Mathematica refuses to simplify a huge expression

Question

today I tried to simplify the following expression:

expr = Sqrt[(Cosh[ν] - Cos[u])/(a Sinh[ν])] /. Thread[{ν, u, ϕ} -> 
    CoordinateTransformData[{"Cartesian" -> {"Toroidal", a}}, "Mapping", {x, y, z}]]

Even FullSimplify doesn't give it a try:

FullSimplify[expr, Assumptions -> {a > 0, Element[{x, y, z}, Reals]}]

after a few seconds returns the original expression. However, this is simply equal to (x^2+y^2)^(-1/4), which can be seen by either manipulating toroidal coordinates, or trying

(expr/(x^2 + y^2)^(-1/4)) /. {a -> 1, x -> 1, y -> 2, z -> 3} // N
1.

(expr/(x^2 + y^2)^(-1/4)) /. {a -> -5, x -> 10, y -> -2, z -> 0.4} // N
1.

What am I doing wrong? How come MMA thinks that the long clumsy expression is simpler than (x^2+y^2)^(-1/4)? Notice that it doesn't even depend on z, however, if you don't specify z, MMA acts like it's not a definite number

(expr/(x^2 + y^2)^(-1/4)) /. {a -> -5, x -> 10, y -> -2} // N
1.4221 Sqrt[-1. (-((1. (77.25 + z^2))/Sqrt[
     100. z^2 + (77.25 + z^2)^2]) + 
    Cosh[0.5 Log[(26.1313 + z^2)/(228.369 + z^2)]]) Csch[
   0.5 Log[(26.1313 + z^2)/(228.369 + z^2)]]]

What's wrong? Thank you.

Answer 1

Mma applies rules it has but nothing reduces the complexity function of the expression it calculates. Perhaps too many variables. If to assign one of the variables $z$ or $a$ a concrete value, say $a=1$, it does simplify. Another way to reduce the number of variables here is putting $r=\sqrt{x^2+y^2}$:

FullSimplify[expr /. y^2 -> r^2 - x^2, Assumptions ->{a>0,r>0,Element[{x, z}, Reals]}]

$ \frac{1}{\sqrt{r}} $