\[ \int \frac {\left (x-\sqrt {a+x^2}\right )^n}{\sqrt {a+x^2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (x -\sqrt {x^{2}+a}\right )^{n}}{n} \]
command
integrate((x-(x^2+a)^(1/2))^n/(x^2+a)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (x - \sqrt {x^{2} + a}\right )}^{n}}{n} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (x - \sqrt {x^{2} + a}\right )}^{n}}{\sqrt {x^{2} + a}}\,{d x} \]________________________________________________________________________________________