integrals with no analytic answer - intuition and proof

Question

It appears that the following integral has no analytic solution:

$$\int_0^\pi e^{\sin(x)} \, dx$$

Intuitively, what is the reason for this integral having no analytic answer? (is there a way to prove it formally?)

More generally, when and why do some integrals have no analytic answer and how can one gain intuition about what types of functions have no analytic integrals?

Answer 1

Essentially if you start with some initial set of functions and consider the functions formed by composing and multiplying them, then the product and chain rules say that differentiating doesn't broaden the class of functions you're considering, whereas there are no analogous rules for integration.

You can prove that the integral of a given elementary function is non-elementary using differential Galois theory.

Edit: see comments.

Answer 2

Actually, your example does have a "closed form", although not elementary: $$ \pi ({\bf L}_0(1) + I_0(1))$$ where ${\bf L}_0$ is a modified Struve function and $I_0$ is a modified Bessel function.