That's the arc length you're stuck with, I'm afraid. But did you really expect the equations defining the involute to be simple?

Things aren't that bad .... work with what you've got.

By the way, the integral for the arc length can be expressed in terms of the inverse hyperbolic sine function. Because I'm lazy, I'll just suggest you fire up the ol' Wolfram Integrator.

To put your mind at rest, you might want to look at this.