See, for example, Rubi, or my earlier project Tilu, for programs that 
> absorbed,

in some sense learning from tables of integrals.
This is not classical machine learning, because the objects being learned 
are
patterns.  So the result for sin(x)dx works for sin(u)du,  as a trivial 
pattern
match.
Categorizing the patterns so there is an effect search method for finding 
one
that matches a problem is important for efficiency.
Also important (and quite a challenge to do effectively) is a
method to simplify expressions.

So there are ways of using "data" productively in 
methods for symbolic definite integration.

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