@Alfred_F Here are higher precision computations; you may play with Digits and epsilon.

restart;
N:=10^12:
Digits:=50:
exact:=evalf(ln(2026/2025)):
J1:=evalf(Int(N*x^N/(x^N+x+2024), x = 0 .. 1, method=_Dexp, epsilon=1e-20)); 
J2:=evalf(Int(t^(1/N)/(t + t^(1/N) + 2024), t = 0 .. 1, method=_Dexp, epsilon=1e-20));
err1=J1-exact;
err2=J2-exact;

      J1 := 0.00049370526798868160514093629790805923985216893049776
      J2 := 0.00049370526798868160514093619869170079706955297924341
      err1 = -4.9352240248370621056973318812232002987*10^(-16) 
      err2 = -4.9352240248380542692817597073827128422*10^(-16)   

If you are using the good old Derive, be careful with numeric computations because it has a very special implementation with fractions instead of the standard floating-point numbers.

@mmcdara The problem with such transforms is that expr and S(expr) could be totally different; just plot expr - S(expr) for  expr:=sin(6-2*sin(x)) + cos(6*sin(-3+x));

You already have a function z(t) in the system. Do you mean another function, say Z(t) = exp(int(y(u),u=0..t)) ?  Maybe u=t..0 ?

@mmcdara The correct mathematical formulation  AREA  ∈  {4, 8}  was given. The and/or expression corresponds to a  common language and it is permitted. I don't see any reason to be too pedantic.

@mmcdara The standard formula for the area of a triangle in coordinate geometry uses abs:
Area of a triangle - Wikipedia

@mmcdara  abs is necessary because we want an unoriented area (Area(ABC) = Area(ACB)).
See the examples:

A,B,C:=[0, 3*t-3], [1, t+1], [2, t+1]; # oriented area = 8 (at t=10)

A,B,C:=[0, 3*t*(1/5)-3], [1, t+1], [2, t+1]; # oriented area = -4 (at t=10)

@acer Thank you. I think that you are also waiting for smarter simplify and evala commands.

@Alfred_F  Let A,B,C be consecutive vertices and M,N the midpoints of AB, BC.
angle(BAC)=Pi/n implies AC = 2*AB*cos(Pi/n) ==> MN = cos(Pi/n) --> 1.

@Alfred_F  Most of the Kamke's problems are easily soved by Maple. See

comparison_maple_mathmatica_des_kamke.pdf

kamke_differential_equations.pdf

@Rouben Rostamian  Technically this is the Gateaux derivative (differential). Unlike the Frechet differential, it is not always linear.

@Kitonum A nice counterexample is the almost regular "tetrahedron" with sides [4,4,4,4,4,7].

@Kitonum I think I have already explained:
1. The orders are distinct. If you want the order xyzuvw, see the P.S. in my answer.
If you want exactly your order, change accordingly.
2. I have checked only the sorted sixtuples. To obtaind all the permutation, these permutations must be added, or, simply take permute instead of choose.

@Kitonum OK, but you have swiched the natural order xyzuvw to the unusual xyzwvu in your code. Just to fool me :-)
(and you were also fooled by my ordering).

@minhthien2016 I have considered only sorted sixtuples. To obtain all of them we must check the permutations of L in the for loop too. But there are too many!

P.S. Why did you detete the extra question in the meantime?

@Kitonum 

min(x+y+z,x+v+w,y+w+u,z+u+v) > max(x+u,y+v,z+w)

(x=DA, y=DB, z=DC, u=BC, v=CA, w=AB in the tetrahedron ABCD).