Definite Logarithmic Integrals Expressed in term of the Log Gamma function with plots

References
https://www.mdpi.com/580662

p11[k_,a_,α_,n_,m_]:={NIntegrate[(((y^m)(((Log[a y]))^k))/(1+α (y^n))),{y,0,∞}],N[((Exp[π I(((m+1)/n))]((((I/n)))^(k-1))(((2π))^(k+1))(α^(-((m+1)/n))))/(n^2))HurwitzLerchPhi[Exp[2π I(((m+1)/n))],-k,((π-I n Log[a]+I Log[α])/(2π))]]}

p11closed[k_,a_,α_,n_,m_]:={Integrate[(((y^m)(((Log[a y]))^k))/(1+α (y^n))),{y,0,∞}],((Exp[π I(((m+1)/n))]((((I/n)))^(k-1))(((2π))^(k+1))(α^(-((m+1)/n))))/(n^2))HurwitzLerchPhi[Exp[2π I(((m+1)/n))],-k,((π-I n Log[a]+I Log[α])/(2π))]}

p11closed[k,a,α,n,m]

p11[1,1,2,2,1/2]

Convert the Lerch function to Hurwitz zeta

{(((y^m)(((Log[a y]))^k))/(1+α (y^n))),((Exp[π I(((m+1)/n))]((((I/n)))^(k-1))(((2π))^(k+1))(α^(-((m+1)/n))))/(n^2))HurwitzLerchPhi[Exp[2π I(((m+1)/n))],-k,((π-I n Log[a]+I Log[α])/(2π))]}/.{n->2+2 m}//FullSimplify

{(((y^m) ((Log[a y])^k))/(1+(y^(2+2 m)) α)),((((((I/(1+m))))^k) (π^(1+k)) HurwitzLerchPhi[(E^(I  π)),-k,((π-I (2+2 m) Log[a]+I Log[α])/(2 π))])/((1+m) (Sqrt[α])))}

Convert the Hurwitz zeta to Log Gamma

D[{(((y^m) ((Log[a y])^k))/(1+(y^(2+2 m)) α)),((((((I/(1+m))))^k) (π^(1+k)) (-(2^k) HurwitzZeta[-k,(1/2) (1+((π-I (2+2 m) Log[a]+I Log[α])/(2 π)))]+(2^k) HurwitzZeta[-k,((π-I (2+2 m) Log[a]+I Log[α])/(4 π))]))/((1+m) (Sqrt[α])))},k]/.{k->0}//FullSimplify

p12[a_,α_,m_]:={NIntegrate[(((y^m) Log[Log[a y]])/(1+(y^(2+2 m)) α)),{y,0,1,∞}],N[((π (Log[((2 I π)/(1+m))]-2 Log[-3 π-2 I (1+m) Log[a]+I Log[α]]+2 Log[-π-2 I (1+m) Log[a]+I Log[α]]+ Log[(((((Gamma[-((π+2 I (1+m) Log[a]-I Log[α])/(4 π))])/(Gamma[-1+((π-2 I (1+m) Log[a]+I Log[α])/(4 π))]))))^2)]))/(2 (1+m) (Sqrt[α])))]}

p12closed[a_,α_,m_]:={Integrate[(((y^m) Log[Log[a y]])/(1+(y^(2+2 m)) α)),{y,0,∞}],((π (Log[((2 I π)/(1+m))]-2 Log[-3 π-2 I (1+m) Log[a]+I Log[α]]+2 Log[-π-2 I (1+m) Log[a]+I Log[α]]+ Log[(((((Gamma[-((π+2 I (1+m) Log[a]-I Log[α])/(4 π))])/(Gamma[-1+((π-2 I (1+m) Log[a]+I Log[α])/(4 π))]))))^2)]))/(2 (1+m) (Sqrt[α])))}

p12closed[a,α,m]

