Thank you very much for suggesting the option,
   method = _d01amc

It did the job perfectly. I needed to compute the integral for 9 different phi functions similar to the one described, but earlier it didn't get any result after 10 minutes.  

The suggestion to write phi exact instead of with floats increased the speed of an integral where the integration curve was finite.

Thank you both very much :D

Thank you very much for suggesting the option,
   method = _d01amc

It did the job perfectly. I needed to compute the integral for 9 different phi functions similar to the one described, but earlier it didn't get any result after 10 minutes.  

The suggestion to write phi exact instead of with floats increased the speed of an integral where the integration curve was finite.

Thank you both very much :D

It is an integral like this that I would like to be able to increase digits for or compute faster. Both things are important for my problem.

where the integral is taken over the curve

for t = 0..infinity.  and the constants are given below.

Here is the code I'm currently using:

restart;
Digits:=10:
phi:=z->(z-1)^.90*(z-2)^.6*(z+1)^.8/(z*(z-2.1)^.90*(z+2)^.4);
g_line:=(t,mu,c)->mu+t+c*t*I;
mu0:=2: c0:=0.1: l:=1:
Int((phi(g_line(t,mu0,c0))/(g_line(t,mu0,c0)-mu0)*exp(-l*g_line(t,mu0,c0))
  *diff(g_line(t,mu0,c0),t)),t=0..infinity);
evalf(Int((phi(g_line(t,mu0,c0))/(g_line(t,mu0,c0)-mu0)*exp(-l*g_line(t,mu0,c0))
  *diff(g_line(t,mu0,c0),t)),t=0..infinity));