$\int \frac{x^2-1}{x^3\sqrt{2x^4-2x^2+1}}dx=$

We have, $I=\int \frac{x^2-1}{x^3\sqrt{2x^4-2x^2+1}}dx$

$\Rightarrow I=\int \frac{x^2-1}{x^5\sqrt{2-2x^{-2}+x^{-4}}}dx$

$\Rightarrow I=\int \frac{x^{-3}-x^{-5}}{\sqrt{2-2x^{-2}+x^{-4}}}dx$

Put, $\sqrt{2-2x^{-2}+x^{-4}}=t\Rightarrow 2-2x^{-2}+x^{-4}=t^2\Rightarrow \left(4x^{-3}-4x^{-5}\right)dx=2tdt$

$\Rightarrow I=\frac{1}{2}\int \frac{tdt}{t}$

$\Rightarrow I=\frac{t}{2}+c$, where $c$ is the constant of integration.

$\Rightarrow I=\frac{1}{2}\sqrt{2-\frac{2}{x^2}+\frac{1}{x^4}}+c$