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