Is it Possible that some Non-Analytically Integrable Functions might actually have Analytical Integrals? Is it Possible that some Non-Analytically Integrable Functions might actually have Analytical Integrals (that have not yet been discovered)?
When reading posts like these (e.g. List of functions not integrable in elementary terms - too bad the main hyperlink in this post is broken), we often hear about functions that are considered "non-analytically integrable" : this means that these functions do not have "closed form and exact" integrals. For these kinds of functions, we are required to integrate them analytically. On another note, I never knew about this - (but if I understand this correctly) there is actually a mathematical theorem called "Liouville's Theorem" that states we can prove certain functions will not have analytical integrals.
For the longest time, I always thought that some functions that are currently Non-Analytically Integrable might actually have Analytical (Exact Closed Form) Integrals - but we just haven't discovered them yet. For example, perhaps in the future, some new theorems in mathematics will be discovered that will provide Analytical Integrals to functions that are currently considered as Non-Analytically Integrable.
From a Statistics and Probability perspective, we often use stochastic sampling methods (e.g. Markov Chain Monte Carlo) to approximate the integrals of posterior probability distribution functions that arise in Bayesian Statistics. Sometimes, these posterior probability distribution functions have "analytically exact closed form solutions - these are called "conjugate priors" and as a result do not require approximations. However, integrating many of these posterior probability distributions will often require some type of approximation method due to their complex and irregular nature.

*

*In the case of these complex and irregular "non-analytically integrable" (posterior probability distribution) functions - is it possible that one day, some new math will be discovered that allows some of these "non-analytically integrable" functions to have closed form analytical integrals?


*Or is the non-existence of analytical integrals for these posterior probability distribution functions is forever guaranteed by the Liouville Theorem?
Thank you!
 A: How can we prove an integral is not elementary?  An easy start (with many references):
"Integration in Finite Terms", Maxwell Rosenlicht,
The American Mathematical Monthly 79 (1972) 963--972.
Stable URL: http://www.jstor.org/stable/2318066
A: We have to differentiate between closed-form expressions and analytic expressions. Both terms should be correctly defined mathematically before they are used. Often, closed form means only expressions of elementary functions.
Liouville's theorem together with Risch algorithm is for functions that lie in a differential field and their antiderivatives that lie in an elementary field extension of that differential field. So it is for closed-form antiderivatives of closed-form functions.
Risch algorithm is a decision algorithm. That means, it can decide if an antiderivative of that class does exist or does not exist.
[Davenport 2007]:
"All integration algorithms for elementary functions rely on Liouville’s principle: that the only new elementary functions which can be introduced are logarithms, and that only with constant coefficients. This theorem remains true even if the integrand is not elementary.
...
In general, one needs a fresh generalisation of Liouville’s Principle for each new function generator introduced. Some such have been proved ..., but even the most general ([Singer 1985]) is far from complete"
[Singer 1985]:
"In Part of this paper, we give an extension of Liouville’s Theorem and give a number of examples which show that integration with special functions involves some phenomena that do not occur in integration with the elementary functions alone. Our main result generalizes Liouville’s Theorem by allowing, in addition to the elementary functions, special functions such as the error function, Fresnel integrals and the logarithmic integral (but not the dilogorithm or exponential integral) to appear in the integral of an elementary function."
The answer to your question is given in [Davenport 2007]: "In general, one needs a fresh generalisation of Liouville’s Principle for each new function generator introduced." That means, some non-analytically integrable functions might actually have integrals in analytic form, if new math will be discovered.
$\ $
[Davenport 2007] Davenport, J. H.: What Might “Understand a Function” Mean? in: Kauers, M., Kerber, M., Miner, Robert, Windsteiger, W.: Towards Mechanized Mathematical Assistants. 14th Symposium, Calculemus 2007, 6th International Conference, MKM 2007, Hagenberg, Austria, June 27-30, 2007. Proceedings. Springer Berlin Heidelberg 2007
[Singer 1985] Singer, M. F.; Saunder, B. D.; Caviness, B. F.: An Extension of Liouville's Theorem on Integration in Finite Terms. SIAM J. Comput. 14 (1985) 966-990
