I can’t compute the above integral
It seems a very complicated integral, with little chance of having an elementary answer. You can try numerical integration for given numerical values of the parameters:
NIntegrate[f[2, u, 2], {u, 0, 2}]
By the way, do you think it is a good idea to use the same symbol f for both the name of a function and the name of an argument?
I would also add the suggestion to use the Piecewise function instead of If. In the documentation page, under Applications, you can find an example which is applying Integrate to a piecewise function.
Looking at the scale of the constants in this function, I also have doubts about using NIntegrate..