SetDirectory[NotebookDirectory[]];
<<MaTeX`
texStyle={};
SetOptions[MaTeX,"BasePreamble"->{"\\usepackage{amsmath}","\\usepackage{xcolor}","\\usepackage{fourier}","\\usepackage{ebgaramond}"},FontSize->11];

frame[legend_]:=Framed[legend,FrameStyle->Thin,RoundingRadius->10,FrameMargins->0];

αλ[λ_,μ_]:=μ/(λ+2μ);
λα[α_,μ_]:=μ(1/α-2);
CWeyl[d_,λ_, μ_]:=((λ+2 μ)^(-d/2)+(d-1) μ^(-d/2)) /(4  Pi)^(d/2)/Gamma[1+d/2];
BDirLiu[d_,λ_, μ_]:=-((d-1)/μ^((d-1)/2)+1/(λ+2μ)^((d-1)/2))/4/(4 Pi)^((d-1)/2)/Gamma[1+(d-1)/2];
BDir[d_?NumericQ,α_?NumericQ, μ_?NumericQ]:=-μ^((1-d)/2)/(2^(d+1)Pi^((d-1)/2)Gamma[(1+d)/2]) (4(d-1)/Pi NIntegrate[τ^(d-2)ArcTan[Sqrt[(1-α τ^(-2))(τ^(-2)-1)]],{τ,Sqrt[α],1}]+α^((d-1)/2)+d-1);
BDirSaVa2[α_, μ_]:=1/(4Pi Sqrt[μ])(-1-Sqrt[α]-4/Pi NIntegrate[ArcTan[Sqrt[(1-α/ζ^2)(1/ζ^2-1)]],{ζ, Sqrt[α],1}]);
BDirSaVa3[λ_, μ_]:=-1/(16Pi)(3λ^2+13λ μ +16μ^2)/(λ^2 μ+5λ μ^2 +6μ^3);
γR[α_]:=Sqrt[Min[x/.Solve[(x^3)-8 (x^2)+16 (-1+α)+8(3-2 α) x ==0,x, Reals]]];
Bfree[d_,α_, μ_]:= μ^((1-d)/2)/(2^(d+1)Pi^((d-1)/2)Gamma[(1+d)/2]) (4(d-1)/Pi NIntegrate[τ^(d-2)ArcTan[(τ^(-2)-2)^2/(4Sqrt[(1-α τ^(-2))(τ^(-2)-1)])],{τ,Sqrt[α],1}]+α^((d-1)/2)+d-5+4γR[α]^(1-d));
BfreeSaVa2[α_, μ_]:=1/(4Pi Sqrt[μ])(4/γR[α]-3+Sqrt[α]+4/Pi NIntegrate[ArcTan[(2-1/ζ^2)^2/(4Sqrt[(1-α/ζ^2)(1/ζ^2-1)])],{ζ, Sqrt[α],1}]);
BfreeSaVa3[λ_, μ_]:=1/(16Pi) (3(λ+2μ)^2-3(λ+2μ)μ+2μ^2)/((λ+2μ)μ(λ+μ));
BDirOdd[k_,α_,μ_]:=-μ^k/(2^(2k+2)Pi^k (k!)) (2/(k!)( D[(2t-1-1/α)/(t-1-1/α)/Sqrt[(1-α t)(1-t)],{t,k}]/.t->0) -2(α/(α+1))^k+α^k+2k);
BfreeOdd[k_,α_,μ_]:=μ^k/(2^(2k+2)Pi^k (k!)) (-8/(k!)( D[(t-2)(2α t^2+(α-3)t+2(1-α))/ (t^3-8t^2+8(3-2α)t+16(α-1))
(1/Sqrt[(1-α t)(1-t)]+4/(t-2)^2),{t,k}]/.t->0) -α^k+2(k+2^(2-k)-1));

tabDirodd=Table[BDirOdd[k,α, μ]//Simplify,{k,1,4}]//TableForm
tabDirodd//TeXForm

tabfreeodd=Table[Simplify[BfreeOdd[k,α, μ],α<1],{k,1,4}]//TableForm
tabfreeodd//TeXForm

xt=Table[{x, MaTeX[x]}, {x,0.2, 1, 0.2}];
yt=Table[{y, MaTeX[y]}, {y,0.5, 1, 0.1}];
xtb=Table[{x, MaTeX[x]}, {x,0.7, 1, 0.1}];
ytb=Table[{y, MaTeX[y]}, {y,0.98, 1, 0.01}];

