I can recommend the book "Integration In Finite Terms: Fundamental Sources".
It gives the background necessary to understand Barry Trager's PhD Thesis.
The last chapter is his thesis.
https://www.amazon.com/Integration-Finite-Terms-Fundamental-Computation/dp/3030987663

A more recent update is Daniel Schultz's "Trager's Algorithm For
Integration Of Algebraic
Functions Revisited"

Tim Daly

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