Decidability of checking an antiderivative?

Question

Let's suppose I have two functions $F$ and $G$ and I'm interested in determining whether

$$F(x) = \int G(x)dx.$$

Let's suppose that my functions are composed of elementary functions (polynomials, exponentials, logs, and trigonometric functions), but not, say, Taylor series.

Is this problem decidable? If not, it is it semidecidable?

(I'm asking because I'm teaching a class on computability and a student asked me if a TM could help you integrate a function whose integral was not currently known. I suspect that the functions we don't know how to integrate are more properly functions whose integral can't be expressed as a combination of the above elementary functions rather than functions for which we don't actually know the integral, but that got me thinking about whether the general problem of checking integrals was decidable.)