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