@Preben Alsholm A human would reduce the integral to a Beta function using the change of variables x^p = t.
Unfortunately, Maple cannot compute it. 
Actually, including the assumption p>1 (mandatory for convergence), the verbose int finds this:

[cook = 2*Beta(1/p, -1/p + 1)/p, FAILS = (distribution, piecewise, series, o, polynomial, ln, lookup, ratpoly, elliptic, elliptictrig, meijergspecial, improper, asymptotic, ftoc, contour), ftocms = 2*GAMMA((p + 1)/p)*GAMMA((p + 1)/p - 2/p), meijerg = 2*Pi/(p*sin(Pi/p))]
 

@Carl Love Thank you. I was aware that the proc can be improved but I have chosen the simplicity, and anyway a Dynkin system is used mainly for infinite families and in this case Maple is useless.
My opinion is that a Maple code implementing a more complicated algorithm (not the case here) should be presented in two flavours: a simple one and an optimized (speed and/or memory) version.

@JAMET But why?

@dharr Yes, it's a group but the group operation is symmetric difference, not union.

However, the group structure is not enough to obtain the algebra; we need the ring having  "+" = `symmetric difference` and "." = `intersection`.

You are confusing a Dynkin system with a monotone class.

In your example it is possible because the space is finite (so we actually have an algebra instead of a sigma-algebra).
We just need a procedure which starts with the family C and takes finite unions of finite intersections of the sets in C and their complements,  until it stabilizes. 

Unfortunately such a proc is not very useful because the resulting (sigma)algebra is in general HUGE.

BTW, for your example X := {1, 2, 3}, C := {{1}, {2}},  the algebra generated by C is obviously the powerset P(X) of X (i.e. it is maximal, containing 2^3 sets). 
 

@Kitonum Actually, the answer is NO. For example, taking f := alpha the integral R-S does not exist but Maple happily "computes" it!

@Kitonum Of course. And this does not work for subexpressions e.g.  expand(f(tan(x+k*Pi))) ... 

@Kitonum It's sad that Maple cannot simplify:
simplify(tan(x+k*Pi))    assuming   k::integer;
simplify(sin(x+2*k*Pi)) assuming   k::integer;
 

So, you have two m x n matrices A and B.
You need a (column) permutation matrix P and a (row) permutation matrix Q  such that Q.A.P = B, if such P, Q exist.
(actually, it seems that you are interested only in P, and for Q the existence is enough).
Is this correct? Do you need all the possibilities for P?
 

@JAMET Then, do not assign X,Y:

[X = (a*m^2 + 2*m^2*p + 2*p)/(2*m^2), Y = a/(2*m)]:
eliminate(%, m);

(a parabola)

Actually, in modern mathematics these symbols are considered redundant.

You've got four answers and no reaction. It's not a polite attitude!

@Earl The method is mentioned in the help page for EulerLagrange (for the case of a single function). They are not important, but they could simplify the computations sometimes.

In Windows both examples work fine.