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