Another thing to think about: I grabbed this from stack exchange:

*In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a finite number of certain "well-known" functions. *

From what I've read analytic solutions and closed-form solutions are the same thing.

I could imagine a case like y=x^2, which is definitely closed-form, to not be closed-form to someone like Ptolemy. How do you calculate a square (or sqrt(x), or sin(x)) in 100AD? The answer seems to be that you get a table that someone else wrote and look it up.