I have a function which has the following form:
$$ f(x)=K_1 \coth (Q_1 Q_2 \sqrt{x})^2 + \frac{1}{x}\left[K_2 + K_3 \coth(Q_1 Q_3\sqrt{x})\sqrt{x}\right]$$
and I want to find $x$ for $f(x)=1$. I'm fairly convinced that this will not work without approximations (feel free to correct me!). I have tried two different approximations in the limit of small $Q_1$, a Taylor series and a Pade approximation, in which I only get a manageable equation (i.e. at maximum a few lines) for first order in $Q_1$ in case of the Taylor expansion and (0,1) order in case of the Pade approximant. These two approximations are unfortunately not very good so I am looking for other approximation methods that I can try.
What I would like to know is whether there are any other appropriate approximations that I could apply to this function?
Maybe useful information: $x$ has a physical meaning such that it should be positive and real