I think [Fricas](https://en.wikipedia.org/wiki/FriCAS) (a homonym-with-slightly-different emphasis for Free CAS = Free Computer Algebra System) has the most complete/carefully thought out "type system" for computer algebra/calculus/general math. It's based on IBM's [Axiom](https://en.wikipedia.org/wiki/Axiom_\(computer_algebra_system\)) which was open sourced (written in Common Lisp). Of the two, `fricas` seems the active community/actually "just works" when you install it. It's graphical interface is unquestionably archaic, but it's CA types and algorithms seem maybe the most solid of the open source options. I believe it is the only such system that guarantees if a symbolic indefinite integral has an answer in terms of elementary functions then it will be found.
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