Characterize the evolution of the reduction system for the tan(ln(x))^2 + 1 denominator

Characterize the evolution of the reduction system generated by Norman’s completion process when integrating with denominator v = tan(ln(x))^2 + 1, specifically for the differential field (Q(t1, t2, t3), d) modelling tan(ln(x)) via dt1 = 1, dt2 = 1/t1, and dt3 = t1 t2, under a monomial order with t3 > t2 > t1; ascertain the structure and long-term behavior (including termination or infinite patterns) of the sequence of reduction rules and critical pairs that arise in this setting.

Background

Norman’s completion-based approach to integration can generate non-terminating sequences of reduction rules for certain denominators. In particular, for v = tan(ln(x))^2 + 1, he reported non-termination and posed an explicit open problem to describe how the reduction system evolves.

In this paper, the authors formalize Norman’s process, develop a refined version, and analyze this specific case by modelling tan(ln(x)) in a differential field. They demonstrate that Norman’s original completion process does not terminate for this denominator, exhibit an infinite pattern of rules, and show that their refined completion process terminates. The open problem remains to fully characterize the evolution of the reduction system in such cases.