{Integrate[((Log[Log[x]])/(1+(x^2))),{x,0,∞}],(π/4)(π I+Log[4(π^2)(((((Gamma[(3/4)])/(Gamma[(1/4)]))))^4)])}

Numerical integration around the singularly at x=1

Plot[Re[((Log[Log[x]])/(1+(x^2)))],{x,0,2},GridLines->{{1},None},GridLinesStyle->Directive[Red,Dashed],Exclusions->None,AxesLabel->TraditionalForm/@{a,"Re(((Log[Log[x]])/(1 + (x^2))))"}]

Plot[Im[((Log[Log[x]])/(1+(x^2)))],{x,0,2},GridLines->{{1},None},GridLinesStyle->Directive[Red,Dashed],Exclusions->None,AxesLabel->TraditionalForm/@{a,"Im(((Log[Log[x]])/(1 + (x^2))))"}]

NIntegrate[((Log[Log[x]])/(1+(x^2))),{x,0,1,∞}]

π HurwitzLerchPhi[-1,0,(1/2)]Log[π I]-π D[HurwitzLerchPhi[-1,k,(1/2)],k]/.{k->0}//N

(π/4)(π I+Log[4(π^2)(((((Gamma[(3/4)])/(Gamma[(1/4)]))))^4)])//N

Reference
https://doi.org/10.3390/math7121148