<s>[INST] Find $\int_{1}^{2}\frac{x^2-1}{x^3\sqrt{2x^4-2x^2+1}}dx$ $$\int_{1}^{2}\frac{x^2-1}{x^3\sqrt{2x^4-2x^2+1}}dx=\frac{u}{v}$$ where $u$ and $v$ are in their lowest form. Find the value of $\dfrac{1000u}{v}$
$$\int_{1}^{2}\frac{x^2-1}{x^3\sqrt{2x^4-2x^2+1}}dx=\int_{1}^{2}\frac{x^2-1}{x^3\sqrt{2x^2(x^2-1)+1}}dx$$ I put $x^2-1=t$ but no benefit. Please guide me.
 [/INST] A couple of hints (not a full solution):
$x\to 2 \sin x$
\begin{align}
\int_{1}^{2}\frac{x^2-1}{x^3\sqrt{2x^4-2x^2+1}}\,dx&=\int_{\pi/6}^{\pi/2}\frac{\cot (x) \left(3-\cot ^2(x)\right)}{4 \sqrt{-12 \cos (2 x)+4 \cos (4 x)+9}}\,dx\\
\end{align}
now you may use  $$\cot (x) \left(3-\cot ^2(x)\right)=\Big[\frac14(24 \sin (2 x)-16 \sin (4 x)) \csc ^2(x)\Big]-\Big[(-12 \cos (2 x)+4 \cos (4 x)+9) \cot (x) \csc ^2(x)\Big]$$
</s>