A Curious Family of Integrals that Always Give \(\pi\)!

I wonder if it is possible to characterize \(\frac{P(x)}{Q(x)}\) such that by inspection we can say: this is \(n\pi\). Travis has commented that the denominator has a special Galois group. Does anybody know who can answer this question at MathSE:
https://math.stackexchange.com/questions/4736889/a-curious-family-of-integrals-that-give-pi?noredirect=1#comment10046166_4736889

#math #calculus #integral #rationalfunctions #pi