Weird issue.

[mathlover2]

The code

(-I*Integral(2*I*pi*exp(6*I*pi*t)/(exp(2*I*pi*t) - 1/2), (t, 0, 1))/(2*pi)).doit()

should, according to Cauchy's integral formula, yield 1/4. Numerical calculation respects this answer, but symbolic calculation returns 0.

EDIT 1

It seems that when we take the first half of the range values of t,
l1 = (-I*Integral(2*I*pi*exp(6*I*pi*t)/(exp(2*I*pi*t) - 1/2), (t, 0, 1/2))/(2*pi))
we have that this portion of the curve is correct. If we take the second half,
l2 = (-I*Integral(2*I*pi*exp(6*I*pi*t)/(exp(2*I*pi*t) - 1/2), (t, 0, 1/2))/(2*pi)),
we have that the real part is negated;

for i in (l2.doit().n(),l2.n(),l1.doit().n(),l1.n()):
    print(i)

prints the following:

-0.0625 - 0.163824569905166*I
0.0625 - 0.163824569905166*I
0.0625 + 0.163824569905166*I
0.0625 + 0.163824569905166*I

EDIT 2

This seems to occur regardless of the algorithm used, with the exception of manual.