On a type of sequence of integrals inspired in the Borwein integral and an integral due to Furdui

This post is inspired in the Borwein integral, and in a problem proposed by Ovidiu Furdui in Crux Mathematicorum, that is the Problem 3707, in page 151.

I've considered integrals of the form $$\int_0^\infty\prod_{k=1}^n\frac{\sin^2\left(\frac{x}{k}\right)}{\pi^2-\left(\frac{x}{k}\right)^2}dx\tag{1}$$ for integers $$n\geq 1$$. My belief is that $$(1)$$ isn't exactly Ovidiu's integral.

Question. Can you show that the case $$n=3$$ the integral in $$(1)$$ is $$\neq 0$$? Is it possible to find $$n>2$$ such that the integral in $$(1)$$ is equals to $$0$$ again? Many thanks.

Please if the integral in $$(1)$$ is Ovidiu's integral, or it is in the literatute, add a comment. I think that isn't the same integral, using Wolfram Alpha online calculator an alternate form of the difference of integrands, for the second case is showed as using the input

sin^2(x)/(pi^2-x^2)sin^2(x/2)/(pi^2-(x/2)^2)- sin^4(x)/((pi^2-x^2)(4pi^2-x^2))

References:

 Borwein integral, from the encyclopedia Wikipedia.

 Problem 3707, Crux Mathematicorum, Volume 38, Number 4, April 2012.

• The web page of Crux Mathematicorum (I believe that is a journal of the Canadian Mathematical Society) is cms.math.ca/crux from which you can see its Digital Archive. Sep 3 '19 at 10:00

$$I_3=\int_0^\infty \frac{ \sin ^2(x)\sin ^2\left(\frac{x}{2}\right)\sin ^2\left(\frac{x}{3}\right)}{\left(\pi ^2-x^2\right) \left(\pi ^2-\frac{x^2}{4}\right) \left(\pi ^2-\frac{x^2}{9}\right)}\,dx=-\frac{27 \sqrt{3}}{320 \pi ^4}=-0.00150029\cdots.$$
and $$I_4=-\frac{9 \left(17 \sqrt{3}+10\right)}{2240 \pi ^6}$$.
• What quick with this answer! Many thanks, I known just the approximation $\approx -0.00150029$ Sep 3 '19 at 11:05