Laplace transform of complicated function

Question

I have the following function for which I would like to find its Laplace transform:

$$f(x) = \sin(x)\left(1 + \frac{\cos(x)}{\sqrt{1+\sin^2(x)}} \right) \tag{1}$$

Naturally, even MATLAB cannot find the analytical Laplace transform of the above. However, is it true to say that the Laplace transform of Eq.(1) is approximated by the Laplace transform of the Fourier series for the first $N$ terms of Eq.(1)?

So in effect I would have:

$$f(x) = \frac{1}{2}a_0 + \sum^N_{n=1}a_n\cos(nx)+b_n\sin(nx)\\ \Rightarrow \mathcal{L}\{f\}(s)= \frac{1}{2s}a_0 + \sum^N_{n=1}\frac{s}{s^2+n^2}(a_n+b_n)$$

Answer 1

Yes, for $\text{Re}(s) > 0$, since this is an absolutely convergent Fourier series.