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