\[ \int \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx \]
Optimal antiderivative \[ \frac {\left (60 x^{2}-8 x -75\right ) \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}+\left (16+6 x \right ) \sqrt {x +\sqrt {x^{2}+1}}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}+\sqrt {x^{2}+1}\, \left (\left (-8+60 x \right ) \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}+6 \sqrt {x +\sqrt {x^{2}+1}}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}\right )}{105 \sqrt {x +\sqrt {x^{2}+1}}}-\arctanh \left (\sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}\right ) \]
command
Integrate[Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]],x]
Mathematica 13.1 output
\[ \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (-75+60 x^2-8 \sqrt {1+x^2}+16 \sqrt {x+\sqrt {1+x^2}}+6 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}+x \left (-8+60 \sqrt {1+x^2}+6 \sqrt {x+\sqrt {1+x^2}}\right )\right )}{105 \sqrt {x+\sqrt {1+x^2}}}-\tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right ) \]
Mathematica 12.3 output
\[ \int \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx \]________________________________________________________________________________________