Differentiation wrt parameter $\int_0^\infty \sin^2(x)\cdot(x^2(x^2+1))^{-1}dx$

Use differentiation with respect to parameter obtaining a differential equation to solve

$$\int_0^\infty \frac{\sin^2(x)}{x^2(x^2+1)}dx$$No complex variables, only this approach. Interesting integral and it should have a nice ODE. I have not found the right way yet. we have singularities at $x=\pm i$.