v=1/(t^2+1);mi=
∞
∫
0
-v
v
csch(πt)
tv∫vt

sinv∫vt-t∫tvtt
In[]:=
Block[{v=1/(t^2+1)},AbsoluteTiming[mi=NIntegrate[Csch[πt]E^((tvIntegrate[v,t]))Sqrt[v^-v]Sin[v(Integrate[v,t]-tIntegrate[tv,t])],{t,0,∞},WorkingPrecision->700,Method->"Trapezoidal",PrecisionGoal->700,MaxRecursion->8]]][[1]]
NIntegrate
:NIntegrate failed to converge to prescribed accuracy after 8 iterated refinements in t in the region {{0,∞}}. NIntegrate obtained 707 and 712 for the integral and error estimates.
Out[]=
1.17161
In[]:=
AbsoluteTiming[ms=N[NSum[(-1)^n(n^(1/n)-1),{n,1,Infinity},Method->"AlternatingSigns",WorkingPrecision->705],700]][[1]]
Out[]=
0.0652941
In[]:=
mi-ms
Out[]=
0.×
-701
10