\[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-2-x^3+2 x^4\right )} \, dx \]
Optimal antiderivative \[ \frac {3 \left (x^{4}-1\right )^{\frac {2}{3}} \left (4 x^{4}-5 x^{3}-4\right )}{40 x^{5}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{x +2 \,2^{\frac {1}{3}} \left (x^{4}-1\right )^{\frac {1}{3}}}\right ) 2^{\frac {1}{3}}}{8}-\frac {\ln \left (-x +2^{\frac {1}{3}} \left (x^{4}-1\right )^{\frac {1}{3}}\right ) 2^{\frac {1}{3}}}{8}+\frac {\ln \left (x^{2}+2^{\frac {1}{3}} x \left (x^{4}-1\right )^{\frac {1}{3}}+2^{\frac {2}{3}} \left (x^{4}-1\right )^{\frac {2}{3}}\right ) 2^{\frac {1}{3}}}{16} \]
command
Integrate[((-1 + x^4)^(2/3)*(3 + x^4)*(-1 - x^3 + x^4))/(x^6*(-2 - x^3 + 2*x^4)),x]
Mathematica 13.1 output
\[ \frac {1}{80} \left (\frac {6 \left (-1+x^4\right )^{2/3} \left (-4-5 x^3+4 x^4\right )}{x^5}+10 \sqrt [3]{2} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^4}}\right )-10 \sqrt [3]{2} \log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^4}\right )+5 \sqrt [3]{2} \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^4}+2^{2/3} \left (-1+x^4\right )^{2/3}\right )\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-2-x^3+2 x^4\right )} \, dx \]________________________________________________________________________________________