Alternate, superior solution to an integral

[GCT]

Hi all, consider the integral
$$I= \int \frac{dx}{cos^2 x( \sqrt{ cos^2 x - cos^2 a} ) }$$
I've solved it with three substitutions (don't have time to post it at the moment) but was wondering if anyone here can provide a superior, more simplistic solution; e.g. one substitution with proper algebraic simplification.

 

[GCT]

the first substitution was $$u=cos \theta$$ the second was $$u=sec w$$, and final substitution was $$z = sin w$$, had to use some trig identities in between and focus in on the simplifications. My current solution is (I'll still have yet to review it close up for dumb mistakes)

$$I= \frac{i}{cos a} [ \frac{( \frac{2(cos^2 x -1)}{cosx}) ^2 ln( \frac{2(cos^2 x -1)}{cosx} ) + ( \frac{2(cos^2 x -1)}{cosx} ) ^4 -1 }{2 ( \frac{2(cos^2 x -1)}{cosx} ) ^2 } ~] + C$$

I'm having some trouble with latex

I = (i/cosa)[2{(cos^2 x -1)/(cosx)}^2 ln {(cos^2 x -1)/(cosx)} + {(cos^2 x -1)/(cosx))^2}^4 - 1]/[2 {(cos^2 x -1)/(cosx)}^2] + C