The following occurred to me while working on some research project, although it was not required, it seems interesting enough to post it for this audience. Anyhow, here it is. Let 
$$f(x)=\left(\frac{x}{e^x-1}\right)^2 + \left(\frac{x+1}{e^{x+1}+1}\right)^2.$$
Prove that $f(x)$ is a strictly decreasing function of $x$ over $\mathbb{R}$.

In fact, the statement holds true if $e$ is replaced by any real number $t>1$.