x^2} \, dx\\ &=\frac {1}{3} \log (1+x)+\frac {5}{6} \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}{3} \log (1+x)+\frac {5}{6} \log \left (1-x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 47, normalized size = 1.15 \[ \frac {1}{6} \left (4 \log \left (x^3+1\right )+\log \left (x^2-x+1\right )-2 \log (x+1)+2 \sqrt {3} \tan ^{-1}\left (\frac {2 x-1}{\sqrt {3}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 34, normalized size = 0.83 \[ \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {5}{6} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{3} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 35, normalized size = 0.85 \[ \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {5}{6} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{3} \, \log \left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 35, normalized size = 0.85 \[ \frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3}+\frac {\ln \left (x +1\right )}{3}+\frac {5 \ln \left (x^{2}-x +1\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 34, normalized size = 0.83 \[ \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {5}{6} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{3} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.96, size = 63, normalized size = 1.54 \[ \frac {5\,\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{6}+\frac {5\,\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{6}+\frac {\ln \left (x+1\right )}{3}-\frac {\sqrt {3}\,\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{6}+\frac {\sqrt {3}\,\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 42, normalized size = 1.02 \[ \frac {\log {\left (x + 1 \right )}}{3} + \frac {5 \log {\left (x^{2} - x + 1 \right )}}{6} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} - \frac {\sqrt {3}}{3} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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