Integral of determinant Good evening. I need help with this task
$$
\int\limits_{-\pi}^\pi\int\limits_{-\pi}^\pi\int\limits_{-\pi}^\pi{\det}^2\begin{Vmatrix}\sin \alpha x&\sin \alpha y&\sin \alpha z\\\sin \beta x&\sin \beta y&\sin \beta z\\\sin \gamma x&\sin \gamma y&\sin \gamma z\end{Vmatrix} \text{d}x\,\text{d}y\,\text{d}z
$$
where $\alpha,\beta,\gamma$ are integers.
Computations are horrible, I gave up.

This is a bit too long to comment. A change in notation $(\alpha, \beta, \gamma) \to (a,b,c)$.
If $a=0$ or $b=0$ or $c=0$ or $a^2 = b^2$ or $b^2=c^2$ or $c^2 = a^2$, the integral is zero.
Let's assume $a,b,c \in \mathbb{Z} \backslash \{0\}$ and are distinct. $$
D(x,y,z,a,b,c) = {\det}^2\begin{Vmatrix}\sin a x&\sin a y&\sin a z\\\sin b x&\sin b y&\sin b z\\\sin c x&\sin c y&\sin c z\end{Vmatrix}\\ = (d_1(b,c,y,z) \sin(ax) + d_2(c,a,y,z) \sin(bx) + d_3(a,b,y,z) \sin(cx))^2
$$
Hence, $$\int_{-\pi}^{\pi}D(x,y,z,a,b,c) dx= \int_{-\pi}^{\pi} \left(d_1^2(b,c,y,z) \sin^2(ax) + d_2^2(c,a,y,z) \sin^2(bx) + d_3^2(a,b,y,z) \sin^2(cx) \right) dx$$ (The cross-terms integrals vanish by our initial assumption)
Hence, $$\int_{-\pi}^{\pi}D(x,y,z,a,b,c) dx= \left(d_1^2(b,c,y,z) + d_2^2(c,a,y,z) + d_3^2(a,b,y,z) \right) \pi$$
Lets now consider $ \displaystyle \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} d_1^2(b,c,y,z) dy dz$. The other two can be computed using symmetry arguments.
$$ \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} d_1^2(b,c,y,z) dy dz =\int_{-\pi}^{\pi} \int_{-\pi}^{\pi} \left(\sin(by) \sin(cz) - \sin(cy) \sin(bz) \right)^2 dy dz $$
Again by our assumption, the integrals of the cross-terms vanish and hence we get
$$ \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} d_1^2(b,c,y,z) dy dz =\int_{-\pi}^{\pi} \int_{-\pi}^{\pi} \left(\sin^2(by) \sin^2(cz) + \sin^2(cy) \sin^2(bz) \right) dy dz = \pi^2 + \pi^2 = 2\pi^2$$
Hence, $$\int_{-\pi}^{\pi} \int_{-\pi}^{\pi} d_1^2(b,c,y,z) dy dz = \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} d_2^2(c,a,y,z) dy dz = \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} d_3^2(a,b,y,z) dy dz = 2\pi^2$$
Hence, $$\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}D(x,y,z,a,b,c) dx dy dz= \left(2 \pi^2 + 2\pi^2 + 2\pi^2 \right) \pi = 6 \pi^3$$
To summarize, $$I(a,b,c) = \begin{cases} 0 & \text{ if $a=0$ or $b=0$ or $c=0$ or $a^2 = b^2$ or $b^2=c^2$ or $c^2 = a^2$}\\ 6 \pi^3 & \text{ if $a,b,c \in \mathbb{Z} \backslash \{0\}$ and are distinct}\\ ? & \text{else}\end{cases}$$
