calculus - Does the integral $int_0^{infty} {1 over {1+x^2 cdot sin^2 x}}$converge?

Does the integral $\int_0^{\infty} {1 \over {1+x^2 \cdot \sin^2 x}}$ converge?

What I tried:

• substituting $x \cdot \sin x$ with $t$ and transforming into arctan of something.


• tried to find a multiplication of useful functions in order to do integration by parts.


• I thought about trying to change the integral into a sum if integrals from $k \pi$ to $(k+1) \pi$ but that didn't really give me anything new (except that it produced a series that converges which says nothing for the sum). also, none of the convergence tests that I know gave any conclusive result.