There is an old worksheet at the Application Center https://www.maplesoft.com/applications/view.aspx?SID=6322

It is clear enough and includes an animation.

Download extrema.mw

restart;
eq1:= f(x) + 2*f(1/x) + 3*f(x/(x-1)) = x:
eq:=eq1, simplify(eval(eq1, x=1/x)):
eq:=eq, simplify(eval(eq1, x=x/(x-1))):
eq:=eq, simplify(eval(eq1, x=-1/(x-1))):
eq:=eq, simplify(eval(eq1, x=1-x)):
eq:=eq, simplify(eval(eq1, x=(x-1)/x)):
solve([eq], indets([eq],function)): f(x) = eval(f(x), %);
simplify( eval(eq1,  f = unapply(rhs(%), x)) );   # check

So, you want the surface integral on the projection of a cuboid on a plane.

V:=<0,0,0>, <1,2,0>, <0,2,0>, <1,0,0>, <0,0,1>, <1,2,1>, <0,2,1>, <1,0,1>:
T:=<-1,0,0>, <0,-1,0>, <0,0,-1>: 
PM:=LinearAlgebra:-ProjectionMatrix([T[2]-T[1], T[3]-T[1]]):
Pr := v -> T[1]+PM.v:
Pr2:= v -> (PM.v)[[1,2]]:  # ConvexHull works in dimension 2.
P:=Pr~([V]):
P2:=Pr2~([V]):
ind:=ComputationalGeometry:-ConvexHull(Matrix(P2)^+):
C:=P[ind]: n:=nops(C):  # the vertices of the projection
Tri:=(v1,v2,v3) -> Surface( v1 + t*(v2+s*(v3-v2)-v1), s=0..1, t=0..1):
# the parametrization of the interior of a triangle
add(VectorCalculus:-SurfaceInt( x^2+y^2, [x,y,z] = Tri(C[1],C[i],C[i+1])), i=2..n-1);

Maple solved your equation with respect to z.
Note that LambertW is a function just like sin or exp.

Just increase Digits (exp(-200) is very small), e.g.

Digits := 20;

1. A faster method is to use Iterator.

with(Iterator):
C:=Combination(5,3):
K:=[1,2,3,4,a]:
seq( K[[seq(c[]+1)]], c=C);

        [1, 2, 3], [1, 2, 4], [1, 3, 4], [2, 3, 4], [1, 2, a], [1, 3, a], [2, 3, a], [1, 4, a], [2, 4, a], [3, 4, a]

2. To print the body of a procedure, use

interface(verboseproc=2);
print(combinat:-subsets);

or, 

showstat(combinat:-subsets);

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]