Current in a Concentration Cell
Yes, your derived expression looks correct to me.

As to the closed form for the integral, WolframAlpha is most likely unable to find one because it is simply not there among the elementary and special functions. This has nothing to do with the equation being simple. Of course it is simple; nature "solves" it quite easily. It is just that the nature's definition of "simple" does not necessarily coincide with ours. Helium atom is simple, so good luck trying to solve the Schrodinger equation for it. Would you argue an atom is not actually that simple? Well, your concentration cell is made up of quite a few atoms.

So it goes.