Formula to inverse a function of integral-splines

Question

I have a function Y = constant + Beta I-Spline(X). All the Beta are > 0, and I use i-splines (integral of M-spline), which guarantees that f(X) = beta*I-Spline(X) is strictly increasing in X. I wonder if there is a way to INVERT f(X) (e.g., a closed form formula for the inverse, given the beta - or if not a closed form, a way to compute it quickly)? Because I need to get X=f^-1(Y), so I need to know the inverse function. So far, I'm estimating f^-1 by estimating f(X) on a grid and doing a reverse interpolation, but that's quite messy, and approximative at some points. I'm working in R.