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 = 0.0220315, antiderivative size = 41, normalized size of antiderivative = 1., 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 292
Rule 31
Rule 634
Rule 618
Rule 204
Rule 628
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*}
Mathematica [A] time = 0.0085541, 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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Maple [A] time = 0., size = 35, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ( 1+x \right ) }{3}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{6}}+{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40549, size = 46, normalized size = 1.12 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9814, size = 112, normalized size = 2.73 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.121567, size = 41, normalized size = 1. \begin{align*} - \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0722, size = 47, normalized size = 1.15 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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