In[]:=
f[b_?NumericQ,h_?NumericQ]:=NIntegrate[Sinh[x]/((Cosh[x]+Cosh[b])*(h^2-x^2)),{x,0,h,Infinity},Method->"PrincipalValue"];g[b_?NumericQ,h_?NumericQ]:=NReSum
m
(-1)
-((b+h)(1+m))

(Gamma[0,-h(1+m)]-
2h(1+m)

Gamma[0,h+hm])
2h
,{m,0,100}+ReSum-
2m
(-1)
2m
h
(-πPolyLog[-1-2m,-
b

]+2Log[h]PolyLog[-1-2m,-
b

]-2PolyGamma[0,2+2m]PolyLog[-1-2m,-
b

]-2
(1,0)
PolyLog
[-1-2m,-
b

])
2(1+2m)!
,{m,0,100},20;
In[]:=
{f[2,2],g[2,2]}
Out[]=
{-0.411183,-0.41118284681331442230+0.×
-21
10
}
In[]:=
Plot[Evaluate@Table[f[b,h],{b,0,3}],{h,0,5},PlotLabels{"b=0","b=1","b=2","b=3"},PlotRange->{Automatic,{-3,0}},ImageSize->600,AxesLabel{h,"Integral[h]"}]//Quiet
Out[]=