Definite Integral of Hyperbolic Tangent and a Rational function expressed as the PolyGamma Function. ​​​
T(a,α)=
∞
∫
0
x(3
2
x
-
2
a
)
3
(
2
x
+
2
a
)
Tanh[xα]x=
α
2
2
π
a
2π
(1)
ψ
aα
π
+
1
2
+aα
(2)
ψ
aα
π
+
1
2
​​where Re(a)>0, Re(α)>0.​​
In[]:=
hypetan[α_,a_]:=NIntegrate
x(3
2
x
-
2
a
)Tanh[xα]
3
(
2
x
+
2
a
)
,{x,0,∞},
α2πPolyGamma1,
1
2
+
αa
π
+αaPolyGamma2,
1
2
+
αa
π

2
2
π
a
//N
In[]:=
hypetan[1+,1+]
Out[]=
{-0.102994-0.289361,-0.102994-0.289361}
Some special cases of T(a,α) in terms of Catalan and Zeta
1. when α=π/4a
2. when α=π/6a
In[]:=
Plot3D
x(
2
a
-3
2
x
)Tanh
πx
4a

3
(
2
a
+
2
x
)
,{a,0,1},{x,0,1/10},ColorFunctionFunction[{x,y,z},Hue[z]]
Out[]=
In[]:=
hypetan1[a_]:=NIntegrate
x(
2
a
-3
2
x
)Tanh
πx
4a

3
(
2
a
+
2
x
)
,{x,0,∞},-
2
3
π
+8(-8Catalan+
2
π
)-56Zeta[3]
32
2
a
//N
In[]:=
hypetan1[1-]
Out[]=
{-4.07063×
-13
10
-0.234881,0.-0.234881}
In[]:=
Plot3D
x(
2
a
-3
2
x
)Tanh
πx
6a

3
(
2
a
+
2
x
)
,{a,0,2},{x,0,1/10},ColorFunctionFunction[{x,y,z},Hue[z]]
Out[]=
In[]:=
hypetan2[a_]:=NIntegrate
x(
2
a
-3
2
x
)Tanh
πx
6a

3
(
2
a
+
2
x
)
,{x,0,∞},-
4
3
π
3
3
+12PolyGamma1,
2
3
-26Zeta[3]
72
2
a
//N
In[]:=
hypetan2[1+]
Out[]=
{-8.92397×
-11
10
+0.204039,0.+0.204039}