If I haven't broken or misunderstood anything then
In[1]:= $Assumptions = {x, y, z} \[Element] Reals;
f = Simplify[1/2*Total[Total[Grad[Simplify[{x, y, z}*Tanh[Norm[{x, y, z}]/2]/Norm[{x, y, z}]], {x, y, z}]^2]]]
Out[1]= ((-1 + x^2+y^2+z^2 + Cosh[2 Sqrt[x^2+y^2+z^2]]) Sech[1/2 Sqrt[x^2+y^2+z^2]]^4)/(8 (x^2+y^2+z^2))
In[2]:= Integrate[f, {x, -Infinity, Infinity}, {y, -Infinity, Infinity}, {z, -Infinity, Infinity}]
Out[2]= $Aborted
In[3]:= Table[{10^q, NIntegrate[f, {x, -10^q, 10^q}, {y, -10^q, 10^q}, {z, -10^q, 10^q}]}, {q, 1, 4}]
Out[3]= {
{10, 131.052},
{100, 1512.39},
{1000, 15325.8},
{10000, 153482.}}
then I think your integral might be headed for infinity.
I did not wait for hours, but I was unable to reproduce any of your errors.
