The function Integrate fails to integrate correctly over a definite domain a certain continuous and differentiable rational function of trigonometric expression (see attached Mathematica 9 notebook with outputs and the comments therein).The corresponding numerical integration via NIntegrate as well as the plot of the function is used as a test. Could this be a bug or do I have a problem in my understanding? Fortunately, I have found a partial workaround which is also indicated in the file. Breaking the integral into two parts and a certain substitution seems to work. I am extremely curious on getting some insights on what is going wrong and how I can avoid this problem.

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