\[ \int \frac {\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[((1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]])/(Sqrt[1 + x^4]*(-1 + x^2 + x^4)),x]
Mathematica 13.1 output
\[ \frac {\text {RootSum}\left [1-2 \text {$\#$1}^2-6 \text {$\#$1}^4+2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (1+x^2+\sqrt {1+x^4}\right )-\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right )+3 \log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^2-3 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^2+\log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^4-\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^4-\log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^6+\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}-6 \text {$\#$1}^3+3 \text {$\#$1}^5+2 \text {$\#$1}^7}\&\right ]}{2 \sqrt {2}} \]
Mathematica 12.3 output
\[ \int \frac {\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx \]________________________________________________________________________________________