I've been struggling with a definite integral as below:
$$ \int_0^1 x^{a-1}(1-x)^{b}(I_x(a+1,b) - 1) dx $$
where $I_{x}$ represents the regularized incomplete beta function.
I'd initially tackled it as an integration by parts problem with
$$dv=x^{a-1}(1-x)^{b}$$ $$u=I_x(a+1,b) - 1$$
however couldn't get a sensible answer at all. I'd really appreciate any help people could offer, or even an opinion on whether an analytical answer is a hopeless case!