\[ \int \frac {1-x^4+x^8}{x^2 \left (-1+x^4\right )^{3/4} \left (-1-x^4+x^8\right )} \, dx \]
Optimal antiderivative \[ -\frac {\left (x^{4}-1\right )^{\frac {1}{4}}}{x}+\frac {\sqrt {-10+10 \sqrt {5}}\, \arctan \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x}{2 \left (x^{4}-1\right )^{\frac {1}{4}}}\right )}{10}-\frac {\sqrt {10+10 \sqrt {5}}\, \arctan \left (\frac {\sqrt {2+2 \sqrt {5}}\, x}{2 \left (x^{4}-1\right )^{\frac {1}{4}}}\right )}{10}-\frac {\sqrt {-10+10 \sqrt {5}}\, \arctanh \left (\frac {\sqrt {-2+2 \sqrt {5}}\, x}{2 \left (x^{4}-1\right )^{\frac {1}{4}}}\right )}{10}+\frac {\sqrt {10+10 \sqrt {5}}\, \arctanh \left (\frac {\sqrt {2+2 \sqrt {5}}\, x}{2 \left (x^{4}-1\right )^{\frac {1}{4}}}\right )}{10} \]
command
Integrate[(1 - x^4 + x^8)/(x^2*(-1 + x^4)^(3/4)*(-1 - x^4 + x^8)),x]
Mathematica 13.1 output
\[ \frac {1}{10} \left (-\frac {10 \sqrt [4]{-1+x^4}}{x}+\sqrt {10 \left (-1+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )-\sqrt {10 \left (1+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )-\sqrt {10 \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )+\sqrt {10 \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )\right ) \]
Mathematica 12.3 output
\[ \int \frac {1-x^4+x^8}{x^2 \left (-1+x^4\right )^{3/4} \left (-1-x^4+x^8\right )} \, dx \]________________________________________________________________________________________