What are some examples of elementary and non-elementary functions?

What are the differences between the two?

How do you take the derivative and integral of non elementary functions?

Especially non elementary trig functions. Is there a formula sheet for this?
I got a message saying I need to write a comment on the steps I have taken. I figured out to derive some non elementary trig function like sin(x\^2). I used the Chain Rule. I took the derivative of the outside "sin" and multiplied of the derivative of the inside "x\^2" and got 2xcos(x\^2).

I figured out how to take the integral and derivatives of functions in the form e\^(ax\^n) for constants "a" and integer "n." I did this by "u" substitution and chain rule.

But I am having trouble with the trig functions. Is there some type of formula sheet or cheat sheet with trig identities that are non-elementary. All the functions in Calc 1-4 were elementary so I don't know what I am doing.
An elementary function is any function (finitely) composed of addition, multiplication, exponentiation, and their inverses (subtraction, division, roots, and logs). All of the trig functions are elementary functions because they can be written in terms of exponents. Likewise, the inverse trig functions can be written in terms of logarithms. The derivative of every elementary function is an elementary function. However, the integral of an elementary function can be composed of elementary functions and special functions, or not representable in either way. Some examples of non-elementary functions are the Lambert W function, the gamma function, the error function, the exponential integral, the polylogarithms, and the sine integral. One of the most common methods of taking non-elementary integrals is by converting the function into an analytic series (such as a Taylor series) and integrating the resulting polynomial (this can restrict the domain of the function). Usually you'll attempt to integrate using other tricks like u-substitution and integration by parts, and if you encounter a non-elementary integral you'll usually put it in a standard form so that the resulting series or special function is in a familiar form. Hope this helps. It wasn't entirely clear what you were looking for or why.