@John2020 The method in my answer can be adapted here too.

restart;
r := x*(diff(theta(t), t))^2+y*(diff(varphi(t), t))^2+z*(diff(theta(t), t$2))+w*diff(varphi(t), t$2)+p*m:
g := (4*(f+T))*(diff(theta(t), t))^2+u*(diff(varphi(t), t))^2+(f+9)*(diff(theta(t), t$2))+4*s*diff(varphi(t), t$2)+p*4*cos(varphi(t)):

eval(r-g, [theta=(t->1+t+t^3), varphi=(t->1+t+t^7)]): # or similar
[coeffs(convert(series(%, t), polynom), t)]:
solve(%, [x,y,z,w,p]):
solve(%[1], [x,y,z,w,p]);
#      [[x = 4 T + 4 f, y = u, z = f + 9, w = 4 s, p = 0]]
simplify(eval(r-g, %[1]));   #check
#                               0

Note that you cannot take m as parameter; it must be p (it appears in both r and g); for a fixed p there is no solution.

@ecterrab  

eq:=sin(x)+cos(x) = a*sin(d*x^2+c*x+b):
solve( identity(eq, x), {a,b,c,d} );  # works
PDEtools:-Solve(eq, {a,b,c,d}, independentof=x); # must be interrupted

@Kitonum Yes, in my search I encountered the solve bug mentioned here

Sys:={x*y*z + y*z + y = -1, x*y*z + x*z + z = 0, x*y*z + x*y + x = 0}:
solve(Sys, [x,y,z]);

       [[x = 0, y = -1, z = 0]]

Anyway, the system with a,b,c has generically 3 solutions. It is impossible for the system to have exactly 3 integer solutions.

@nm More likely to be generic or defined in the code read at the beginning.

BTW, the algorithm is implemented in modern Maple, see  ?DEtools,kovacicsols

@janhardo No, it does not. Maple ignores the properties of ithprime(i), e.g. the fact that  ithprime(i) ~ i*ln(i)  for i --> oo.

@janhardo You should write

@jud (a)   int(F,...) works after a simplify and gives 4 - Pi  because you have a square minus a circle, not a circle.
Unfortunately, changing to the correct "f<0" the simplify is not enough.

(b) MultiInt works only for explicit functions, just like int; you can use of course piecewise or Heaviside as above (but the resulting function will be discontinuous, so, more difficult for Maple to handle).

@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?