Trigonometric Indefinite Integration

Question

Integrate:

$$\int\frac{\cos5x+\cos4x}{1-2\cos 3x}\; dx$$

I tried using sums and products formula but couldn't make it. How to approach this problem?

Answer 1

Hint: $$\cos x =\frac {e^{ix} + e^{-ix}}{2} $$ Or

Multiply and divide by $\sin \left (\frac {3x}{2}\right )$

Answer 2

Hint. By using De Moivre's formula $$ e^{ix}=\cos x+i \sin x $$ one may prove that $$ \frac{\cos 5x+\cos 4x}{1-2\cos 3x} =-\cos x-\cos (2x) $$ then the integral is easier to evalute.