@sursumCorda 

The following might do what you are looking for (in charactaristic 0):

Let be sym(p) = Sum( pi(p), pi in symmetric group S_n) / n! the symmetrization of the polynomial in the n variables.

Then p = sym(p) + ( p - sym(p) ) = symmetric part + remainder

Here pi(p)(x1, ..., xn) = p( pi(x(pi(1)))), ..., pi(x(pi(n)))) ), i.e. the polynomial using permuted variables.

do you mean something like this?

nicepo:=(a,b)->(a)[b]:
pochhammer(1/2,n): eval(%, pochhammer=nicepo);

where I would not use it anyway ...

@mmcdara I do not mind at all ...

The following gives a formal solution. "RootOf" is difficult for newbies, just read it as
"all the roots of ...", it is about a polynom of degree 5, so no explicit formal solution
is known to Maple.

subs(int= Int, N):         # do not use lowercase here
subs([p=2, Zeta=1/6,M = 1, m = 1, kappa=1], %): # data
subs(varphi = phi, %):     # I dislike "varphi"
theIntegrand:=op(1, %);

simplify(theIntegrand): map(factor, %);
Int(%, phi): value(%);

For a numerical answer you need to feed upper and lower bound as numerical value, at least.

For a symbolical answer Maple needs informations to compare singularities of the integrand and integration bounds - though it will find the anti-derivative

Edit: I just see  mmcdara's answer ...

@Hullzie16 

So you want a minimum or a zero of the function f:= b -> H(alpha,b) - G(alpha,b)

What is H, what is G and what are the allowed ranges?

@Hullzie16 what do you want to show in the final plot of your sheet? How would you write it if having a procedure F(alpha,b)?

For me your question is unclear. Could you post or upload some coded example?

@sursumCorda It works now (strange ...) and needs ~ 5 minutes

I was not aware of that component. After 1/2 hour I cancelled the execution.

Anyway I think it would be better to understand what vv is actually doing in https://www.mapleprimes.com/questions/237110-Solve-Linear-System-Of-Congruences#answer297368

for me it works ...

@Jesús Guillera what is the reason for these questions and why p^3, p = prime ?

I have doubts that one can install Maple on 2 machines with the same license number and hope it works properly. It is still not clear what you did.

You may give the source for your code and - if possible - the expected result.

Edit:
note that the last step in the loop has s_k=1 and that H(x, t, 1, u) = - u.
Hence the last update for u_approx results in 0.

Yes, trial-and-error, points away from (0,0) but close to singularities

[37.6324, -0.356503], [-39.4716, 0.365055], to be refined with Digits=50 ...