Help evaluating or upper-bounding integral $\int_{-\infty}^{\infty} \frac{ab\operatorname{sinc}^2(cx)}{a+b\operatorname{sinc}^2(cx)}\;dx$ I am trying to evaluate the following definite integral:
$$\int_{-\infty}^\infty \frac{ab\operatorname{sinc}^2(cx)}{a+b\operatorname{sinc}^2(cx)}\;dx$$
where $\operatorname{sinc}(x)=\dfrac{\sin x}{x}$ is the Sinc function and $a$, $b$, and $c$ are positive constants.  I would also be happy with a reasonably tight upper bound.  Does anyone have any ideas?
 A: I an assuming $a,b,c>0$. Let $\delta=\sqrt{b/a}$. Then
$$
\int_{-\infty}^\infty \frac{a\,b\,\operatorname{sinc}^2(c\,x)}{a+b\,\operatorname{sinc}^2(c\,x)}\,dx=\frac{b}{c}\int_{-\infty}^\infty\frac{\sin^2x}{x^2+\delta^2\sin^2x}\,dx=\frac{b}{c}\,I(\delta).
$$
Integrating the inequalities
$$
\frac{\sin^2x}{x^2+\delta^2}\le\frac{\sin^2x}{x^2+\delta^2\sin^2x}\le\frac{\sin^2x}{x^2}
$$
we get
$$
\frac{1-e^{-2\delta}}{2\,\delta}\,\pi\le I(\delta)\le\pi.
$$
This bounds are good only for for small $\delta$. For Large $\delta$ try the following:
$$
\begin{align*}
I(\delta)&=2\int_0^\delta\frac{\sin^2x}{x^2+\delta^2\sin^2x}\,dx+2\int_\delta^\infty\frac{\sin^2x}{x^2+\delta^2\sin^2x}\,dx\\
&\le\frac{2\,\delta}{1+\delta^2}+2\int_\delta^\infty\frac{\sin^2x}{x^2}\,dx\\
&=\frac{2\,\delta }{\delta ^2+1}+\frac{1-\cos2\,\delta}{\delta}+{\pi -2\,\operatorname{Si}(2\,\delta )},
\end{align*}
$$
where $\operatorname{Si}$ is the sine integral. in the graph, $I(\delta)$ is in red and the upper and lower bound in blue.

