\[ \int \frac {-b^2+a x^2}{\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx \]
Optimal antiderivative \[ \frac {2 x}{\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}-\frac {4 \sqrt {b}\, \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}\, \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}\right )}{\sqrt {a}}+\frac {2 \sqrt {2}\, \sqrt {b}\, \arctan \left (\frac {\sqrt {a}\, x \sqrt {2}}{2 \sqrt {b}\, \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}-\frac {\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}\, \sqrt {2}}{2 \sqrt {b}}\right )}{\sqrt {a}} \]
command
Integrate[(-b^2 + a*x^2)/((b^2 + a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]]),x]
Mathematica 13.1 output
\[ \frac {2 x}{\sqrt {b+\sqrt {b^2+a x^2}}}-\frac {4 \sqrt {b} \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}}+\frac {\sqrt {2} \sqrt {b} \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \]
Mathematica 12.3 output
\[ \int \frac {-b^2+a x^2}{\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx \]________________________________________________________________________________________