Maybe I'm misunderstanding, but the table says that the "root of a polynomial that is not an algebraic solution" doesn't count as closed-form.

https://en.wikipedia.org/wiki/Closed-form_expression

"Commonly, the basic functions that are allowed in closed forms are nth root, exponential function, logarithm, and trigonometric functions.[a] However, the set of basic functions depends on the context. For example, if one adds polynomial roots to the basic functions, the functions that have a closed form are called elementary functions."

It's only maths, don't expect things to be so black and white.