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