au=0.129;
bu=3.641;
cu=15.024;
du=68.373;
bd=0.3522;

nu =2/ (((Gamma[1+bu] Gamma[au+s])/(Gamma[1+au+bu+s]))+((du Gamma[1+bu] Gamma[1+au+s])/(Gamma[2+au+bu+s]))+((cu Gamma[1+bu] Gamma[0.5` +1.` au+1.` s])/(Gamma[1.5` +1.` au+bu+1.` s])))/.s->0;
xuv=nu (x^au)(((1-x))^bu)(1+cu (x^0.5)+du x);
nd =1/(((Gamma[1+bd+bu] Gamma[au+s])/(Gamma[1+au+bd+bu+s]))+((du Gamma[1+bd+bu] Gamma[1+au+s])/(Gamma[2+au+bd+bu+s]))+((cu Gamma[1+bd+bu] Gamma[0.5` +1.` au+1.` s])/(Gamma[1.5` +1.` au+bd+bu+1.` s])))/.s->0;

xdv=nd/nu (((1-x))^bd)xuv;
x=Exp[-v];
Un=LaplaceTransform[xuv,v,s];
%/.s->0
%%/.s->1
Dn=LaplaceTransform[xdv,v,s];
%/.s->0
%%/.s->1

Λ=lan^2;
Q0=4;


"Q=Input["Q"]"; 
αQ=((1/(β0*(y-Log[Λ])))-((β1)/(((β0))^3))((Log[y-Log[Λ]])/(((y-Log[Λ]))^2))+(1/((((β0))^3)(((y-Log[Λ]))^3)))(((((β1))^2)/(((β0))^2))((((y-Log[Λ]))^2)-(y-Log[Λ])-1)+(((β2))/((β0)))));
αQ0=((1/(β0*(y0-Log[Λ])))-((β1)/(((β0))^3))((Log[y0-Log[Λ]])/(((y0-Log[Λ]))^2))+(1/((((β0))^3)(((y0-Log[Λ]))^3)))(((((β1))^2)/(((β0))^2))((((y0-Log[Λ]))^2)-(y0-Log[Λ])-1)+(((β2))/((β0)))));

AlfasQ=αQ;
AlfasQ0=αQ0;
(*αQ=4Pi((1/(β0*(y-Log[Λ]))));
αQ0=4Pi((1/(β0*(y0-Log[Λ]))));*)
taQ=Integrate[1/(4Pi)AlfasQ,y]/.y->Log[Q];
taQ0=Integrate[1/(4Pi)AlfasQ0,y0]/.y0->Log[Q0];
ta2Q=Integrate[1/(((4Pi))^2)(AlfasQ^2),y]/.y->Log[Q];
ta2Q0=Integrate[1/(((4Pi))^2)(AlfasQ0^2),y0]/.y0->Log[Q0];
ta3Q=Integrate[1/(((4Pi))^3)(AlfasQ^3),y]/.y->Log[Q];
ta3Q0=Integrate[1/(((4Pi))^3)(AlfasQ0^3),y0]/.y0->Log[Q0];
ta2n=ta2Q-ta2Q0;
ta3n=ta3Q-ta3Q0;
ta=taQ-taQ0;
cf=(4/3);
ca=3;
 tf=f;
tr=1/2;
(*lan=0.2*)





pqq0=4-8/3(1/(s+1)+1/(s+2)+2(EulerGamma+PolyGamma[0,s+1]));

