Evaluate: `int1/(cos3x-cosx)\ dx`

Explanation

`int frac{1}{cos 3 x-cos x} d x `

`=int frac{1}{4 cos ^{3} x-4 cos x} d x[because cos 3 x=4 cos ^{3} x-3 cos x] `

`=int frac{1}{4 cos x(cos ^{2} x-1)} d x `

`=frac{-1}{4} int frac{1}{cos x sin ^{2} x} d x `

`=frac{-1}{4} int(frac{sin ^{2} x+cos ^{2} x}{cos x sin ^{2} x} d x) `

`=frac{-1}{4}[int sec x d x+int cot x cosec x d x] `

`=frac{-1}{4}(ln |sec x+tan x|-cosec x)+C `

`=frac{1}{4}cosec x-ln |sec x+tan x|)+C`