Given $ 0 < {\tau_{1}} < 1$, ${\alpha} > 1$ and ${\varepsilon} > 0$, is it possible to find the ${\tau_{2}}$ that maximizes
$ R = \dfrac{\tau_{1}-\tau_{2}}{\theta^{\alpha}}-\dfrac{\alpha}{\alpha-1}\theta^{1-\alpha}\left(\tau_{1}(1-\tau_{1})^{\varepsilon}-\tau_{2}(1-\tau_{2})^{\varepsilon}\right) $,
where $\theta=\dfrac{(\varepsilon+1)(\tau_{1}-\tau_{2})}{(1-\tau_{2})^{\varepsilon+1}-(1-\tau_{1})^{\varepsilon+1}}$?
I've tried Wolfram Alpha and ChatGPT but they both give very complicated expressions for $R'(\tau_{2})$ that I have not been able to solve for ${\tau_{2}}$.
If it's not possible to find the maximum in terms of ${\tau_{2}}$, is it possible to show that $R'(\tau_{2}) < 0$ given $ {\tau_{2}} < {\tau_{1}}$?
This is for a research paper in economics where $R$ is tax revenue and the $\tau$'s are tax rates.