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\int_0^{2\pi} cosx/(a-cosx), a-parameter (say a>0)

i've converted it into a complex contour integral over z=e^(ix):

~ \int_{|z|=1} dz (z^2+1)/[z(z^2-2az+1)]

which is easily evaluated for a>1. my question regards a<1 - i am not sure how to solve it in this case, because the the 2 poles

z_1=a+Sqrt[a^2-1], z_2=a-Sqrt[a^2-1]

are exactly on the unit circle and off the real axis. thanks for any suggestions!