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