I am working on a fitting model and I need to use a sigmoid function in the following integral
$$\int_0^R S(x)\cdot x \cdot J_{0}(x) dx $$ where $S(x)$ is the sigmoid function and $J_0(x)$ is the bessel function. I know there are some famous sigmoid functions available, namely: Logistic function, Error Function, Arctangent Function, and some algebraic functions. But I don't know which one is computationally faster(for evaluating this integral numerically). Is there any other sigmoid function out there which is faster? Also, I should mention that the sigmoid function needs to be modified for my program. By modified I mean, that I should be able to change the centre point and the width of the function. I need to evaluate this integral for fitting thus this needs to be evaluated a few hundreds of time. This is the main reason why I want the function to be fast(computationally).