In[]:=
Integrate
Log[Log[x]]
1+
2
x
,{x,0,∞},
π
4
π+Log4
2
π
4
Gamma
3
4

Gamma
1
4


Out[]=

∞
∫
0
Log[Log[x]]
1+
2
x
x,
1
4
ππ+Log
4
2
π
4
Gamma
3
4

4
Gamma
1
4


Numerical integration around the singularly at x=1
In[]:=
PlotRe
Log[Log[x]]
1+
2
x
,{x,0,2},GridLines{{1},None},GridLinesStyleDirective[Red,Dashed],ExclusionsNone,AxesLabelTraditionalForm/@a,"Re​
Log[Log[x]]
1+
2
x
​"
Out[]=
In[]:=
PlotIm
Log[Log[x]]
1+
2
x
,{x,0,2},GridLines{{1},None},GridLinesStyleDirective[Red,Dashed],ExclusionsNone,AxesLabelTraditionalForm/@a,"Im​
Log[Log[x]]
1+
2
x
​"
Out[]=
In[]:=
NIntegrate
Log[Log[x]]
1+
2
x
,{x,0,1,∞}
Out[]=
-0.520886+2.4674
In[]:=
πHurwitzLerchPhi-1,0,
1
2
Log[π]-πDHurwitzLerchPhi-1,k,
1
2
,k/.{k0}//N
Out[]=
-0.520886+2.4674
In[]:=
π
4
π+Log4
2
π
4
Gamma
3
4

Gamma
1
4

//N
Out[]=
-0.520886+2.4674
References:
​https://doi.org/10.3390/math7121148