Optimal. Leaf size=48 \[ \frac {1}{2 x^2}-\frac {1}{6} \log \left (x^2+x+1\right )+\frac {1}{3} \log (1-x)-\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {1593, 325, 200, 31, 634, 618, 204, 628} \[ \frac {1}{2 x^2}-\frac {1}{6} \log \left (x^2+x+1\right )+\frac {1}{3} \log (1-x)-\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 200
Rule 204
Rule 325
Rule 618
Rule 628
Rule 634
Rule 1593
Rubi steps
\begin {align*} \int \frac {1}{-x^3+x^6} \, dx &=\int \frac {1}{x^3 \left (-1+x^3\right )} \, dx\\ &=\frac {1}{2 x^2}+\int \frac {1}{-1+x^3} \, dx\\ &=\frac {1}{2 x^2}+\frac {1}{3} \int \frac {1}{-1+x} \, dx+\frac {1}{3} \int \frac {-2-x}{1+x+x^2} \, dx\\ &=\frac {1}{2 x^2}+\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}{2 x^2}+\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 {1}{2 x^2}-\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 = 0.01, size = 48, normalized size = 1.00 \[ \frac {1}{2 x^2}-\frac {1}{6} \log \left (x^2+x+1\right )+\frac {1}{3} \log (1-x)-\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 46, normalized size = 0.96 \[ -\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + x^{2} \log \left (x^{2} + x + 1\right ) - 2 \, x^{2} \log \left (x - 1\right ) - 3}{6 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.97, size = 38, normalized size = 0.79 \[ -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{2 \, x^{2}} - \frac {1}{6} \, \log \left (x^{2} + x + 1\right ) + \frac {1}{3} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 38, normalized size = 0.79 \[ -\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{3}+\frac {\ln \left (x -1\right )}{3}-\frac {\ln \left (x^{2}+x +1\right )}{6}+\frac {1}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.27, size = 37, normalized size = 0.77 \[ -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{2 \, x^{2}} - \frac {1}{6} \, \log \left (x^{2} + x + 1\right ) + \frac {1}{3} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.11, size = 51, normalized size = 1.06 \[ \frac {\ln \left (x-1\right )}{3}+\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\frac {1}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.16, size = 48, normalized size = 1.00 \[ \frac {\log {\left (x - 1 \right )}}{3} - \frac {\log {\left (x^{2} + x + 1 \right )}}{6} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{3} + \frac {1}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________