I need to find an upper bound for $f: \mathbb{R}\setminus \{0\} \to \mathbb{R}$ where $f$ is defined by $$f(x):= \sqrt{\frac{2x+e^{-2x}-1 }{4x^2}}-\sqrt{\frac{1-4xe^{-2x}-e^{-4x} }{4x^2(1-e^{-2x})}}$$ for each $x \in \mathbb{R}\setminus \{0\}$.
My attempt
Let $x \in \mathbb{R}\setminus \{0\}$. Then we have that
\begin{align} f(x)&\leq \sqrt{ \frac{2x+e^{-2x}-1 }{4x^2} - \frac{1-4xe^{-2x}-e^{-4x} }{4x^2(1-e^{-2x})}}\\ &=\sqrt{\frac{x(1+e^{-2x})-(1-e^{-2x})}{2x^2(1-e^{-2x})}}. \end{align}
I don't know how I can continue. Can you help me, please?