We extend the shell and kernel reductions for hyperexponential functions over a field of rational functions to a monomial extension. Both of the reductions are combined into one algorithm. As an application, we present an additive decomposition in rationally hyperexponential towers. The decomposition yields an alternative algorithm for computing elementary integrals over such a tower. Preliminary experiments show that the alternative can be used to compute elementary integrals that are not found by the integrators in the latest versions of Maple and Mathematica.