Hrushovski and Loeser have shown that Hrushovski and Kazhdan?s motivic integration provides a direct approach to relate the motivic Milnor fiber and invariants of the classical Milnor fiber, e.g. to show that their Euler characteristics coincide. Their work relies on the existence of both a purely additive integration morphism and one that respect volume forms. I will present a new integration morphism on some bounded sets that allows to relate those two morphisms. This is joint work in progress with Yimu Yin.