(I've got a bit more time now, and since Mike has not chimed in, I'll offer an answer.)
The point is, there are things you MIGHT do, but how can we possibly help you? What do you know or not know about numerical integration? What characteristics does your function have? Are the limits really infinite, or are they finite? The problem is, one could write a book about integration. Oh, thats right, there are already lots of books on the subject. But without knowing something about what you have to solve, the best answer is to buy a faster computer.
There are no n-fold integrations in MATLAB, but I recall that Mike Hosea has posted a higher degree integration tool on the FEX, here:
It handles 4,5,6 fold integrations, I think even allowing infinite limits.
That is the good news. The bad news is to expect it to be seriously slow. Iterated integration tools like this tend to take a lot of function evaluations. The reason is simple. To understand, lets make a simple test...
function [f,nx] = testfun(x)
persistent nevals
if isempty(nevals)
nevals = 0;
end
nevals = nevals + numel(x);
f = 1/sqrt(2*pi)*exp(-x.^2/2);
if nargout > 1
nx = nevals;
end
That little function counts the number of function evaluations integral makes for a simple Gaussian.
integral(@testfun,-inf,inf)
ans =
1
[f,nx] = testfun(0)
f =
0.39894
nx =
211
So to compute the integral of ONE gaussian in one dimension, it took 211 function evaluations.
But you have a 5-fold integral. Each of these integrations is of similar complexity, so you might expect to need
so about 4e11 function evaluations. That is 400 billion evals. If your function has 600 lines of code, how long will it take to execute? If it is a bit slow, and you get only 1 eval per second, then it would take
YEARS to finish. Even if one call to your 600 line function takes 0.001 seconds to execute, that is still over 13 YEARS to terminate.
Even if some of those integrations are simpler, and take only half as many calls, go get a cup of coffee and a good book, as you will be waiting for a bit. "War and Peace" might be a good choice of books for this one, or better yet, the complete works of Shakespeare, and expect to re-read whatever book you have quite a few times over. The good news is when it gets done, you will be a true Shakespeare scholar.
My point about not telling us anything is still valid. Are there symmetries you can take advantage of? Or, if your function has characteristics like that of a Gaussian as you show, do you know anything about Gauss-Hermite quadrature? This could easily be a boon for you, or a bust. I cannot know, since you have told us nothing.
