Laplace transform of $f(g(t))$

Question

I was wondering if there is some easy way or useful set of rules which can be applied to find the Laplace transform of $f(g(t))$. For example, is there a trivial way to find the Laplace transform of say: $$\phi(t)=atan[\frac{0.4cos(t)+0.5cos(t+\pi/4)}{2cos(t-\pi/4)+0.5cos(t+\pi/3)}]$$ where the $f$ function is the arctangent and the $g(t)$ function is the expression inside the arctangent?