This seems very interesting, I have only just skimmed it so far. Am I wrong in thinking this might relate to (representations for) Geometric Algebra?
The calculus for differential forms is a part of Geometric algebra, so I think there is a relation.
Egison's strong point is it can handle tensor-valued p-forms due to its ability to handle both of tensor index notation and differential forms at the same time.
Here is a link for a new paper that discusses the integration of tensor index notation including the support for differential forms.
https://arxiv.org/abs/1804.03140