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