Background: In "Stability under integration of sums of products of real globally subanalytic functions and their logarithms", by R. Cluckers and D.J. Miller, it is shown that the integral of a semialgebraic function $f:\mathbb{R}^m\times\mathbb{R}^n\to \mathbb{R}$ over $\mathbb{R}^n$ may be represented as a log-analytic function over $\mathbb{R}^m$. (A log-analytic function is a function belonging to the algebra generated by subanalytic functions and logarithms of positive subanalytic functions- in particular, there exists a subanalytic stratification of the domain so that on each piece, the function resides in the algebra generated by analytic functions and logarithms of positive analytic functions.)
In "Integration of semialgebraic functions and integrated Nash functions," by T. Kaiser, this result is improved and the class of functions that may arise as integrals is pared down a bit, but it is still written that the sort of decomposition of the domain described above is an analytic decomposition.
Question: What prevents this result from being sharpened to say that the decomposition of the domain may be done semi-algebraically? Is there is a known counterexample to this hope, or is this something that's conjectured to be true?
