Evaluate $\int\frac{\cos(5x) + \cos(4x)}{1-2\cos(3x)}dx$

Question

I need help evaluating the indefinite integral $$\int\frac{\cos(5x) + \cos(4x)}{1-2\cos(3x)}dx.$$

Answer 1

HINT:

Set $x=2y$ and

use Prosthaphaeresis Formula and $\cos3A=4\cos^3A-3\cos A$

$$\dfrac{\cos10y+\cos8y}{1-2\cos6y}=\dfrac{2\cos9y\cos y}{1-2(2\cos^23y-1)}$$

$$=\dfrac{2\cos y(4\cos^33y-3\cos3y)}{-(4\cos^23y-3)}$$

$$=-2\cos y\cos3y=-[\cos(3y-y)+\cos(3y+y)]$$