calculus and analysis - Bug in mathematica analytic integration?

I found Mathematica provides me a wrong answer for a relatively simple analytically solvable integral:

Integrate[Exp[-I θ]/(1 + b Cos[θ]), {θ, 0, 2 π}, 
     Assumptions -> b < 1 && b > -1]

Provides as a result (2π)/b, which is incorrect. The integral can be computed easily writing the integrand as a geometric series.

Additionally, you can compute it numerically

NIntegrate[Exp[-I θ]/(1 + b Cos[θ]), {θ, 0, 2 π}]

which produces the right result, different from (2π)/b.

Is this a very bad bug? or am I doing something wrong? What could be the reason for this error?

Comments

plotting - Plot 4D data with color as 4th dimension

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