Optimal. Leaf size=41 \[ \frac {\tan ^{-1}\left (\frac {1-2 x^3}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{6} \log \left (x^6-x^3+1\right )+\log (x) \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1474, 800, 634, 618, 204, 628} \begin {gather*} -\frac {1}{6} \log \left (x^6-x^3+1\right )+\frac {\tan ^{-1}\left (\frac {1-2 x^3}{\sqrt {3}}\right )}{3 \sqrt {3}}+\log (x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 618
Rule 628
Rule 634
Rule 800
Rule 1474
Rubi steps
\begin {align*} \int \frac {1-x^3}{x \left (1-x^3+x^6\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1-x}{x \left (1-x+x^2\right )} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {1}{x}-\frac {x}{1-x+x^2}\right ) \, dx,x,x^3\right )\\ &=\log (x)-\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{1-x+x^2} \, dx,x,x^3\right )\\ &=\log (x)-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,x^3\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,x^3\right )\\ &=\log (x)-\frac {1}{6} \log \left (1-x^3+x^6\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x^3\right )\\ &=\frac {\tan ^{-1}\left (\frac {1-2 x^3}{\sqrt {3}}\right )}{3 \sqrt {3}}+\log (x)-\frac {1}{6} \log \left (1-x^3+x^6\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 44, normalized size = 1.07 \begin {gather*} \log (x)-\frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3+1\&,\frac {\text {$\#$1}^3 \log (x-\text {$\#$1})}{2 \text {$\#$1}^3-1}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-x^3}{x \left (1-x^3+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.46, size = 34, normalized size = 0.83 \begin {gather*} -\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}\right ) - \frac {1}{6} \, \log \left (x^{6} - x^{3} + 1\right ) + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.59, size = 35, normalized size = 0.85 \begin {gather*} -\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}\right ) - \frac {1}{6} \, \log \left (x^{6} - x^{3} + 1\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 35, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{3}-1\right ) \sqrt {3}}{3}\right )}{9}+\ln \relax (x )-\frac {\ln \left (x^{6}-x^{3}+1\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.97, size = 38, normalized size = 0.93 \begin {gather*} -\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}\right ) - \frac {1}{6} \, \log \left (x^{6} - x^{3} + 1\right ) + \frac {1}{3} \, \log \left (x^{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.86, size = 36, normalized size = 0.88 \begin {gather*} \ln \relax (x)-\frac {\ln \left (x^6-x^3+1\right )}{6}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}}{3}-\frac {2\,\sqrt {3}\,x^3}{3}\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.15, size = 41, normalized size = 1.00 \begin {gather*} \log {\relax (x )} - \frac {\log {\left (x^{6} - x^{3} + 1 \right )}}{6} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{3}}{3} - \frac {\sqrt {3}}{3} \right )}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________