Evaluate: `int1/(4cosx-1)\ dx`

Explanation

`cosx=frac{1-tan^2 x/2}{1+tan^ x/2}`

`=int frac{1}{-1+4frac{1-tan^2 x/2}{1+tan^2 x/2}}dx`

`=int frac{1+tan^2 x/2}{-1(1+tan^2 x/2)+4(1-tan^2 x/2)}dx`

On replacing `(1 + tan^2x)/2` in numerator by `sec^2x/2`

`=int frac{sec^2 x/2}{-5tan^2 x/2+3}dx`

On putting `tan (x/2) = t` and `(1/2) sec^2(x/2) dx = 2dt`

`=1/5 int frac{1}{3/5-t^2}dt`

`=1/sqrt 15 log abs(frac{sqrt 3-sqrt 5tan x/2}{sqrt 3+sqrt 5tan x/2})+C`