Integral of determinant

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, $$I(a,b,c) = \begin{cases} 0 & \text{ if $a=0$ or $b=0$ or $c=0$ or $a^2 = b^2$ or $b^2=c^2$ or $c^2 = a^2$}\\ 6 \pi^3 & \text{ if $a,b,c \in \mathbb{Z} \backslash \{0\}$ and are distinct}\\ ? & \text{else}\end{cases}$$

Consider the vector $x = [x_1,...,x_N].$ Let $A(x)$ and $B(x)$ be two $N\times N$ matrices such that $(A)_{ij} = f_i(x_j)$ and $(B)_{ij} = g_i(x_j)$ where $f_i(z)$ and $g_i(z)$ are any functions, $i=1,\cdots ,N;$

Then $\int_{[a,b]^N} det(A)det(B) dx = N! det(C)$

where $(C)_{ij} = \int_{[a,b]} f_i(z)g_j(z) dz$

In your case $A=B, a=-\pi, b=\pi, f_i(z) = g_i(z) = \sin(\alpha_i z)$

Since $\alpha_i$ are integers, the matrix $C$ is diagonal with diagonal elements equal to $\int_{-\pi}^{\pi} \sin^2(\alpha_i z) dz = \pi$

Therefore the integral you are looking for gives $N! \pi^N$ For $N=3$ you have $6\pi^3$

If for some $i$ and $j$, $|\alpha_i|=|\alpha_j|$ the matrices $A$ and $B$ are rank deficient and therefore their determinant is $0$. It follows that the integral is $0$

This can be generalized for any $N$ and for real coefficients $\alpha_i$

Exact solution and Mathematica code to produce it (< 10 seconds computing time):

M = {{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]}};
Expand[Det[M]^2] /. Plus -> List;
Total[Integrate[%, {x, -Pi, Pi}, {y, -Pi, Pi}, {z, -Pi, Pi}]]

It looks terrible but for a computer no problem: $-\frac{8 \left(\pi -\frac{\sin (2 c \pi )}{2 c}\right) (b \cos (b \pi ) \sin (a \pi )-a \cos (a \pi ) \sin (b \pi ))^2}{\left(a^2-b^2\right)^2}-\frac{8 (2 c \pi -\sin (2 c \pi )) (b \cos (b \pi ) \sin (a \pi )-a \cos (a \pi ) \sin (b \pi ))^2}{\left(a^2-b^2\right)^2 c}+\frac{16 (a \cos (a \pi ) \sin (b \pi )-b \cos (b \pi ) \sin (a \pi )) (2 c \cos (c \pi ) \sin (a \pi )-2 a \cos (a \pi ) \sin (c \pi )) (b \cos (b \pi ) \sin (c \pi )-c \cos (c \pi ) \sin (b \pi ))}{\left(a^2-b^2\right) \left(a^2-c^2\right) \left(b^2-c^2\right)}+\frac{16 (2 b \cos (b \pi ) \sin (a \pi )-2 a \cos (a \pi ) \sin (b \pi )) (a \cos (a \pi ) \sin (c \pi )-c \cos (c \pi ) \sin (a \pi )) (c \cos (c \pi ) \sin (b \pi )-b \cos (b \pi ) \sin (c \pi ))}{\left(a^2-b^2\right) \left(a^2-c^2\right) \left(c^2-b^2\right)}+\frac{16 (a \cos (a \pi ) \sin (b \pi )-b \cos (b \pi ) \sin (a \pi )) (a \cos (a \pi ) \sin (c \pi )-c \cos (c \pi ) \sin (a \pi )) (2 c \cos (c \pi ) \sin (b \pi )-2 b \cos (b \pi ) \sin (c \pi ))}{\left(a^2-b^2\right) \left(a^2-c^2\right) \left(b^2-c^2\right)}-\frac{(\sin (2 a \pi )-2 a \pi ) (2 b \pi -\sin (2 b \pi )) \left(\pi -\frac{\sin (2 c \pi )}{2 c}\right)}{2 a b}-\frac{(\sin (2 a \pi )-2 a \pi ) \left(\pi -\frac{\sin (2 b \pi )}{2 b}\right) (2 c \pi -\sin (2 c \pi ))}{2 a c}-\frac{\left(\pi -\frac{\sin (2 a \pi )}{2 a}\right) (\sin (2 b \pi )-2 b \pi ) (2 c \pi -\sin (2 c \pi ))}{2 b c}-\frac{8 \left(\pi -\frac{\sin (2 b \pi )}{2 b}\right) (c \cos (c \pi ) \sin (a \pi )-a \cos (a \pi ) \sin (c \pi ))^2}{\left(a^2-c^2\right)^2}-\frac{8 (2 b \pi -\sin (2 b \pi )) (c \cos (c \pi ) \sin (a \pi )-a \cos (a \pi ) \sin (c \pi ))^2}{b \left(a^2-c^2\right)^2}-\frac{8 \left(\pi -\frac{\sin (2 a \pi )}{2 a}\right) (c \cos (c \pi ) \sin (b \pi )-b \cos (b \pi ) \sin (c \pi ))^2}{\left(b^2-c^2\right)^2}-\frac{8 (2 a \pi -\sin (2 a \pi )) (c \cos (c \pi ) \sin (b \pi )-b \cos (b \pi ) \sin (c \pi ))^2}{a \left(b^2-c^2\right)^2}$

Perhaps someone can simplify it from here?

Edit: Confirmed

FullSimplify[%,
    a \[Element] Integers &&
    b \[Element] Integers &&
    c \[Element] Integers]

outputs $6\pi^3$.