function defined as an integral involving Bessel functions

Question

i need to analyze a function of the form

$$F(x,y) = \int_0^{1} e^{-(1+s)\alpha x}\sinh((1-s)\beta y) I_0(\sqrt{(x^2-y^2)s}) ds $$

Where $I_0$ is the modified Bessel function.

$x>y$ always.

$\alpha$ and $\beta$ positive.

I need to explore the convexity of the function, monotony on the variables, bounds, asymptotics, some analysis on the derivatives (sign) etc.

Thanks.