Airy T-bar integral

airydelta[p_]:=-ArcTan[p(AiryAi[0-p^2]/AiryAiPrime[0-p^2])]

deltapade[p_]:= ((a1 p+a2 (p^2)+a3 (p^3)+a4 (p^4)+a5 (p^5))/(1+b1 p+b2 (p^2)))

a:=Pi/4

b:=2/3

c:=3^(-1/3)Gamma[1/3]/Gamma[2/3]

d:=1-6Gamma[7/6]^3/Pi^(3/2)

a1:=N[c]

a2:=N[(b-d)a/c]

a3:=N[b]

a4:=N[(b-d)a*b/c^2]

a5:=N[(b-d)b/c]

b1:=N[(b-d)a/c^2]

b2:=N[(b-d)/c]

n := N[1/(8Pi^2)]

N[n c^(-2)]

ruintpade[u_,r_,z_]:= Pi^(-3)r (Sqrt[1-(u^2)]) Cos[2 z r u-2 deltapade[r u]]

u:=0.5

LVpaderiesz0ur[z_,lambda_]:=NIntegrate[ruintpade[u,r,z],{r,0,lambda},Method->"LevinRule"]

LV0u05=Table[LVpaderiesz0ur[-1,lambda],{lambda,1,25}]

LV0ur1=ListPlot[LV0u05]

u:=0.7

LV0u07=Table[LVpaderiesz0ur[-1,lambda],{lambda,1,25}]

LV0ur2=ListPlot[LV0u07]

u:=0.8

LV0u08=Table[LVpaderiesz0ur[-1,lambda],{lambda,1,25}]

LV0ur3=ListPlot[LV0u08]

u:=0.9

LV0u09=Table[LVpaderiesz0ur[-1,lambda],{lambda,1,25}]

LV0ur4=ListPlot[LV0u09]

u:=0.95

LV0u095=Table[LVpaderiesz0ur[-1,lambda],{lambda,1,25}]

LV0ur5=ListPlot[LV0u095]

GraphicsRow[{LV0ur1,LV0ur2}]

GraphicsRow[{LV0ur3,LV0ur4}]

GraphicsRow[{LV0ur1,LV0ur5}]

u:=0.5

CCpaderiesz0ur[z_,lambda_]:=NIntegrate[ruintpade[u,r,z],{r,0,lambda},Method->"ClenshawCurtisRule"]

CC0u05=Table[CCpaderiesz0ur[-1,lambda],{lambda,1,25}]

CC0ur1=ListPlot[CC0u05]

u:=0.7

CCpaderiesz0ur[z_,lambda_]:=NIntegrate[ruintpade[u,r,z],{r,0,lambda},Method->"ClenshawCurtisRule"]

CC0u07=Table[CCpaderiesz0ur[-1,lambda],{lambda,1,25}]

CC0ur2=ListPlot[CC0u07]

u:=0.8

CCpaderiesz0ur[z_,lambda_]:=NIntegrate[ruintpade[u,r,z],{r,0,lambda},Method->"ClenshawCurtisRule"]

CC0u08=Table[CCpaderiesz0ur[-1,lambda],{lambda,1,25}]

CC0ur3=ListPlot[CC0u08]

u:=0.9

CCpaderiesz0ur[z_,lambda_]:=NIntegrate[ruintpade[u,r,z],{r,0,lambda},Method->"ClenshawCurtisRule"]

CC0u09=Table[CCpaderiesz0ur[-1,lambda],{lambda,1,25}]

CC0ur4=ListPlot[CC0u09]

u:=0.95

CCpaderiesz0ur[z_,lambda_]:=NIntegrate[ruintpade[u,r,z],{r,0,lambda},Method->"ClenshawCurtisRule"]

CCu095=Table[CCpaderiesz0ur[-1,lambda],{lambda,1,25}]

CC0ur5=ListPlot[CCu095]

GraphicsRow[{CC0ur1,CC0ur2}]

GraphicsRow[{CC0ur3,CC0ur4}]

GraphicsRow[{CC0ur1,CC0ur5}]

Comparision Between Levin and ClenshawCurtis for padereisz0ur

GraphicsRow[{LV0ur1,CC0ur1}]

GraphicsRow[{LV0ur2,CC0ur2}]

GraphicsRow[{LV0ur3,CC0ur3}]

GraphicsRow[{LV0ur4,CC0ur4}]

GraphicsRow[{LV0ur5,CC0ur5}]