Is there any special function corresponding to $\int_{-1}^1 \frac{1-e^{b(x-a)}}{(x-a)^2}\sqrt{1-x^2}dx$?

Question

I try to get an expression for this difficult integral:

$$\int_{-1}^1 \frac{1-e^{b(x-a)}}{(x-a)^2}\sqrt{1-x^2}dx\quad b>0,\;a>1$$

It could also be written in terms of trigonometric functions with $x=\cos \theta$.

Can it be, by any chance, written in terms of known special functions?

EDIT:

Actually, the hard bit is $$\int_{-1}^1 \frac{e^{bx}}{(x-a)^2}\sqrt{1-x^2}dx\quad b>0,\;a>1$$

Written as an integral on a complex contour (the unit circle), the pole $z=0$ is essential.

$$\oint e^{b(\sqrt{z}+1/\sqrt{z})/2}\left(\frac{z-1}{z+1-2a\sqrt{z}}\right)^2\frac{dz}{iz}\quad b>0,\;a>1$$

Any idea to compute this residue?

Answer 1

With high confidence, no.

Notice that as the function has variable parameters, the definite integral amounts to an indefinite one.

And Wolfram Alpha capitulates.