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][1] 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.   

  [1]: https://math.stackexchange.com/questions/1970571/integral-of-product-of-two-inverse-regularized-incomplete-beta-functions