How could the following integration of a 6-dimensional Gaussian be achieved? Are there some techniques? (MMA code below)
$$\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }a \cdot e^{-\frac{1}{2} \left(z_1^2+z_2^2+z_3^2+z_4^2+z_5^2+z_6^2\right)}dx_6dx_5dx_4dx_3dx_2dx_1$$ with
$a =\sqrt{(z_1 z_5 - z_2 z_4)^2 + (z_3 z_4 - z_1 z_6)^2 + (z_2 z_6 - z_3 z_5)^2} \\ z1 = p1 + x1 \\ z2 = p2 + x2 \\ z3 = p3 + x3 \\ z4 = p4 + x4 \\ z5 = p5 + x5 \\ z6 = p6 + x6 $
where $p_1$,...,$p_6$ are free variables.
MMA code:
z1 = p1 + x1;
z2 = p2 + x2;
z3 = p3 + x3;
z4 = p4 + x4;
z5 = p5 + x5;
z6 = p6 + x6;
a = Sqrt[(z1*z5 - z2*z4)^2 + (z3*z4 - z1*z6)^2 + (z2*z6 - z3*z5)^2];
Integrate[a*E^((-1/2)*(z1^2 + z2^2 + z3^2 + z4^2 + z5^2 + z6^2)), {x1, -Infinity, Infinity}, {x2, -Infinity, Infinity}, {x3, -Infinity, Infinity},
{x4, -Infinity, Infinity}, {x5, -Infinity, Infinity}, {x6, -Infinity, Infinity}]
