Structure theorems for parallel integration

We introduce structure theorems that refine Liouville’s Theorem on integration in closed form for general derivations on multivariate rational function fields. By predicting the arguments of the new logarithms that can appear in integrals, as well as the denominator of its rational part, those theorems provide theoretical backing for the Risch–Norman integration method. They also generalize its applicability to non-monomial extensions, for example the Lambert W function.