Optimal. Leaf size=39 \[ -\frac{1}{2} \log \left (x^2+x+1\right )-\log (1-x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0395694, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {1875, 31, 634, 618, 204, 628} \[ -\frac{1}{2} \log \left (x^2+x+1\right )-\log (1-x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1875
Rule 31
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x (1+2 x)}{1-x^3} \, dx &=\frac{1}{3} \int \frac{-3-3 x}{1+x+x^2} \, dx+\int \frac{1}{1-x} \, dx\\ &=-\log (1-x)-\frac{1}{2} \int \frac{1}{1+x+x^2} \, dx-\frac{1}{2} \int \frac{1+2 x}{1+x+x^2} \, dx\\ &=-\log (1-x)-\frac{1}{2} \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}}-\log (1-x)-\frac{1}{2} \log \left (1+x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.010297, size = 53, normalized size = 1.36 \[ \frac{1}{6} \log \left (x^2+x+1\right )-\frac{2}{3} \log \left (1-x^3\right )-\frac{1}{3} \log (1-x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 33, normalized size = 0.9 \begin{align*} -\ln \left ( -1+x \right ) -{\frac{\ln \left ({x}^{2}+x+1 \right ) }{2}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40839, size = 43, normalized size = 1.1 \begin{align*} -\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 ) - \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50109, size = 108, normalized size = 2.77 \begin{align*} -\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 ) - \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.12768, size = 41, normalized size = 1.05 \begin{align*} - \log{\left (x - 1 \right )} - \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05645, size = 45, 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}{2} \, \log \left (x^{2} + x + 1\right ) - \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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