Evaluate the following integral:

re-writing the given equation as

$\int \frac{1-\frac{3}{x}+\frac{1}{x^{2}}}{x^{2}+1+\frac{1}{x^{2}}} d x$

$\int \frac{1+\frac{1}{x^{2}}}{\left(x-\frac{1}{x}\right)^{2}+3} d x-\int \frac{3 x}{x^{4}+x^{2}+1} d x$

Substituting $\mathrm{tas} \mathrm{x}-\frac{1}{\mathrm{x}}$ and $\mathrm{z}$ as $\mathrm{x}^{2}$

$\left(1+\frac{1}{\mathrm{x}^{2}}\right) \mathrm{dx}=\mathrm{dt}$ and $2 \mathrm{xdx}=\mathrm{dz}$

$\int \frac{\mathrm{dt}}{(\mathrm{t})^{2}+3}-\frac{3}{2} \int \frac{\mathrm{dz}}{\mathrm{z}^{2}+\mathrm{z}+1}$

$\int \frac{\mathrm{dt}}{(\mathrm{t})^{2}+3}-\frac{3}{2} \int \frac{\mathrm{dz}}{\left(\mathrm{z}+\frac{1}{2}\right)^{2}+\frac{3}{4}}$

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

$\frac{1}{\sqrt{3}} \arctan \left(\frac{\mathrm{t}}{\sqrt{3}}\right)-\sqrt{3} \arctan \left(\frac{2 \mathrm{z}+1}{\sqrt{3}}\right)+\mathrm{c}$

Substituting $t$ as $x-\frac{1}{x}$ and $z$ as $x^{2}$

$\frac{1}{\sqrt{3}} \arctan \left(\frac{x-\frac{1}{x}}{\sqrt{3}}\right)-\sqrt{3} \arctan \left(\frac{2 x^{2}+1}{\sqrt{3}}\right)+c$