real analysis - How to calculate the improper integral $int_{0}^{infty}Big(frac{x}{mathrm{e}^x-mathrm{e}^{-x}}-frac{1}{2}Big)frac{mathrm{d}x}{x^2}$

How to calculate the improper integral

$$\int_{0}^{\infty}\Big(\frac{x}{\mathrm{e}^x-\mathrm{e}^{-x}}-\frac{1}{2}\Big)\frac{\mathrm{d}x}{x^2}$$
Do you have same idea?