TITLE: How do you integrate $\int \frac{\cos(4x)}{\cos(x)}dx$?
QUESTION [5 upvotes]: I tried using trigonometric formulas for turning it into 2$\int \frac{\cos^2(2x)}{\cos(x)}dx - \int \frac{1}{\cos(x)}dx$ and can solve the second one, but still no idea of how to proceed with $\int \frac{\cos^2(2x)}{\cos(x)}dx$. Any suggestion?

REPLY [4 votes]: Hint:
By Werner Formulas
$$\cos(n+2)x+\cos nx=2\cos x\cos(n+1)x$$
For $\cos x\ne0,$
$$\implies\int\dfrac{\cos(n+2)x}{\cos x}\ dx=2\int\cos(n+1)x\ dx-\int\dfrac{\cos nx}{\cos x}\ dx$$
Set $n=2$ and then $n=0$
Finally use Integral of the secant function