In[]:=
prec=2000;Quiet[Print["integrating without parts took ",(Timing[(cc=NIntegrate[(Exp[Log[t]/t-Pit/I]),{t,1,InfinityI},WorkingPrecisionprec,Method"Trapezoidal",MaxRecursionFloor[Log[prec]/Log[2]]]-I/Pi)])[[1]],"sec. ","check with 200K ",N[ReMBK200k-Re[cc],20]]];Table[Quiet[Print["integrating by parts for ",4n+1," iteration(s) took ",Timing[Block[{d,h,g,$MaxExtraPrecision=200},f[x_]=x^(1/x);d=n*4+1;h[n_]:=Sum[StirlingS1[n,k]*Sum[(-j)^(k-j)*Binomial[k,j],{j,0,k}],{k,1,n}];h[0]=1;g=2I/Pi-Sum[-I^(n+1)h[n]/Pi^(n+1),{n,1,d}];expplus1:=NIntegrate[Simplify[Exp[IPi*x]*D[f[x],{x,d+1}]],{x,1,InfinityI},WorkingPrecisionprec,PrecisionGoalprec,Method"Trapezoidal",MaxRecursionFloor[Log[prec]/Log[2]]];c[n_]=(-g-(1/Pi)^(d+1)*expplus1)]][[1]]," sec.","check with 200K ",N[ReMBK200k-Re[c[n]],20]]],{n,0,1}];Print[]
integrating without parts took 17.5938sec. check with 200K 0.×
-2001
10
integrating by parts for 1 iteration(s) took 25.1094 sec.check with 200K 0.×
-2002
10
integrating by parts for 5 iteration(s) took 35. sec.check with 200K 0.×
-2002
10