I would like to know whether there is a closed-form solution in $x$ for the equation:
\begin{align*} &e^{-(\alpha\delta+(\gamma-\phi)\frac{\alpha}{\alpha - 1})x} -\frac{\alpha(r+\delta+\phi-\gamma)}{r+\alpha\delta}e^{-(r+\alpha\delta)x} - \frac{r+\alpha(\gamma-\phi-r)}{r+\alpha\delta} = 0,& \end{align*} with following requirements for parameter values: \begin{align*} 0<\alpha<1\\ 0<\delta<<1\\ -1<<\gamma<0\\ 0<\phi<<1\\ 0<r<<1\\ r>(\gamma+\phi)\frac{\alpha}{\alpha-1}\\ and\quad x>0, \end{align*} where "<<" means much greater (alternatively, one may assume that $\delta < \alpha, |\gamma|<\alpha,\phi<\alpha$ and $r<\alpha$). Here, "closed-form solution" can be understood in very broad sense, meaning that all kind of special functions are allowed (to be precise, analytic expression for the solution would be enough). Thank you for your time.