Are there ways of telling whether or not an integral will have a closed expression?

Question

I was looking at some of the great integral posts that have graced this website and I am wondering if there is a way to tell whether or not a definite integral will have a closed form. If there is no "one-size-fits-all" criterion, are there any general rules? (e.g. integrals of polynomial functions will have closed forms)

Answer 1

One way to find an expression for a definite integral is to first find the corresponding indefinite integral. This is already a difficult problem.

There is an algorithm for computing indefinite integrals in terms of elementary functions, when such representations exist: Risch's algorithm. It is a very complex algorithm which is now included in the major symbolic mathematics programs.

Note that even the integral of a rational function may not have a closed form in any way that may be useful because you won't have closed forms for the coefficients of the partial fractions decomposition needed for that (since there are no formulas for the roots of most polynomials of degree 5 or higher).

Risch's algorithm assumes that all necessary constants are available.

However, sometimes it is possible to find an expression for a definite integral without first finding the corresponding indefinite integral, even when the indefinite integral has no closed form. The most famous example is perhaps the Gaussian integral: $$\int_{-\infty}^{+\infty} e^{-x^2}\,dx = \sqrt{\pi}$$

I don't think there are any general rules in this case.