<s>[INST] Integrate $\int \frac{x^2-1}{x^3 \sqrt{2x^4-2x^2+1}} \mathop{dx}$ Evaluate $$\int \frac{x^2-1}{x^3 \sqrt{2x^4-2x^2+1}} \mathop{dx}$$
I tried $u=\sqrt{2x^4-2x^2+1}$, $u=\dfrac{1}{x}$ and $u=\sqrt{x}$ but none of these worked.
A friend gave this and said it's from IIT JEE and answer is $$\dfrac{\sqrt{2x^4-2x^2+1}}{2x^2}+C$$
I would like a hint or suggestion.
 [/INST] Following @AdityaDwivedi's suggestion, $dt=4x^{-5}(x^2-1)dx$ so the integral is $\int\frac{dt}{4\sqrt{2+t}}=\tfrac12\sqrt{2+t}+C$. As to how you'd come up with this idea, note the original integral is $\int\frac{x^{-3}-x^{-5}}{\sqrt{2-2x^{-2}+x^{-4}}}dx$, which suggests the proposed substitution, or better still $u=2-2x^{-2}+x^{-4}$.
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