In[]:=
ClearAll[u,MKB1];​​​​prec=1000;​​​​Print[Timing[MKB1=-IQuiet[NIntegrate[Exp[Log[1+Iu]/(1+Iu)-Piu],{u,0,Infinity},WorkingPrecision->prec,Method->"DoubleExponential",MaxRecursion->Floor[Log2[prec]]]]-I/Pi][[1]]," ",N[MKB-MKB1,20]]
1.20313 0.×
-1001
10
+0.×
-1001
10

In[]:=
ClearAll[u,MKB1];​​​​prec=2000;​​​​Print[Timing[MKB1=-IQuiet[NIntegrate[Exp[Log[1+Iu]/(1+Iu)-Piu],{u,0,Infinity},WorkingPrecision->prec,Method->"Trapezoidal",MaxRecursion->Floor[Log2[prec]]]]-I/Pi][[1]]," ",N[MKB-MKB1,20]]
7.5 0.×
-2001
10
+0.×
-2001
10

In[]:=
ClearAll[t,MKB1];​​​​prec=3000;​​​​Print[Timing[MKB1=-IQuiet[NIntegrate[Exp[Log[1+It/(1-t)]/(1+It/(1-t))]/(Exp[Pit/(1-t)](1-t)^2),{t,0,1},WorkingPrecision->prec,Method->"DoubleExponential",MaxRecursion->Floor[Log[prec]/Log[2]]]]-I/Pi][[1]]," ",N[MKB-MKB1,20]]
31.6094 0.×
-3001
10
+0.×
-3001
10

In[]:=
ClearAll[t,MKB1];​​​​prec=4000;​​​​Print[Timing[MKB1=-IQuiet[NIntegrate[Exp[Log[1+It/(1-t)]/(1+It/(1-t))]/(Exp[Pit/(1-t)](1-t)^2),{t,0,1},WorkingPrecision->prec,Method->"DoubleExponential",MaxRecursion->Floor[Log[prec]/Log[2]]]]-I/Pi][[1]]," ",N[MKB-MKB1,20]]
57.5625 0.×
-4001
10
+0.×
-4001
10

ClearAll[u,MKB1];​​​​prec=4000;​​​​Print[Timing[MKB1=-IQuiet[NIntegrate[Exp[Log[1+Iu]/(1+Iu)-Piu],{u,0,Infinity},WorkingPrecision->prec,Method->"Trapezoidal",MaxRecursion->Floor[Log2[prec]]]]-I/Pi][[1]]," ",N[MKB-MKB1,20]]​​​​
44.8125 0.×
-4001
10
+0.×
-4001
10

​
In[]:=
ClearAll[u,MKB1];​​​​prec=5000;​​​​Print[Timing[MKB1=-IQuiet[NIntegrate[Exp[Log[1+Iu]/(1+Iu)-Piu],{u,0,Infinity},WorkingPrecision->prec,Method->"Trapezoidal",MaxRecursion->Floor[Log2[prec]]]]-I/Pi][[1]]," ",N[MKB-MKB1,20]]​​​​
165.219 0.×
-5001
10
+0.×
-5001
10

In[]:=
ClearAll[u,MKB1];​​​​prec=10000;​​​​Print[Timing[MKB1=-IQuiet[NIntegrate[Exp[Log[1+Iu]/(1+Iu)-Piu],{u,0,Infinity},WorkingPrecision->prec,Method->"Trapezoidal",MaxRecursion->Floor[Log2[prec]]]]-I/Pi][[1]]," ",N[MKB-MKB1,20]]​​​​
1189.5 0.×
-10001
10
+0.×
-10001
10

​​​​
In[]:=
ClearAll[t,MKB1];​​​​prec=20000;​​​​Print[Timing[MKB1=-IQuiet[NIntegrate[Exp[Log[1+It/(1-t)]/(1+It/(1-t))]/(Exp[Pit/(1-t)](1-t)^2),{t,0,1},WorkingPrecision->prec,Method->"DoubleExponential",MaxRecursion->Floor[Log[prec]/Log[2]]]]-I/Pi][[1]]," ",N[MKB-MKB1,20]]
11009.3 0.×
-20001
10
+0.×
-20001
10

In[]:=
prec=40000;Quiet[(Print[AbsoluteTiming[(cc=NIntegrate[(Exp[Log[t]/t-Pit/I]),{t,1,InfinityI},WorkingPrecision->prec,Method->"Trapezoidal",MaxRecursion->Floor[Log[prec]/Log[2]]]-I/Pi)][[1]]])]
92237.1
In[]:=
N[MKB-cc,20]
Out[]=
0.×
-40001
10
+0.×
-40001
10

In[]:=
DateString[]
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