Compute the flux of the vector field

xy\sin(z)\vec{i}+\cos(xz)\vec{j}+y\cos(z)\vec{k}

across the surface of the ellipsoid \frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{16}=1

Attempt:
I was thinking of making the following substitutions:

x=\frac{\sin(\phi)\cos(\theta)}{2}

y=\frac{\sin(\phi)\sin(\theta)}{3}

z=\frac{cos(\phi)}{4}

now calculating (r_\phi \times r_\theta) gives:

\frac{\sin^2(\phi)\cos(\theta)}{12}\vec{i}+\frac{\  sin^2(\phi)\sin(\theta)}{8}\vec{j}+\frac{\sin(\phi  )\cos(\phi)}{6}\vec{k}

now putting everything into \iint_S F\cdot(r_\phi \times r_\theta) \ dS

gives me:

\int_{\theta=0}^{\theta=2\pi}\int_{\phi=0}^{\phi=\  pi} \left(\frac{\sin(\phi)\cos(\theta)}{2}\right)\left  (\frac{\sin(\phi)\sin(\theta)}{3}\right)\sin\left(  \frac{cos(\phi)}{4}\right)\cdot\left(\frac{\sin^2(  \phi)\cos(\theta)}{12}\right) {\color{blue}+\cos\left(\left(\frac{\sin(\phi)\cos  (\theta)}{2}\right)\left(\frac{cos(\phi)}{4}\right  )\right)\cdot\left(\frac{\sin^2(\phi)\sin(\theta)}  {8}\right)} +\left(\frac{\sin(\phi)\sin(\theta)}{3}\right)\cos  \left(\frac{cos(\phi)}{4}\right)\cdot\left(\frac{\  sin(\phi)\cos(\phi)}{6}\right) \ d\phi \ d\theta

now everything except for the part in blue is integrable, where the first and third part give 0, is there a simpler way of approaching this problem?