Complete Reduction for Derivatives in a Primitive Tower

A complete reduction $\phi$ for derivatives in a differential field is a linear operator on the field over its constant subfield. The reduction enables us to decompose an element $f$ as the sum of a derivative and the remainder $\phi(f)$. A direct application of $\phi$ is that $f$ is in-field integrable if and only if $\phi(f) = 0.$ In this paper, we present a complete reduction for derivatives in a primitive tower algorithmically. Typical examples for primitive towers are differential fields generated by (poly-)logarithmic functions and logarithmic integrals. Using remainders and residues, we provide a necessary and sufficient condition for an element from a primitive tower to have an elementary integral, and discuss how to construct telescopers for non-D-finite functions in some special primitive towers.