\[ \int \frac {\left (-1+x^4\right )^2}{\left (1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx \]
Optimal antiderivative \[ \frac {x^{2} \sqrt {x^{4}+1}\, \left (6 x^{6}+9 x^{2}\right )+x^{2} \left (6 x^{8}+12 x^{4}+4\right )}{2 x \sqrt {x^{4}+1}\, \left (2 x^{6}+2 x^{2}\right ) \sqrt {x^{2}+\sqrt {x^{4}+1}}+2 x \left (2 x^{8}+3 x^{4}+1\right ) \sqrt {x^{2}+\sqrt {x^{4}+1}}}-\frac {3 \arctanh \left (\frac {x}{\sqrt {x^{2}+\sqrt {x^{4}+1}}}\right )}{2}+\frac {\arctanh \left (\frac {\sqrt {2}\, x \sqrt {x^{2}+\sqrt {x^{4}+1}}}{1+x^{2}+\sqrt {x^{4}+1}}\right ) \sqrt {2}}{2} \]
command
Integrate[(-1 + x^4)^2/((1 + x^4)^2*Sqrt[x^2 + Sqrt[1 + x^4]]),x]
Mathematica 13.1 output
\[ \frac {1}{2} \left (\frac {x \left (4+12 x^4+6 x^8+9 x^2 \sqrt {1+x^4}+6 x^6 \sqrt {1+x^4}\right )}{\left (1+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}} \left (1+2 x^4+2 x^2 \sqrt {1+x^4}\right )}-3 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (-1+x^4\right )^2}{\left (1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx \]________________________________________________________________________________________