\[ \int \frac {\left (-1+x^8\right ) \left (1+x^8\right )}{\sqrt [4]{-1-x^4+x^8} \left (1-3 x^8+x^{16}\right )} \, dx \]
Optimal antiderivative \[ -\frac {x}{2 \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}}}+\frac {\arctan \left (\frac {2^{\frac {3}{4}} x \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}}}{x^{2} \sqrt {2}-\sqrt {x^{8}-x^{4}-1}}\right ) 2^{\frac {1}{4}}}{8}-\frac {\arctanh \left (\frac {2 \,2^{\frac {1}{4}} x \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}}}{2 x^{2}+\sqrt {2}\, \sqrt {x^{8}-x^{4}-1}}\right ) 2^{\frac {1}{4}}}{8} \]
command
Integrate[((-1 + x^8)*(1 + x^8))/((-1 - x^4 + x^8)^(1/4)*(1 - 3*x^8 + x^16)),x]
Mathematica 13.1 output
\[ \frac {1}{8} \left (-\frac {4 x}{\sqrt [4]{-1-x^4+x^8}}+\sqrt [4]{2} \text {ArcTan}\left (\frac {2^{3/4} x \sqrt [4]{-1-x^4+x^8}}{\sqrt {2} x^2-\sqrt {-1-x^4+x^8}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{-1-x^4+x^8}}{2 x^2+\sqrt {2} \sqrt {-1-x^4+x^8}}\right )\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (-1+x^8\right ) \left (1+x^8\right )}{\sqrt [4]{-1-x^4+x^8} \left (1-3 x^8+x^{16}\right )} \, dx \]________________________________________________________________________________________