Edit: The previous answer was wrong because it began with the incorrect series expansion.
The correct trigonometric series is the following:
$$\frac{1}{(p^2-1)}+\frac{p^2+1}{(p^2-1)}\sum_{k=1}^{\infty}\frac{\cos{kx}}{p^k}=\frac{p\cos{x}}{p^2-2p\cos{x}+1};~where~p>1,$$
and
$$\frac{1}{p^2-1}+\frac{2}{p^2-1}\sum_{k=1}^{\infty}\frac{\cos{kx}}{p^k}=\frac{1}{p^2-2p\cos{x}+1};~where~p>1.$$
Subtracting the two gives,
$$\sum_{k=1}^{\infty}\frac{\cos{kx}}{p^k}=\frac{p\cos{x}-1}{p^2-2p\cos{x}+1};~where~p>1.$$
With $p=e^{2x}$,
$$\begin{align}
\int_{0}^{\infty}\mathrm{d}x\,\frac{x^2(e^{2x}\cos{x}-1)}{e^{4x}-2e^{2x}\cos{x}+1}
&=\int_{0}^{\infty}\mathrm{d}x\,\sum_{k=1}^{\infty}x^2e^{-2kx}\cos{kx}\\
&=\sum_{k=1}^{\infty}\int_{0}^{\infty}\mathrm{d}x\,x^2e^{-2kx}\cos{kx}\\
&=\sum_{k=1}^{\infty}\left[\frac{\partial^2}{\partial k^2}\int_{0}^{\infty}\mathrm{d}x\,\frac14e^{-2kx}\cos{nx}\right]_{n=k}\\
&=\sum_{k=1}^{\infty}\left[\frac{\partial^2}{\partial k^2}\frac14\cdot\frac{2k}{4k^2+n^2}\right]_{n=k}\\
&=\sum_{k=1}^{\infty}\left[\frac{4k(4k^2-3n^2)}{(4k^2+n^2)^3}\right]_{n=k}\\
&=\sum_{k=1}^{\infty}\frac{4}{125k^3}\\
&=\frac{4}{125}\zeta{(3)}.
\end{align}$$
Thanks to Tunk-Fey for pointing out the erroneous value I had beforehand.