(RefLink[Off, paclet:ref/Off])[NIntegrate::nlim]; figbLtobi=Plot[Table[BDirLiu[d,λα[α,1], 1]/BDir[d,α,1],{d,{2,3,4,5}}]//Evaluate,{α,0,1},PlotRange->{0.4,1},Ticks->{xt,yt}, AxesLabel->MaTeX[{"\\alpha","b^\\mathrm{Liu}_\\mathrm{Dir}/b_\\mathrm{Dir}"}],
PlotLegends->LineLegend[97,MaTeX[{"d=2","d=3","d=4","d=5"}],LegendFunction->None,LegendLayout->{"Column",1}]];
(* ignore warnings! *)

figbLtobii=Plot[Table[BDirLiu[d,λα[α,1], 1]/BDir[d,α,1],{d,{2,3,4,5}}]//Evaluate,{α,0.7,1},PlotRange->{0.978,1},Ticks->{xtb,ytb} ];
(* ignore warnings! *)

figbLiutob=Show[figbLtobi, Graphics[{Arrowheads[0.05], Arrow[{{0.85,0.99},{0.6,0.77}}]}],Epilog->Inset[Framed[Show[figbLtobii, ImageSize->150],ContentPadding->False,FrameMargins->Tiny, RoundingRadius->10], {0.6,0.6} ]
]

Export["figbLiutob.pdf",figbLiutob]

figbLtobiFree=Plot[{-BDirLiu[2,λα[α,1], 1]/Bfree[2,α,1],-BDirLiu[3,λα[α,1], 1]/Bfree[3,α,1],-BDirLiu[4,λα[α,1], 1]/Bfree[4,α,1],-BDirLiu[5,λα[α,1], 1]/Bfree[5,α,1]},{α,0,1},AxesLabel->MaTeX[{"\\alpha","b^\\mathrm{Liu}_\\mathrm{free}/b_\\mathrm{free}"}],
PlotLegends->LineLegend[97,MaTeX[{"d=2","d=3","d=4","d=5"}],LegendFunction->None,LegendLayout->{"Column",1}]]

Export["figbLtobfree.pdf",figbLtobiFree]

yt2=Table[{y, MaTeX[y]}, {y,-3, -2.25, 0.25}];

figbsi=Plot[Table[2^(d+1)Pi^((d-1)/2)BDir[d,α,1],{d,{2,3,4,5}}]//Evaluate,{α,0,1}, Ticks->{xt,yt2},AxesLabel->MaTeX[{"\\alpha","2^{d+1}\\pi^{(d-1)/2} \\mu^{(d-1)/2} b_\\mathrm{Dir}"}] ];
(* ignore warnings! *)

figgamma=Plot[γR[α],{α,0,1}, PlotStyle->Black,AxesLabel->MaTeX[{"\\alpha","\\gamma_R"}], Ticks->{Table[{x, MaTeX[x]}, {x,{0.5,1}}],Table[{x, MaTeX[x]}, {x,{0.5,1}}]}];

yt3=Table[{y, MaTeX[y]}, {y,3,12, 3}];

figbsii=Plot[
{2^(2+1)Pi^((2-1)/2)Bfree[2,α,1],2^(3+1)Pi^((3-1)/2)Bfree[3,α,1],2^(4+1)Pi^((4-1)/2)Bfree[4,α,1],2^(5+1)Pi^((5-1)/2)Bfree[5,α,1]},{α,0,1},Ticks->{xt,yt3},AxesOrigin->{0,1},PlotRange->{{0,1},{0,13}},AxesLabel->MaTeX[{"\\alpha","2^{d+1}\\pi^{(d-1)/2} \\mu^{(d-1)/2} b_\\mathrm{free}"}],
Epilog->Inset[Framed[Show[figgamma, ImageSize->80],ContentPadding->False,FrameMargins->Tiny, RoundingRadius->10], Scaled[{0.05,1}], Scaled[{0,1}] ] ];

figbs=GraphicsColumn[{GraphicsRow[{figbsi,figbsii}],LineLegend[97,MaTeX[{"d=2\\qquad","d=3\\qquad","d=4\\qquad","d=5"}],LegendFunction->None,LegendLayout->{"Row",1}] }]

Export["figbs.pdf",figbs]

γR[0]//N
γR[1]