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