In my research I encountered the following integral $$J = \int_0^1 I_t^{-1}(a_1,b_1) \: I_t^{-1}(a_2,b_2) \: \mathrm{d}t$$ which I would like to evaluate as a closed form expression, that is, as a function of the parameters $a_{1}, b_{1}, a_{2}, b_{2} >0$. In words, this is the integral of the product of two inverse regularized incomplete beta functions.

This was posted in Math StackExchange about two months back but received no answer. That link shows my naive attempt via integration by parts. If a closed form (or lack of it) is well known, I would appreciate a reference. Any ideas or approximation for the integral are welcome.