I need to integrate complex, oscillatory function that is only known at equidistant grid points. Simpson's not accurate enough, what are my choices? I have a complex function and I need to integrate its real part (which oscillates highly).
I only know the function values at equidistant grid points in $[a, b]$. 
I have many such points, enough that I would normally expect a highly accurate result from trapezoid or simpson's rule ... but both schemes are only accurate up to 4th or 5th digit. This is accurate, but not accurate enough for my purposes.
What can I do to improve the accuracy? Is there some other method I can try, or adapt the above ones? 
Note again: I only know the function values at equidistant grid points. So I can't use Gaussian quadrature, nor composite Simpson's rule.
I am willing to trade computational time for more accuracy. Is there anything I can do?

Newton-Cotes integration should be your friend.
