Recently I met an integral as follow: $$\int_0^{2\pi}\cdots\int_0^{2\pi}\left(\prod\limits_{1\leq i<j\leq9}\sin\frac{\theta_i-\theta_j}{2}\right)\left(\prod\limits_{i=1}^9(1+\cos(\theta_i-\theta_{i+1}))\right)\left(\prod\limits_{i=1}^9(1+\cos(\theta_i-\theta_{i+2}))\right)d\theta_1\cdots d\theta_9,$$ where $\theta_{10}=\theta_1,\theta_{11}=\theta_2$.

I want to know whether the above integral is $0$ or not. I do not know how to connect this integral with some known integral. So I ask it here for some ideas.

Well, my trick was seen through by Fedor Petrov.

Originally I wanted to prove the antisymmetrization of $$f(x_1,x_2,\cdots,x_9)=\prod\limits_{i=1}^9(x_i+x_{i+1})^2\prod\limits_{i=1}^9(x_i+x_{i+2})^2,\ \text{where the indices are module}\ 9,$$ is nonzero, just like what I asked before, see my previous question. Then I did some transformations as follow: $$\text{the antisymmetrization of}\ \prod\limits_{i=1}^9(x_i+x_{i+1})^2\prod\limits_{i=1}^9(x_i+x_{i+2})^2\ \text{is nonzero}$$ $$\Longleftrightarrow$$ $$\text{the constant term of this Laurent polynomial}\\\prod\limits_{1\leq i<j\leq 9}(\frac{1}{z_i}-\frac{1}{z_j})\prod\limits_{i=1}^9(z_i+z_{i+1})^2\prod\limits_{i=1}^9(z_i+z_{i+2})^2\ \text{is nonzero}$$ $$\Longleftrightarrow$$ $$\int_0^{2\pi}\cdots\int_0^{2\pi}\prod\limits_{1\leq i<j\leq9}(e^{-i\theta_i}-e^{-i\theta_j})\prod\limits_{i=1}^9(e^{i\theta_i}+e^{i\theta_{i+1}})^2\prod\limits_{i=1}^9(e^{i\theta_i}+e^{i\theta_{i+2}})^2d\theta_1\cdots d\theta_9\neq0$$ $$\Longleftrightarrow$$ $$\int_0^{2\pi}\cdots\int_0^{2\pi}\left(\prod\limits_{1\leq i<j\leq9}\sin\frac{\theta_i-\theta_j}{2}\right)\left(\prod\limits_{i=1}^9(1+\cos(\theta_i-\theta_{i+1}))\right)\left(\prod\limits_{i=1}^9(1+\cos(\theta_i-\theta_{i+2}))\right)d\theta_1\cdots d\theta_9\neq 0,$$ where the indices above are all module $9$. That is how my above question came from.

I thought maybe it is easy to turn into an integral, but now it seems that it is not easy to see whether the above integral is zero or not. So come back to my original question:

Is the antisymmetrization of $$f(x_1,x_2,\cdots,x_9)=\prod\limits_{i=1}^9(x_i+x_{i+1})^2\prod\limits_{i=1}^9(x_i+x_{i+2})^2$$ nonzero?