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