Why does Mathematica give the wrong answer when integrating?

This bug appears to have been known for over a year now! https://groups.google.com/forum/?fromgroups#!searchin/comp.soft-sys.math.mathematica/integrate%2420bessel%2420function%2420bug%7Csort%3adate/comp.soft-sys.math.mathematica/ofJh_oRFAB4/TFcbpQsKw3MJ Edit To explain why I consider these bugs to be equivalent, I'll re-write the two integrals using the integration variable x. In the present question, we have $$I_1(c)=\int_0^{2\pi}\exp(i\cos(x-c))\cos x\,dx$$ On the other hand, the MathGroup post I linked here considers the integral $$I_2(c)=\int_{-\pi}^{\pi}\exp(i\cos(x-c))\exp(ix-ic)\,dx$$ where I specialized to the case giving the same Bessel argument in the result. Pulling out the constant factor, using the periodicity of the integrand and making a substitution of variables, one finds $$I_2(c)=e^{-ic}\left[I_1(c)+i\, I_1(c-\frac{\pi}{2})\right]$$ Therefore, if Wolfram fixed the bug in the calculation of $I_2$ but still gets an incorrect result for $I_1$, that fix was itself incorrect. This argument generalizes to integer Bessel indices other than 1 (which is what this question has). Finally, the question is whether it is a bug fix to return the integrals unevaluated. I would argue that because these are such ubiquitous integrals, Mathematica should definitely know what to do with them. But I may not have all the information as to why this bug is so hard to fix... I just wanted to briefly clarify my thinking in case anyone cares.