\[ \int \frac {-1+x^6}{\left (1+x^6\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \left (\frac {\sqrt {3}\, a x}{a x +2 \left (a^{3} x^{3}+x^{6}+1\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3 a}+\frac {\ln \left (-a x +\left (a^{3} x^{3}+x^{6}+1\right )^{\frac {1}{3}}\right )}{3 a}-\frac {\ln \left (a^{2} x^{2}+a x \left (a^{3} x^{3}+x^{6}+1\right )^{\frac {1}{3}}+\left (a^{3} x^{3}+x^{6}+1\right )^{\frac {2}{3}}\right )}{6 a} \]
command
Integrate[(-1 + x^6)/((1 + x^6)*(1 + a^3*x^3 + x^6)^(1/3)),x]
Mathematica 13.1 output
\[ -\frac {2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{1+a^3 x^3+x^6}}\right )-2 \log \left (a \left (a x-\sqrt [3]{1+a^3 x^3+x^6}\right )\right )+\log \left (a^2 x^2+a x \sqrt [3]{1+a^3 x^3+x^6}+\left (1+a^3 x^3+x^6\right )^{2/3}\right )}{6 a} \]
Mathematica 12.3 output
\[ \int \frac {-1+x^6}{\left (1+x^6\right ) \sqrt [3]{1+a^3 x^3+x^6}} \, dx \]________________________________________________________________________________________