\[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^4} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[((-1 + x^4)*Sqrt[1 + Sqrt[1 + x^2]])/(1 + x^4),x]
Mathematica 13.1 output
\[ \frac {2 x \left (2+\sqrt {1+x^2}\right )}{3 \sqrt {1+\sqrt {1+x^2}}}-\frac {1}{2} \text {RootSum}\left [1+4 \text {$\#$1}^4+4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\frac {x}{\sqrt {1+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{2 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ] \]
Mathematica 12.3 output
\[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^4} \, dx \]________________________________________________________________________________________