Evaluate integrals in terms of Log Gamma with plots

p12[1,1,1]

p12closed[1,1,1]//FullSimplify

Plot[Re[((y Log[Log[y]])/(1+(y^4)))],{y,0,2},GridLines->{{1},None},GridLinesStyle->Directive[Red,Dashed],Exclusions->None,AxesLabel->TraditionalForm/@{a,"Re(\!\(\*FractionBox[\(y\\ Log[Log[y]]\), \(1 + \*SuperscriptBox[\(y\), \(4\)]\)]\))"}]

Plot[Im[((y Log[Log[y]])/(1+(y^4)))],{y,0,2},GridLines->{{1},None},GridLinesStyle->Directive[Red,Dashed],Exclusions->None,AxesLabel->TraditionalForm/@{a,"Im(\!\(\*FractionBox[\(y\\ Log[Log[y]]\), \(1 + \*SuperscriptBox[\(y\), \(4\)]\)]\))"}]

p12[1,1,1/2]

p12closed[1,1,1/2]//FullSimplify

Plot[Re[(((Sqrt[y]) Log[Log[y]])/(1+(y^3)))],{y,0,2},GridLines->{{1},None},GridLinesStyle->Directive[Red,Dashed],Exclusions->None,AxesLabel->TraditionalForm/@{a,"Re(\!\(\*FractionBox[\(\*SqrtBox[\(y\)]\\ Log[Log[y]]\), \(1 + \*SuperscriptBox[\(y\), \(3\)]\)]\))"}]

Plot[Im[(((Sqrt[y]) Log[Log[y]])/(1+(y^3)))],{y,0,2},GridLines->{{1},None},GridLinesStyle->Directive[Red,Dashed],Exclusions->None,AxesLabel->TraditionalForm/@{a,"Im(\!\(\*FractionBox[\(\*SqrtBox[\(y\)]\\ Log[Log[y]]\), \(1 + \*SuperscriptBox[\(y\), \(3\)]\)]\))"}]

p12[1,1,-1/2]

p12closed[1,1,-1/2]//FullSimplify

Plot[Re[((Log[Log[y]])/((Sqrt[y]) (1+y)))],{y,0,2},GridLines->{{1},None},GridLinesStyle->Directive[Red,Dashed],Exclusions->None,AxesLabel->TraditionalForm/@{a,"Re(\!\(\*FractionBox[\(Log[Log[y]]\), \(\*SqrtBox[\(y\)]\\ \((1 + y)\)\)]\))"}]

Plot[Im[((Log[Log[y]])/((Sqrt[y]) (1+y)))],{y,0,2},GridLines->{{1},None},GridLinesStyle->Directive[Red,Dashed],Exclusions->None,AxesLabel->TraditionalForm/@{a,"Im(\!\(\*FractionBox[\(Log[Log[y]]\), \(\*SqrtBox[\(y\)]\\ \((1 + y)\)\)]\))"}]

p12[1,1,2]

p12closed[1,1,2]//FullSimplify

Plot[Re[(((y^2) Log[Log[y]])/(1+(y^6)))],{y,0,2},GridLines->{{1},None},GridLinesStyle->Directive[Red,Dashed],Exclusions->None,AxesLabel->TraditionalForm/@{a,"Re(\!\(\*FractionBox[\(\*SuperscriptBox[\(y\), \(2\)]\\ Log[Log[y]]\), \(1 + \*SuperscriptBox[\(y\), \(6\)]\)]\))"}]

Plot[Im[(((y^2) Log[Log[y]])/(1+(y^6)))],{y,0,2},GridLines->{{1},None},GridLinesStyle->Directive[Red,Dashed],Exclusions->None,AxesLabel->TraditionalForm/@{a,"Im(\!\(\*FractionBox[\(\*SuperscriptBox[\(y\), \(2\)]\\ Log[Log[y]]\), \(1 + \*SuperscriptBox[\(y\), \(6\)]\)]\))"}]

p12[1,2,0]

p12closed[1,2,0]//FullSimplify

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

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

p12[1,1/2,0]

p12closed[1,1/2,0]//FullSimplify

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

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