Evaluate $\displaystyle \int \frac{1}{\left(\mathrm{e}^{x}+\mathrm{e}^{-x}\right)\left(\mathrm{e}^{x}-\mathrm{e}^{-x}\right)} \mathrm{d} x$ \[\log \sqrt{2} \]

Let $\displaystyle \mathrm{I}=\int_{\log \sqrt{2}}^{\log \sqrt{3}} \frac{1}{\left(\mathbf{e}^{\mathbf{x}}+\mathbf{e}^{-\mathbf{x}}\right)\left(\mathbf{e}^{\mathbf{x}}-\mathbf{e}^{-\mathbf{x}}\right)} \mathbf{d x}=\int_{\log \sqrt{2}}^{\log \sqrt{3}} \frac{\mathbf{e}^{2 \mathbf{x}}}{\left(\mathbf{e}^{2 \mathbf{x}}\right)^{2}-\mathbf{1}} \mathbf{d x}$ Put $\displaystyle \mathrm{e}^{2 x}=t \Rightarrow \mathrm{e}^{2 x} \mathrm{d} x=\frac{1}{2} \mathrm{d} t$, Upper limit $\displaystyle =3$, Lower Limit $\displaystyle =2$\left.\therefore \mathrm{I}=\frac{1}{2} \int_{2}^{3} \frac{1}{t^{2}-1} \mathrm{d} t=\frac{1}{4} \log \left|\frac{t-1}{t+1}\right|\right]_{2}^{3} $$