In[]:=
HoldFormIntegrate[Sinh[x]/((Cosh[x]+Cosh[b])*(h^2-x^2)),{x,0,Infinity}]==-Sum
m
(-1)
-b(1+m)

π
MeijerG
1
2
,{},
1
2
,
1
2
,{0},
1
4
2
h
2
(1+m)

2h
,{m,0,Infinity}+Sum-
2m
h
Log[h]PolyLog[-1-2m,-
b

]
Gamma[2+2m]
+
2m
h
PolyGamma[0,2+2m]PolyLog[-1-2m,-
b

]
Gamma[2+2m]
+
2m
h
(1,0)
PolyLog
[-1-2m,-
b

]
Gamma[2+2m]
,{m,0,Infinity}==
Log[h]Sech
b-h
2
Sech
b+h
2
Sinh[h]
4h
-
(1,0)
PolyLog
[0,-
b-h

]-
(1,0)
PolyLog
[0,-
b+h

]
2h
+Sum
2m
h
PolyGamma[0,2+2m]PolyLog[-1-2m,-
b

]
Gamma[2+2m]
,{m,0,Infinity}-Sum
m
(-1)
-b(1+m)

π
MeijerG
1
2
,{},
1
2
,
1
2
,{0},
1
4
2
h
2
(1+m)

2h
,{m,0,Infinity}==
Log[h]Sech
b-h
2
Sech
b+h
2
Sinh[h]
4h
-
(1,0)
PolyLog
[0,-
b-h

]-
(1,0)
PolyLog
[0,-
b+h

]
2h
+Sum
2m
h
PolyGamma[0,2+2m]PolyLog[-1-2m,-
b

]
Gamma[2+2m]
,{m,0,Infinity}+Sum
m
(-1)
-b(1+m)

CoshIntegral[h(1+m)]Sinh[h+hm]
h
-
m
(-1)
-b(1+m)

Cosh[h+hm]SinhIntegral[h(1+m)]
h
,{m,0,Infinity}//TraditionalForm
Out[]//TraditionalForm=
∞
∫
0
sinh(x)
(cosh(x)+cosh(b))(
2
h
-
2
x
)
x-
∞
∑
m=0
m
(-1)
-b(1+m)

π
2,1
G
1,3
1
4
2
h
2
(1+m)
1
2
1
2
,
1
2
,0
2h
+
∞
∑
m=0
-
2m
h
log(h)
Li
-1-2m
(-
b

)
Γ(2+2m)
+
2m
h
(0)
ψ
(2+2m)
Li
-1-2m
(-
b

)
Γ(2+2m)
+
2m
h
(1,0)
PolyLog
(-1-2m,-
b

)
Γ(2+2m)

log(h)sech
b-h
2
sech
b+h
2
sinh(h)
4h
-
(1,0)
PolyLog
(0,-
b-h

)-
(1,0)
PolyLog
(0,-
b+h

)
2h
+
∞
∑
m=0
2m
h
(0)
ψ
(2+2m)
Li
-1-2m
(-
b

)
Γ(2+2m)
-
∞
∑
m=0
m
(-1)
-b(1+m)

π
2,1
G
1,3
1
4
2
h
2
(1+m)
1
2
1
2
,
1
2
,0
2h

log(h)sech
b-h
2
sech
b+h
2
sinh(h)
4h
-
(1,0)
PolyLog
(0,-
b-h

)-
(1,0)
PolyLog
(0,-
b+h

)
2h
+
∞
∑
m=0
2m
h
(0)
ψ
(2+2m)
Li
-1-2m
(-
b

)
Γ(2+2m)
+
∞
∑
m=0
m
(-1)
-b(1+m)

Chi(h(1+m))sinh(h+hm)
h
-
m
(-1)
-b(1+m)

cosh(h+hm)Shi(h(1+m))
h
In[]:=
f11[b_,h_]:=N
Log[h]Sech
b-h
2
Sech
b+h
2
Sinh[h]
4h
-
(1,0)
PolyLog
[0,-
b-h

]-
(1,0)
PolyLog
[0,-
b+h

]
2h
+Sum
2m
h
PolyGamma[0,2+2m]PolyLog[-1-2m,-
b

]
Gamma[2+2m]
,{m,0,100}-Sum
m
(-1)
-b(1+m)

π
MeijerG
1
2
,{},
1
2
,
1
2
,{0},
1
4
2
h
2
(1+m)

2h
,{m,0,100},20//Chop;f12[b_,h_]:=N
Log[h]Sech
b-h
2
Sech
b+h
2
Sinh[h]
4h
-
(1,0)
PolyLog
[0,-
b-h

]-
(1,0)
PolyLog
[0,-
b+h

]
2h
+Sum
2m
h
PolyGamma[0,2+2m]PolyLog[-1-2m,-
b

]
Gamma[2+2m]
,{m,0,100}+Sum
m
(-1)
-b(1+m)

CoshIntegral[h(1+m)]Sinh[h+hm]
h
-
m
(-1)
-b(1+m)

Cosh[h+hm]SinhIntegral[h(1+m)]
h
,{m,0,100},20//Chop;
In[]:=
{f11[2,2],f12[2,2]}
Out[]=
{-0.41118284681331442230,-0.41118284681331442230}