$\int \frac{Cosx}{Cos3x}dx=$

$\cos 3x=4{\cos }^{3}x-3\cos x=$

$=\cos x\left(4{\cos }^{3}x-3\right)$

$\int \frac{\cos x}{\cos 3x}dx=\int \frac{1}{4{\cos }^{2}x-3}dx$

$=\int \frac{{\sec }^{2}x}{4-3\left(1+{\tan }^{2}x\right)}dx$

$=\int \frac{{\sec }^{2}x}{1-3{\tan }^{2}x}dx$

$=\frac{1}{2\sqrt{3}}\log \left|\frac{1+\sqrt{3}\tan x}{1-\sqrt{3}\tan x}\right|+c$