Classification of integrals not expressible in elementary functions?

Question

Let's consider the smallest set of functions (with real coefficients) closed under addition, multiplication, division, root extraction, and finite compositions containing

• constants
• exponential and logarithmic functions
• trigonometric functions and their inverses

This set is generally known as "elementary functions". It is known that they are closed under all algebraic operations and differentiation, but not closed under inversion and not closed under integration (+infinite sums).

I am wondering if there is any classification, with a proof, of indefinite integrals (antiderivatives) not expressible in elementary functions. If so, we can introduce a minimal number of special functions to integrate elementary functions (once, not repeatedly).

Also, I am not sure about the appropriate tags for this question that are capable of attracting people specialized in related areas. You are welcome to edit accordingly.

Answer 1

You are talking about 'differential galois theory' or 'the galois theory of (linear) differential equations.' A quick glance at "filter" didn't seem to show any appropriate tags.You could start by looking up those terms in Wikipedia where you will find a bibliography with several fairly recent books.

Answer 2

Just as in classical Galois theory one can classify exactly when a root can be written in terms of the 'elementary' operations $+,-,*,/,\sqrt[n]{}$, once can use Differential Galois theory to analyze when various solutions to differential equations can be written in terms of elementary functions. Since indefinite integration and solving differential equations are equivalent, this allows us to prove when an elementary antiderivative does not exists. For example, this paper proves that the Gaussian is such a function.