Question: How to make the numerical evaluation of sums of integrals faster?

I want to perform a numerical evaluation of sums of integrals of relatively complicated functions. I know about the evalf(Int( )) and evalf(Sum( )) commands to numerically evaluate both sums and integrals individually. My question is: what is the time-efficient way to numerically evaluate a sum of integrals? 

Here is a simplified sketch of what I have.

Say I define my complicated function F of the variable x (which will be integrated over) and of some constant parameter n.

I am interested in numerically evaluating in a time efficient way the following sum of integrals of F:

Where should I apply the evalf() command(s)? Should I go evalf(Sum( evalf(Int( )))) or evalf(Sum( Int())) or sum( evalf(Int( ))), or something else? I am not too worried about the accuracy here: it is for plots mainly. How to make this numercal evaluation fast?

Bonus question. If now I make F also depend on t, and wish to define a function G(t) out of a linear combination of such sums of integrals: is the method the same? I can have G(t) defined numerically with a t dependance. For example:

Thanks a lot!

PS: F is a complicated function in the sense that it is rationnal in some (non-usual) polynomials defined by a Rodrigues Rormula. The integrand has no singularity on the domain of integration. I have Maple 2018.