How to calculate the improper integral $\int_{0}^{\infty}\Big(\frac{x}{\mathrm{e}^x-\mathrm{e}^{-x}}-\frac{1}{2}\Big)\frac{\mathrm{d}x}{x^2}$
Evaluate $\int_{0}^{+\infty }{\left( \frac{x}{{{\text{e}}^{x}}-{{\text{e}}^{-x}}}-\frac{1}{2} \right)\frac{1}{{{x}^{2}}}\text{d}x}$