@janhardo It's of course a typo, should be b.

expr := I*( ln(-z*I + b) - ln(z*I + b) ):
expr = combine(expr, symbolic):
simplify(eval(%, [z=-1, b=-1]));

        -(3*Pi)/2 = Pi/2

@janhardo You should use the symbolic option only in the rare cases when you don't care about the branch of the log.
Check here e.g. z=-1, a=-1.

@Carl Love Thank you Carl for this. For me it was a surprise that the presence of the lexical variable `a` is not visible in the body of the procedure F:

showstat(F);

F := proc(x)
   1   (3.5)*x*(1-x)
end proc

The only suspect element being (3.5) instead of 3.5.
The presence of `a` is revealed only in dismantle(eval(F)); and in op(7,eval(F));

BTW, I would have defined  f := unapply(F(x), x);

@Kitonum 
Vol(19, 17, 16, 8, 10, 12) = 210
Vol(8, 10, 12, 19, 17, 16) = 21
# using Vol(DA,DB,DC,BC,CA,AB)

@Kitonum Good idea to consider only "distinct" tetrahedra. 
Strangely, you missed a tetrahedron, Vol(19, 17, 16, 8, 10, 12) = 210.

@mmcdara I agree that the arithmetic progression version is more interesting (otherwise there are too  many solutions).
(I did not notice that the edges were in progression.)
It seems that indeed 6..11 is the only solution (provided that the gcd of the terms is 1, of course, otherwise take any multiple).

@minhthien2016  I have used Maple 2021 syntax.
Replace:
nr++       with    nr := nr+1

break 6  with    error "Stopped at nrmax"

@acer I wonder why 0[expr] simplifies to 0 but 1[expr] remains indexed. Is it documented?

You said: I know how to solve it using either FindMaximum or Langrage multiplier if I assign numerical values to T1, T2, R, Rthhot, Rthcold, and Z.

Please upload a worksheet with your numerical example.

@tomleslie Actually the book is not a bad one. It has a different purpose. 
From the  preface:

It is appropriate to stress here that this book is NOT for learning how to use Maple, but rather for learning how to write and construct computer programs. We expect that the general programming principles learned in this book will greatly help students in their other programming courses (e.g., in C, Pascal, FORTRAN) particularly in their ability to analyze and write complex algorithms.

@mmcdara The "directional expansion" works if f is in the class C^n (near [0,0,...,0]) and the result is a polynomial.
Otherwise, the expansion fails or is non-polynomial. mtaylor does basically the same but also checks the polynomiality.

For example,

f:=(x,y)->sin(x^2*y/(x^2+y^2)):
normal(eval(series(f(t*x,t*y),t=0),t=1)); # non-polynomial

because f is not differentiable at [0,0], (although continuous, provided that f(0,0)=0).

@nm Just differentiate (1) wirh respect to t and then take t=1. Using Maple:

f(t*x, t^p*y) = t^(p-1)*f(x,y):
eval(diff(%, t), t=1); p = solve(%, p);

See a nicer solution here.

@acer My answer contains a simplified explanation based on:
-- the information in the help ?simplify/size
-- the output generated by the simplify commands using printlevel:=40;
 For example, we see here:

--> enter \`simplify/size/size/object\`, args = _z1^2*_z2^2+_z1^2*_z3^2

--> enter \`simplify/size/size/object\`, args = _z1^2*(_z2^2+_z3^2)

@mmcdara In newer versions, the size option is by default in simplify.