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lang-mma
2*%pi/b * (2*sqrt(1-b^2)+b^2-2)/(sqrt(1-b^2)+b^2-1). When computing with +i instead of -i in the exponential, it gives instead2*%pi/b * (sqrt(1-b^2)-1)/sqrt(1-b^2), but it's actually the same. $\endgroup$Manipulateto see the dependence onbof the singularities of the integrand in the complexThetaplane:Manipulate[ContourPlot[Abs[Exp[-I Theta]/(1 + b Cos[Theta]) /. {Theta -> ThetaR + I ThetaI}], {ThetaR, -1, 2 Pi + 1}, {ThetaI, -Pi - 1, Pi + 1}, PlotRange -> {Automatic, 30}, Contours -> 50, Epilog -> {Red, Thick, Line[{{0, 0}, {2 Pi, 0}}]}], {{b, 0.5}, -1, 1}]. You need to ensure that the path of integration goes appropriately around the singularities in order to get the intended result. $\endgroup$Seriesapproach is offered only as a way of helping Mathematica to compute the answer that you seek. As for the complex roots problem, I don't have any inside knowledge about how Mathematica does its integrations or how it handles singularities that may (or may not) be important. See the answer from @artes below for more details on this, where I agree with the statement that this is all due to "imperfectness of symbolic integration". When there are singularities lurking around, you have to appeal to the underlying physics (or whatever) of the problem to decide how to handle them. $\endgroup$Limit. Will try to sort it out (usual caveat: roughly even odds it sorts me out instead). $\endgroup$