\[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {x+\sqrt {1+x^2}}} \, dx \]
Optimal antiderivative \[ \frac {\left (32 x^{2}+3 x +8\right ) \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}+\left (-2+16 x \right ) \sqrt {x +\sqrt {x^{2}+1}}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}+\sqrt {x^{2}+1}\, \left (\left (3+32 x \right ) \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}+16 \sqrt {x +\sqrt {x^{2}+1}}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}\right )}{24 \left (x +\sqrt {x^{2}+1}\right )^{\frac {3}{2}}}-\frac {\arctanh \left (\sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}\right )}{8} \]
command
Integrate[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/Sqrt[x + Sqrt[1 + x^2]],x]
Mathematica 13.1 output
\[ \frac {1}{24} \left (\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (8+32 x^2+3 \sqrt {1+x^2}-2 \sqrt {x+\sqrt {1+x^2}}+16 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}+x \left (3+32 \sqrt {1+x^2}+16 \sqrt {x+\sqrt {1+x^2}}\right )\right )}{\left (x+\sqrt {1+x^2}\right )^{3/2}}-3 \tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )\right ) \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {x+\sqrt {1+x^2}}} \, dx \]________________________________________________________________________________________