I recently tried to evaluate
$$\int e^{\beta\arccos(a/(a+x))}dx$$
(everything constant except $x$)
and got a complicated answer involving a hypergeometric series with complex arguments.
Can anyone come up with a way of turning $f(x)=arccos(a/(a+x))$ into a function that, when put in the place of $f(x)$ makes the original integral easier? NOTE: the approximation only needs to be good for small $x$ (note that a taylor series at 0 involves $arccos'(1)$ etc (which is $-\infty$) so I need something different).
Thanks!