\[ \int \frac {\left (1+x^2+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[((1 + x^2 + x^4)^2*Sqrt[x^2 + Sqrt[1 + x^4]])/(Sqrt[1 + x^4]*(-1 + x^2 + x^4)^2),x]
Mathematica 13.1 output
\[ -\frac {2 x \sqrt {x^2+\sqrt {1+x^4}} \left (1+2 x^6+2 \sqrt {1+x^4}+3 x^2 \left (1+\sqrt {1+x^4}\right )+x^4 \left (3+2 \sqrt {1+x^4}\right )\right )}{5 \left (-1+x^2+x^4\right ) \left (1+2 x^4+2 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 )+2 \sqrt {2} \text {RootSum}\left [1-2 \text {$\#$1}^2-6 \text {$\#$1}^4+2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-8 \log \left (1+x^2+\sqrt {1+x^4}\right )+8 \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}{-\text {$\#$1}-6 \text {$\#$1}^3+3 \text {$\#$1}^5+2 \text {$\#$1}^7}\&\right ]+\frac {\text {RootSum}\left [1-2 \text {$\#$1}^2-6 \text {$\#$1}^4+2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {163 \log \left (1+x^2+\sqrt {1+x^4}\right )-163 \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 )-59 \log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^2+59 \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+13 \log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^4-13 \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 ]}{5 \sqrt {2}} \]
Mathematica 12.3 output
\[ \int \frac {\left (1+x^2+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )^2} \, dx \]________________________________________________________________________________________