Evaluate the following integral:

re-writing the given equation as

$\int \frac{1+\frac{1}{x^{2}}}{x^{2}+7+\frac{1}{x^{2}}} d x$

$\int \frac{1+\frac{1}{x^{2}}}{\left(x-\frac{1}{x}\right)^{2}+9} d x$

Assume $t=x-\frac{1}{x}$

$\mathrm{dt}=\left(1+\frac{1}{\mathrm{x}^{2}}\right) \mathrm{d} \mathrm{x}$

$\int \frac{d t}{(t)^{2}+9}$

Using identity $\int \frac{1}{x^{2}+1} d x=\arctan (x)$

$\frac{1}{3} \arctan \left(\frac{t}{3}\right)+c$

Substituting $\mathrm{t}$ as $\mathrm{x}-\frac{1}{\mathrm{x}}$

$\frac{1}{3} \arctan \left(\frac{\mathrm{x}-\frac{1}{\mathrm{x}}}{3}\right)+\mathrm{c}$