Consider the equation (such that all variables are positive real numbers)
$$S=S_0e^{t\left(\mu-\frac{1}{\omega}\right)-\frac{t\sigma^2}{2}}-\frac{2\Lambda\left(1- e^{\frac{2t\left(\mu-\frac{1}{\omega}\right)-t\sigma^2}{2}}\right)}{\omega\left(2\left(\mu-\frac{1}{\omega}\right)-\sigma^2\right)}.$$
Is there an explicit solution for $\omega$ (possibly in Lambert form)? Otherwise, could a suitable approximation be made (since Newton's method seems tedious in this case)? Any help would be much appreciated. For context/derivation see Determining Properties of Stochastic Differential Equation ...
As stated in the comments, an approximation of the form
$$\omega=K\tanh^{-1}\left(\frac{\left(\Lambda-S\right)e^{\frac{t\sigma^{2}}{2}}}{\Lambda\ e^{\frac{t\sigma^{2}}{2}}-S_{0}e^{t\mu}}\right)$$
seems sufficient.