Step-by-step Solution:

The integration can be written as: \[ \int \frac{\left(x^2 - 1\right) dx}{x^5 \sqrt{2 - \frac{2}{x^2} + \frac{1}{x^4}}} = \int \frac{\left(\frac{1}{x^3} - \frac{1}{x^5}\right) dx}{\sqrt{2 - \frac{2}{x^2} + \frac{1}{x^4}}}. \] Now, let: \[ 2 - \frac{2}{x^2} + \frac{1}{x^4} = t \quad \Rightarrow \quad \left(\frac{4}{x^3} - \frac{4}{x^5}\right) dx = dt. \] Simplifying: \[ \left(\frac{1}{x^3} - \frac{1}{x^5}\right) dx = \frac{dt}{4}. \] The integration becomes: \[ \int \frac{1}{4} \frac{dt}{\sqrt{t}} = \frac{1}{2} \sqrt{t} + C. \] Substitute back for \( t \): \[ \frac{1}{2} \sqrt{t} + C = \frac{1}{2} \sqrt{2 - \frac{2}{x^2} + \frac{1}{x^4}} + C. \]