\[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (2+x^3+2 x^4\right )}{x^6 \left (4-x^3+4 x^4\right )} \, dx \]
Optimal antiderivative \[ \frac {3 \left (x^{4}+1\right )^{\frac {2}{3}} \left (8 x^{4}+15 x^{3}+8\right )}{80 x^{5}}-\frac {3 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{x +2 \,2^{\frac {2}{3}} \left (x^{4}+1\right )^{\frac {1}{3}}}\right ) 2^{\frac {2}{3}}}{32}+\frac {3 \ln \left (-x +2^{\frac {2}{3}} \left (x^{4}+1\right )^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{32}-\frac {3 \ln \left (x^{2}+2^{\frac {2}{3}} x \left (x^{4}+1\right )^{\frac {1}{3}}+2 \,2^{\frac {1}{3}} \left (x^{4}+1\right )^{\frac {2}{3}}\right ) 2^{\frac {2}{3}}}{64} \]
command
Integrate[((-3 + x^4)*(1 + x^4)^(2/3)*(2 + x^3 + 2*x^4))/(x^6*(4 - x^3 + 4*x^4)),x]
Mathematica 13.1 output
\[ \frac {3}{320} \left (\frac {4 \left (1+x^4\right )^{2/3} \left (8+15 x^3+8 x^4\right )}{x^5}-10\ 2^{2/3} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2\ 2^{2/3} \sqrt [3]{1+x^4}}\right )+10\ 2^{2/3} \log \left (-x+2^{2/3} \sqrt [3]{1+x^4}\right )-5\ 2^{2/3} \log \left (x^2+2^{2/3} x \sqrt [3]{1+x^4}+2 \sqrt [3]{2} \left (1+x^4\right )^{2/3}\right )\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (2+x^3+2 x^4\right )}{x^6 \left (4-x^3+4 x^4\right )} \, dx \]________________________________________________________________________________________