This
t\geq \frac{1}{e^x-1},\quad x>0
is equivalent to
1\geq e^{-x}+\frac{1}{1+t}
which can be written almost verbatim
1 >= exp(-x) + inv_pos(1+t)
However your variant with x^2 seems much harder. Maybe you can reformulate the whole model somehow to avoid the square.