By the divergence theorem,
[tex]\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\iiint_{[0,\pi/2]^3}\mathrm{div}\vec F\cdot\mathrm dV[/tex]
We have
[tex]\mathrm{div}\vec F(x,y,z)=\cos x\cos^2y+3\sin^2y\cos y\cos^4z+5\sin^4z\cos z\cos^6x[/tex]
Then the volume integral is
[tex]\displaystyle\int_0^{\pi/2}\int_0^{\pi/2}\int_0^{\pi/2}\mathrm{div}\vec F\,\mathrm dx\,\mathrm dy\,\mathrm dz=\boxed{\frac{19\pi^2}{64}}[/tex]
