\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+\sqrt {1+x}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[(-1 + x^2)/((1 + x^2)*Sqrt[1 + Sqrt[1 + x]]*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]),x]
Mathematica 13.1 output
\[ \frac {8}{15} \left (-4+3 \sqrt {1+x}-4 \sqrt {1+\sqrt {1+x}}\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}-\text {RootSum}\left [2-8 \text {$\#$1}^4+8 \text {$\#$1}^6+14 \text {$\#$1}^8-32 \text {$\#$1}^{10}+24 \text {$\#$1}^{12}-8 \text {$\#$1}^{14}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\text {$\#$1}\right )}{\text {$\#$1}-\text {$\#$1}^3-4 \text {$\#$1}^5+8 \text {$\#$1}^7-5 \text {$\#$1}^9+\text {$\#$1}^{11}}\&\right ] \]
Mathematica 12.3 output
\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+\sqrt {1+x}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx \]________________________________________________________________________________________