We describe differential rational normal forms of a rational function and their properties. Based on these normal forms, we present an algorithm which, given a hyperexponential function T(x), constructs two hyperexponential functions T1(x) and T2(x) such that T(x) = T1'(x) + T2(x) and T2(x) is minimal in some sense. The algorithm can be used to accelerate the differential Gosper's algorithm and to compute right factors of the telescopers.