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Direct integration formulas for barycentric rational representations

Develop direct analytical formulas or numerical schemes to evaluate indefinite integrals of barycentric rational representations, thereby avoiding conversion to partial fractions.

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Background

For numerical integration of rational approximants, the paper notes the absence of formulas to directly integrate barycentric representations, requiring conversion to partial fractions. A direct barycentric integration approach would streamline workflows, reduce computational overhead, and potentially improve numerical stability in integral evaluations.

References

We are unaware of formulas that evaluate $\int \kern -1pt r(x) \kern .7pt dx$ directly in the barycentric representation, so as an alternative, one can convert a rational approximation to partial fractions and then integrate these term by term.

— Applications of AAA rational approximation (2510.16237 - Nakatsukasa et al., 17 Oct 2025) in Section 5 Computing derivatives and integrals