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