real analysis - $int_0^1int_0^1binom{text{something}}{text{something}}binom{text{something}}{text{something}}cdot text{something }dxdy$ with closed-form

I've calculated an approximation of integrals like than $$\int_0^1\int_0^1\binom{f(x)}{f(y)}\binom{f(y)}{f(x)}dxdy\tag{1}$$ for simple functions $f(x)$. I don't know if some of these were in the literature or have a nice closed-form.

Question . I would like to know how to create, if it is feasible, nice examples of double integrals of binomials similar than $(1)$. Do you know how to calculate a nice example using different functions
$$\int_0^1\int_0^1\binom{\text{something}}{\text{something}}\binom{\text{something}}{\text{something}}\cdot \text{something }dxdy\,?\tag{2}$$
If you know an example from the literature with a nice closed-form, please answer this question as a reference request, then I am going to try search such literature and read the example. Many thanks.

Your closed-form can be expressed as a series of special functions (
I'm especially interested in how to create such an example).