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