How to solve this Laplace transform? $f(t)=e^{-2t}\cos^2 3t - 3t^2 e^{3t}$

Question

Find the laplace transform of

$$f(t)=e^{-2t}\cos^2 3t - 3t^2 e^{3t}$$

The answer is $$\frac{1}{2(s+2)}+ \frac{1}{2} \frac{s+2}{s^2 + 4s + 40} - \frac{6}{(s-3)^3}.$$

This took me about an hour to solve, which seems ridiculously long. I probably did things inefficiently, how can this be solved?

My method: Separate using linearity and then integrate by parts, with partial fraction decomposition.

Answer 1

Remember that

$$\cos^2(3t) = \frac{1+\cos(6t)}{2}$$

and

$$\mathcal{L}(t^n f(t)) = (-1)^n \frac{d^n}{ds^n} F(s).$$