How to perform the following integration $\int \frac{\cos 5x+5\cos 3x+10\cos x}{\cos 6x+ 6\cos 4x+ 15\cos 2x +10}dx$ $$\int \frac{\cos 5x+5\cos 3x+10\cos x}{\cos 6x+ 6\cos 4x+ 15\cos 2x +10}dx$$
How to simplify the expression given? I tried using formulas for $\cos 2x$ and $\cos 3x$. I also tried the using $\cos x + \cos y = 2cos(\frac{x+y}{2})\cos(\frac{x+y}{2})$. 
The options for answers are given are in form of half angles of trigonometric functions.

If you write $$\cos w = \frac{1}{2}\left(e^{iw}+e^{-iw}\right),$$ you find the numerator is $$\frac{1}{2}\left(e^{ix}+e^{-ix}\right)^5 = 16\cos^5(x)$$
The denominator is, likewise:
$$\frac{1}{2}\left(e^{ix}+e^{-ix}\right)^6=32\cos^6(x)$$
So the integrand is $\frac{1}{2\cos x}$.
This all comes because $1,5,10,10,5,1$ is the fifth row of Pascal's triangle, and $1,6,15,20,15,6,1$ is the sixth row.
