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