Additive normal forms and integration of differential fractions

This paper presents two new normal forms for fractions of differential polynomials, as well as algorithms for computing them. The first normal form allows to write a fraction as the derivative of a fraction plus a nonintegrable part. The second normal form is an extension of the first one, involving iterated differentiations. The main difficulty in this paper consists in defining normal forms which are linear operations over the field of constants, a property which was missing in our previous works. Our normal forms do not require fractions to be converted into polynomials, a key feature for further problems such as integrating differential fractions, and more generally solving differential equations.