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