Fourier coefficients of $ 4\sin\left(x\right)+\cos\left(x\right)-2\sin\left(4x\right) $

Question

Define a function $ f:\mathbb{R}\to \mathbb{C} $ by:

for the segment $ [-\pi,\pi) $:

$f(x) = 4\sin\left(x\right)+\cos\left(x\right)-2\sin\left(4x\right) $

and for the rest of $ \mathbb{R} $ define $ f $ such that it would be periodic with period $ 2\pi $.

Now I want to find the fourier coefficients of $ f $.

I tried to use the formula

$ \tilde{f}\left(k\right)=\frac{1}{2\pi}\intop_{-\pi}^{\pi}\left(4\sin x+\cos x-2\sin\left(4x\right)\right)e^{-ikx}dx $

But it gets really complicated. Is there anything Im missing here ?

Thanks in advance.