Optimal. Leaf size=41 \[ \frac {1}{6} \log \left (x^2-x+1\right )-\frac {1}{3} \log (x+1)-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {292, 31, 634, 618, 204, 628} \[ \frac {1}{6} \log \left (x^2-x+1\right )-\frac {1}{3} \log (x+1)-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x}{1+x^3} \, dx &=-\left (\frac {1}{3} \int \frac {1}{1+x} \, dx\right )+\frac {1}{3} \int \frac {1+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 )-\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 {1}{6} \log \left (1-x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 0.98 \[ \frac {1}{6} \log \left (x^2-x+1\right )-\frac {1}{3} \log (x+1)+\frac {\tan ^{-1}\left (\frac {2 x-1}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 34, normalized size = 0.83 \[ \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{6} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{3} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.89, size = 35, normalized size = 0.85 \[ \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{6} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{3} \, \log \left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 35, normalized size = 0.85 \[ \frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3}-\frac {\ln \left (x +1\right )}{3}+\frac {\ln \left (x^{2}-x +1\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 34, normalized size = 0.83 \[ \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{6} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{3} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 46, normalized size = 1.12 \[ -\frac {\ln \left (x+1\right )}{3}-\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 41, normalized size = 1.00 \[ - \frac {\log {\left (x + 1 \right )}}{3} + \frac {\log {\left (x^{2} - x + 1 \right )}}{6} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} - \frac {\sqrt {3}}{3} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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