The integro-differential closure of a commutative differential ring

An integro-differential ring is a differential ring that is closed under an integration operation satisfying the fundamental theorem of calculus. Via the Newton--Leibniz formula, a generalized evaluation is defined in terms of integration and differentiation. The induced evaluation is not necessarily multiplicative, which allows to model functions with singularities and leads to generalized shuffle relations. In general, not every element of a differential ring has an antiderivative in the same ring. Starting from a commutative differential ring and a direct decomposition into integrable and non-integrable elements, we construct the free integro-differential ring. This integro-differential closure contains all nested integrals over elements of the original differential ring. We exhibit the relations satisfied by generalized evaluations of products of nested integrals. Investigating these relations of constants, we characterize in terms of Lyndon words certain evaluations of products that determine all others. We also analyze the relation of the free integro-differential ring with the shuffle algebra. To preserve integrals in the original differential ring for computations in its integro-differential closure, we introduce the notion of quasi-integro-differential rings and give two adapted constructions of the integro-differential closure. Finally, in a given integro-differential ring, we consider the internal integro-differential closure of a differential subring and identify it as quotient of the free integro-differential ring by certain constants.