http://axelvogt.de/temp/Part1.pdf

http://axelvogt.de/temp/Part1.mws

PS: that site is a joke ...

Find attched the stuff for the relations

Part1.mws

Part1.pdf

Fine. Roughly: then there are no parameters and you want to prove an identity between 3-dim integrals.

After integrating w.r.t. 'a' those are 2-dim. Your Heaviside actually seems to mean to integrate over a (ellipsoid) range, a curve. A proper math answer/solution would be to reduce that to dim=1 by Stoke's / Green's theorem (for which I am not clever enough, one needs the parametric curve to proceed) instead of asking Maple for a brute solution.

I would simply not use it (or would comment the code)

http://www.maplesoft.com/support/help/Maple/view.aspx?path=LinearAlgebra%2fAdd

@Dima 

some numerical experiments along the line given by Carl ... and then guessing, a sketch is given as separate answer

@Dima 

Is it (roughly) k2 = k1, k3 = -2*k1 (and numerically confirmed with modest error) ?

@Alejandro Jakubi 

Yes, there was one by Markiyan at http://www.mapleprimes.com/questions/200726-Numerical-Integration, which now vanished, my answer also vanished. After some minutes they appear again. I was not able to upload a file which persists and with IE an edit is not possible (while I got warnings about Facebooks, that certifcates are wrong) and what ever ... not a pleasure at all.

MP_3_triple_int.mws

That does not solve it. However note, there is a symmetry in a, so one needs it only up to Pi.

Then not, that you add min and max and as far as I can see that is for Reals, so also that simplifies to jusr a Real.

The last is the Heaviside - it is the same in all of the 3 - but I stopped here whether one can use it.

So just find a sketchy sheet to proceed further.

The extremum is in phi = -0.01259116530, achieving 8.098949326, which would be difficult for a 'brute' way to 'show' it.

I would not do that - just use Kitonum's way of 'brute' plotting

BTW: in my attached sheet the formal solution is not always correct (for 0 < phi it is), but the correction to a much more simple integral is still valid (your Heaviside more or less only restricts the range of integration, so better eliminate its use)

It seems that Maple makes an error by evaluating the integral if phi is negative ...

At least for the example the is an explicit solution,
 
g := phi -> 
  114171/6494*exp(-395201/510*(phi-733/577129)^2+1/400)*
  ((phi+733/577129)^2)^(1/4)*
  BesselK(1/4,395201/510*(phi+733/577129)^2);

I have not looked in your parametric case, but the attached worksheet should help you.

MP_200734.mws

For me - as well - the upload does not work in all cases if using Firefox

The help for ?numpoints says:

"Specifies the minimum number of points to be generated.  The default is 200. Note: plot employs an adaptive plotting scheme which automatically does more work where the function values do not lie close to a straight line. Hence, plot often generates more than the minimum number of points."

Please provide the concrete task (=integral under question), either as as ASCII code or uploaded worksheet.