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I think the following articles can give a clue: http://www.tandfonline.com/doi/abs/10.1080/10652460701688125?journalCode=gitr20 (Closed-form evaluation of some families of cotangent and cosecant integrals, by D. Cvijović) and http://www.sciencedirect.com/science/article/pii/S0377042702003588 (Integral representations of the Riemann zeta function for odd-integer arguments, by D. Cvijović and J. Klinowski).