what is integration of 1/(a^2-2*a*cos(x)+1) its lower limit is 0 and upper limit is pie

I= dx/[a² -2acosx +1]       lim 0 to pi

put cosx = 1-tan²x/2/1+tan²x/2

I= sec²x/2 dx / [(a-1)² + (a+1)²tan²x/2]

now put tanx/2 = t

    sec²x/2dx =2dt

I = 2dt / [(a-1)²+(a+1)²t²)           

I =2dt/(1+a)² .[(a-1/a+1)² + t²]

this integral is same as  1/a²+x² dx & its integral is tan^-1(x/a)/a

so

 I =2/(a²-1)tan^-1[(a+1)t/(a-1)]     

 I = 2/(a²-1)tan^-1[(a+1)tanx/2/(a-1)]               lim 0 tp pi

taking limit

 I =2/(a²-1) [ tan^-1infinity -tan^-10]

    =pi/(a²-1)         ans