integration - Is it true that $frac{1}{pi^{2n+1}} int_0^{theta} ln^{2n}left(frac{sin x}{sinleft(theta-xright)}right),dx$ is a rational...

I was trying to evaluate $\displaystyle \int_0^{\frac{\pi}{6}}\ln^2\left(2\sin x\right)\,dx$ in an elementary way (no complex variable) so i have considered:

$\displaystyle \int_0^{\frac{\pi}{6}} \ln^2\left(\frac{\sin x}{\sin\left(\frac{\pi}{6}-x\right)}\right)\,dx$.

Using lindep a function in PARI GP i have conjectured that this integral is equal to a rational times $\pi^3$*.
Then i have considered:

$\displaystyle \frac{1}{\pi^5}\int_0^{\frac{\pi}{6}} \ln^4\left(\frac{\sin x}{\sin\left(\frac{\pi}{6}-x\right)}\right)\,dx,\frac{1}{\pi^7} \int_0^{\frac{\pi}{6}} \ln^6\left(\frac{\sin x}{\sin\left(\frac{\pi}{6}-x\right)}\right)\,dx$ and it seems that these integrals are rational numbers.

then i have considered:

$\displaystyle \frac{1}{\pi^5}\int_0^{\frac{\pi}{7}} \ln^4\left(\frac{\sin x}{\sin\left(\frac{\pi}{7}-x\right)}\right)\,dx,\frac{1}{\pi^7} \int_0^{\frac{\pi}{7}} \ln^6\left(\frac{\sin x}{\sin\left(\frac{\pi}{7}-x\right)}\right)\,dx$

same things happen.

Then i have considered:

$\displaystyle \frac{1}{\pi^3}\int_0^{\sqrt{2}} \ln^2\left(\frac{\sin x}{\sin\left(\sqrt{2}-x\right)}\right)\,dx$.

and lindep doesn't show that this number is rational. (it's not a proof).

i have tested much more values ($\frac{\pi}{7}+\frac{1}{10000}$ for example)

My question:

is it true that:

$0< \theta <\pi$, a real

for all $n$, natural integer

$\displaystyle \frac{1}{\pi^{2n+1}} \int_0^{\theta} \ln^{2n}\left(\frac{\sin x}{\sin\left(\theta-x\right)}\right)\,dx$ is a rational

if only if $\theta=r\pi$, $0< r<1$ a rational.

*: i think i have a proof for this.

PS:

The idea of this came after reading: Evaluation of $\int_0^{\pi/3} \ln^2\left(\frac{\sin x }{\sin (x+\pi/3)}\right)\,\mathrm{d}x$