evalf( Int(h, Omega, epsilon=1e-4, method=_CubaCuhre));

gives 4.12122491078804

(To have a probability you forgot probably to divide by (2*Pi)^3.)

You cannot write like this. diff(z, x)  simplifies to 0, because z is interpreted as an independent variable (just like x).
If z depends on x, you must write z(x). 
In this case,
diff(w(x,y,t,z(x)), x); 
==>  D[1](w)(x, y, t, z(x)) + D[4](w)(x, y, t, z(x))*diff(z(x), x) 

You may use e.g.
alias(z=z(x));
so that the argument x is omitted, but you must be careful (I personally do not use this facility in such situations).

You have a small mistake. Use:

Note that here the origin is at the minimal point of the catenary.

You have a linear ordinary differential equation of order 1. The solution contains an integral. This integral, in most cases, cannot be expressed in terms of elementary functions.
Note that in your example, if e.g. theta is a negative integer, the solution is elementary.

# You must provide correct assumptions
restart;
with(inttrans):
#with(gfun);
#assume(alpha + beta = 1):
f := a*t^(-alpha) + b*t^(-beta):
inttrans[laplace](f, t, s) assuming alpha<1,beta<1;

This cannot be simplified, even if alpha+beta=1

P:=proc(pmax::posint)
   local p,q,u,v,n,sn, U, R, nR:=0;
   for q to pmax-1 do for p from q+1 to pmax do
     n:=p!/q!; sn:=isqrt(n);
     U:=NumberTheory:-Divisors(n);
     for u in U while u<sn do
       v:=n/u;
       if issqr((v-u)/2) then  nR:=nR+1; R[nR]:=[p, q, (u+v)/2, isqrt((v-u)/2)] fi;
     od:
   od od;
  print('num'=nR);
  entries(R, nolist)
  end proc:

P(13);
                            num = 56
[4, 1, 5, 1], [5, 1, 11, 1], [7, 1, 71, 1], [8, 1, 524, 22],  [8, 1, 204, 6], [8, 1, 201, 3], [9, 1, 684, 18], 
[11, 1, 7911, 69], [10, 1, 2304, 36], [3, 2, 2, 1], [12, 1, 22716, 78], [6, 2, 19, 1], [6, 2, 21, 3], 
[7, 2, 51, 3], [7, 2, 131, 11], [5, 3, 6, 2], [6, 3, 11, 1], [10, 2, 1620, 30], [11, 2, 4905, 45], [10, 3, 916, 22], 
[11, 3, 2956, 38], [7, 3, 29, 1], [8, 3, 104, 8], [8, 4, 76, 8], [13, 3, 32444, 62], [12, 3, 8935, 5], 
[12, 3, 12060, 90], [9, 4, 209, 13], [9, 4, 284, 16], [8, 4, 41, 1], [8, 4, 44, 4], [11, 5, 601, 13], 
[12, 5, 2511, 39], [13, 5, 10449, 87], [10, 6, 71, 1], [9, 4, 124, 4], [12, 4, 4905, 45], [8, 5, 31, 5], 
[9, 5, 55, 1], [11, 6, 236, 4], [11, 6, 244, 8], [11, 7, 336, 18], [8, 7, 3, 1], [11, 6, 431, 19], 
[11, 6, 2316, 48], [11, 6, 249, 9], [11, 6, 276, 12], [13, 8, 659, 23], [12, 9, 61, 7], [9, 8, 5, 2], 
[12, 8, 109, 1], [11, 7, 89, 1], [13, 7, 1231, 23],  [11, 7, 191, 13], [11, 7, 96, 6], [13, 9, 131, 1]
 

Download sys1-vv.mw

H0:=x->piecewise(x<=0,0,x<1/2,1,x<1,-1,0):
H_:=(k,m,x)-> H0(2^k*x-m):
H:=(n,x)->`if`(n=1,1, H_(ilog2(n), n-2^ilog2(n),x)):
seq(plot(H(n,x),x=0..1, color=red),n=1..7);

f:=(x^2)-6*x+p+3: g:=4*x^2-(p-8)*x+7:
minimize(f, x) = minimize(g, x)  assuming p::real;

           -6 + p = 4*(p/8 - 1)^2 - (p - 8)*(p/8 - 1) + 7

solve(%, p);
                            12, -12

It is a surprising bug. Maplesoft should fix it as soon as possible. (It probably comes due to the changes in add in Maple 2021).

A workaround is to replace add(p[], p=P)  with `+`(seq(p[], p=P));
Note that just adding compile=false (tom's suggestion)  is not enough, because removing the seq,  another error appears.

P := CartesianProduct([1,2,3,4],[1,2,3,4], compile=false);
#  seq(p[], p=P);
   add(p[], p=P);  # ==> error

The library must exist, or create a new one.

restart;
MyPackage := module() export f1, f2; local loc1; option package; 
  f1 := proc() loc1 end proc; 
  f2 := proc( v ) loc1 := v end proc; 
  loc1 := 2; 
end module:
mylib:="C:/tmp/test.mla";
LibraryTools:-Create(mylib);

#                   mylib := "C:/tmp/test.mla"

LibraryTools:-Save(MyPackage, mylib);
LibraryTools:-ShowContents(mylib); # optional
#    [["MyPackage.m", [2021, 9, 15, 12, 20, 8], 41984, 126]]

restart;
libname:=libname, "C:/tmp/test.mla":
with(MyPackage);
#                            [f1, f2]

f1();
#                               2

f2(123);
#                              123

f1();
#                              123

restart;
nat_param:=proc(ff, xx ::(name=anything), yy::(name=anything), t::name)
  local K,xt,yt, ode;
  xt:= diff(ff, op(1,xx)); 
  yt:= diff(ff, op(1,yy));
  K := 1/sqrt(xt^2 + yt^2);
  ode := {'diff'( op(1,xx),t ) = -K*yt, 'diff'( op(1,yy),t ) = K*xt};
  ode := subs( [op(1,xx)=op(1,xx)(t), op(1,yy)=op(1,yy)(t) ], ode);
  dsolve(ode union {op(1,xx)(0)=op(2,xx), op(1,yy)(0)=op(2,yy)},numeric, output=listprocedure);
  eval([lhs(xx(t)),lhs(yy(t))],%)[]
end:

f:=x1^4 + x2^4 + 0.4*sin(7*x1) + 0.3*sin(4*Pi*x2) - 1: 
xx,yy:=nat_param(f, x1=0,x2=1, t):
L:=fsolve(xx(t)=xx(0),t=8..10): N:=6:
plots:-display(seq(plot([xx,yy,k*L/N..(k+1)*L/N],color=COLOR(HUE,k/(N+2))), k=0..N-1), thickness=6);

op(1, xLp_tmp)

It's a bug (but actually the generators are correct).

restart;
with(GroupTheory):
H:=PermutationGroup({[[2, 3], [4, 5]], [[2, 5], [3, 4]]}, degree = 5, supergroup = PermutationGroup({[[2, 3, 4]], [[1, 2], [4, 5]]}, degree = 5));
#                    H := ((23)(45)(25)(34))

MinPermRepDegree(H);
#"GENS" = [[Perm([[1, 2]])], [Perm([[3, 4]])]]

#Error, (in GroupTheory:-Subgroup) not all the provided generators belong to the group

Workaround:

GroupOrder(H);
 #                              4

H1:=DirectProduct(CyclicGroup(2)$2):
AreIsomorphic(H, H1);
#                              true

MinimumPermutationRepresentationDegree(H1);
#"GENS" = [[Perm([[1, 2]])], [Perm([[3, 4]])]]

#                               4