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