\[ \int \frac {\sqrt {1-x^2}}{\left (-1+x^2\right ) \sqrt {a+b x^2}} \, dx \]
Optimal antiderivative \[ -\frac {\EllipticF \left (x , \sqrt {-\frac {b}{a}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}}{\sqrt {b \,x^{2}+a}} \]
command
integrate((-x^2+1)^(1/2)/(x^2-1)/(b*x^2+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {{\rm ellipticF}\left (x, -\frac {b}{a}\right )}{\sqrt {a}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b x^{2} + a} \sqrt {-x^{2} + 1}}{b x^{4} + {\left (a - b\right )} x^{2} - a}, x\right ) \]