Optimal. Leaf size=40 \[ -\frac{1}{6} \log \left (x^2+x+1\right )+\frac{1}{3} \log (1-x)+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0211335, antiderivative size = 40, 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} \[ -\frac{1}{6} \log \left (x^2+x+1\right )+\frac{1}{3} \log (1-x)+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 292
Rule 31
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x}{-1+x^3} \, dx &=\frac{1}{3} \int \frac{1}{-1+x} \, dx-\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 = 0.0083162, size = 40, normalized size = 1. \[ -\frac{1}{6} \log \left (x^2+x+1\right )+\frac{1}{3} \log (1-x)+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 33, normalized size = 0.8 \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 ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.43189, size = 43, normalized size = 1.08 \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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.61868, size = 112, normalized size = 2.8 \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.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.128426, size = 41, normalized size = 1.02 \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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.09073, size = 45, 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 ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]