\[ \int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx \]
Optimal antiderivative \[ \frac {3 \arcsin \left (x -\sqrt {x^{2}-1}\right ) \sqrt {2}}{8}+\frac {\left (3 x +\sqrt {x^{2}-1}\right ) \sqrt {1-x^{2}+x \sqrt {x^{2}-1}}}{4} \]
command
Integrate[Sqrt[1 - x^2 + x*Sqrt[-1 + x^2]],x]
Mathematica 13.1 output
\[ \frac {1}{8} \left (\frac {2 \left (-1+x^2\right ) \left (3 x+\sqrt {-1+x^2}\right )}{\sqrt {1-x^2+x \sqrt {-1+x^2}} \left (-1+x^2+x \sqrt {-1+x^2}\right )}-3 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {-1+x^2}}{\sqrt {1-x^2+x \sqrt {-1+x^2}}}\right )\right ) \]
Mathematica 12.3 output
\[ \int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx \]________________________________________________________________________________________