\[ \int \frac {\left (1+x^4\right )^2}{\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx \]
Optimal antiderivative \[ \frac {x \left (x^{4}-3\right )}{2 \left (x^{4}-1\right ) \sqrt {x^{2}+\sqrt {x^{4}+1}}}+\frac {\sqrt {14+10 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-2+2 \sqrt {2}}\, x \sqrt {x^{2}+\sqrt {x^{4}+1}}}{1+x^{2}+\sqrt {x^{4}+1}}\right )}{4}-\frac {\sqrt {14+10 \sqrt {2}}\, \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}\, x \sqrt {x^{2}+\sqrt {x^{4}+1}}}{1+x^{2}+\sqrt {x^{4}+1}}\right )}{4}+\frac {\arctanh \left (\frac {\sqrt {2}\, x \sqrt {x^{2}+\sqrt {x^{4}+1}}}{1+x^{2}+\sqrt {x^{4}+1}}\right ) \sqrt {2}}{2}-\frac {\sqrt {-14+10 \sqrt {2}}\, \arctanh \left (\frac {\sqrt {-2+2 \sqrt {2}}\, x \sqrt {x^{2}+\sqrt {x^{4}+1}}}{1+x^{2}+\sqrt {x^{4}+1}}\right )}{4}-\frac {\sqrt {-14+10 \sqrt {2}}\, \arctanh \left (\frac {\sqrt {2+2 \sqrt {2}}\, x \sqrt {x^{2}+\sqrt {x^{4}+1}}}{1+x^{2}+\sqrt {x^{4}+1}}\right )}{4} \]
command
Integrate[(1 + x^4)^2/((-1 + x^4)^2*Sqrt[x^2 + Sqrt[1 + x^4]]),x]
Mathematica 13.1 output
\[ \frac {1}{4} \left (\frac {2 x \left (-3+x^4\right )}{\left (-1+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}-\sqrt {2 \left (7+5 \sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\sqrt {2 \left (7+5 \sqrt {2}\right )} \text {ArcTan}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right )+2 \sqrt {2} \tanh ^{-1}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {2 \left (-7+5 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {2 \left (-7+5 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right )\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (1+x^4\right )^2}{\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx \]________________________________________________________________________________________