$$ \int_{1}^{\infty} \frac{\sin^2 (\mu \sqrt{x^2 -1})}{(x+1)^{\frac{9}{2}} (x-1)^{\frac{3}{2}}} \,dx $$ Note: $\mu$ here is an extremely small constant.

I have tried:

- Estimating the integral by Taylor expansion of $\sin^2(\mu \sqrt(x^2 - 1)$ but the it diverges after few terms.
- I have also tried to split the integral into two regions, solving the two integrals differently and then adding them in the hope that the arbitrary parameter $\alpha$ will vanish (asymptotic matching/splitting). While I have been able to solve the two integrals but $\alpha$ does not vanish: $$\int_{1}^{\alpha} f(x)\,dx + \int_{\alpha}^{\infty} f(x)\,dx$$
Approximations in for the two regions:

$1$ to $\alpha$ region: $x \tilde = 1$ and hence $(x+1) \tilde= 2$

$\alpha$ to $\infty$ region: $(x^2 -1) \tilde= x^2$