### Abstract

We present new results on real integration of oscillating functions, for example, Fourier transforms of subanalytic functions; it is joint work with Aschenbrenner and Rolin. We begin to control very good the transcendental functions we have to add, (namely, oscillating versions of basic Abelian integrals), in order to describe these parameterized integrals. The method is in principle algorithmic, with a similar algorithm as to compute motivic oscillating integrals. Yet, conjectures linking real and motivic integrals remain unsolvable.