OK, I think I might have it, but it involved Laurent expansions...and I'm not very good at these kind of things.
So the integrand has a singularity about 0, but is holomorphic on any punctured disc about zero. Therefore we can expand about z=0 and look for the coefficient of the term.
So reading of the terms,
I can see now why only odd powers of t contribute but I can't see the quick way of finding all the coefficients of these powers, can anyone see how? Or possibly even a better way of approaching the problem?