\[ \int \frac {-1+x^8}{\sqrt [4]{1+x^4} \left (1+x^8\right )} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[(-1 + x^8)/((1 + x^4)^(1/4)*(1 + x^8)),x]
Mathematica 13.1 output
\[ \frac {1}{4} \left (2 \left (\text {ArcTan}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )\right )+\text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{1+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right ) \]
Mathematica 12.3 output
\[ \int \frac {-1+x^8}{\sqrt [4]{1+x^4} \left (1+x^8\right )} \, dx \]________________________________________________________________________________________