β0=11-(2/3)f;
β1=102-(38/3)f;
β2=(2857/2)-(5033/18)f+(325/54)(f^2);
β3=29243-6946.3 f+405.089 (f^2)+1093/729 (f^3);

pnsp1=cf tf (-(2/(3 (((1+s))^2)))-(2/(9 (1+s)))-(2/(3 (((2+s))^2)))+(22/(9 (2+s)))+((20 HarmonicNumber[s])/9)+(4/3) Zeta[2,1+s])+(cf^2) (-(1/(((1+s))^3))-(5/(1+s))-(1/(((2+s))^3))+(2/(((2+s))^2))+(5/(2+s))+(1/(((1+s))^2))2 (EulerGamma+(1/(1+s))+PolyGamma[0,1+s]-(1+s) PolyGamma[1,2+s])+(1/(((2+s))^2))2 (EulerGamma+(1/(2+s))+PolyGamma[0,2+s]-(2+s) PolyGamma[1,3+s])-4 (HarmonicNumber[s] PolyGamma[1,1+s]-(1/2) PolyGamma[2,1+s])+3 Zeta[2,1+s])+ca cf (-(1/(((1+s))^3))+(5/(6 (((1+s))^2)))+(53/(18 (1+s)))-(1/(((2+s))^3))+(5/(6 (((2+s))^2)))-(187/(18 (2+s)))+((Pi^2)/(6 (1+s)))+((Pi^2)/(6 (2+s)))-((67 HarmonicNumber[s])/9)+(1/3) (Pi^2) HarmonicNumber[s]-(11/3) Zeta[2,1+s]+2 Zeta[3,1+s]);
%/.s->1//N

ppsqq1=cf (-(ca/2)+cf) (-(2/(((1+s))^3))-(2/(((1+s))^2))+(4/(1+s))+((Pi^2)/(3 (1+s)))+(2/(((2+s))^3))-(2/(((2+s))^2))-(8/(2+s))-((Pi^2)/(3 (2+s)))+(5/(3+s))-(13/(9 (4+s)))+(25/(36 (5+s)))-(41/(100 (6+s)))+(4/(25 (7+s)))-HarmonicNumber[1+(s/2)]-(1/3) (Pi^2) (-HarmonicNumber[(1/2) (-1+s)]+HarmonicNumber[(s/2)])+HarmonicNumber[((1+s)/2)]+4 (-HarmonicNumber[(s/2)]+HarmonicNumber[((1+s)/2)])+(4/9) (-HarmonicNumber[1+(s/2)]+HarmonicNumber[((3+s)/2)])+(1/4) (-HarmonicNumber[2+(s/2)]+HarmonicNumber[((3+s)/2)])+(4/25) (-HarmonicNumber[2+(s/2)]+HarmonicNumber[((5+s)/2)])+(1/(((1+s))^3))(-8+(1+s) Log[16]+2 (1+s) PolyGamma[0,1+(s/2)]-2 (1+s) PolyGamma[0,((1+s)/2)]-(((1+s))^2) PolyGamma[1,1+(s/2)]+(((1+s))^2) PolyGamma[1,((1+s)/2)])-(1/(((2+s))^3))0.9998400000000001` ((16/(((1+s))^2))+((12 s)/(((1+s))^2))+(2+s) Log[16]-2 (2+s) PolyGamma[0,1+(s/2)]+2 (2+s) PolyGamma[0,((1+s)/2)]+(((2+s))^2) PolyGamma[1,1+(s/2)]-(((2+s))^2) PolyGamma[1,((1+s)/2)])-(1/(((3+s))^3))1.9702` ((164/((((1+s))^2) (((2+s))^2)))+((284 s)/((((1+s))^2) (((2+s))^2)))+((188 (s^2))/((((1+s))^2) (((2+s))^2)))+((60 (s^3))/((((1+s))^2) (((2+s))^2)))+((8 (s^4))/((((1+s))^2) (((2+s))^2)))-4 (3+s) Log[2]-2 (3+s) PolyGamma[0,1+(s/2)]+2 (3+s) PolyGamma[0,((1+s)/2)]+(((3+s))^2) PolyGamma[1,1+(s/2)]-(((3+s))^2) PolyGamma[1,((1+s)/2)])-(1/(((4+s))^3))1.801` ((2176/((((1+s))^2) (((2+s))^2) (((3+s))^2)))+((4392 s)/((((1+s))^2) (((2+s))^2) (((3+s))^2)))+((3504 (s^2))/((((1+s))^2) (((2+s))^2) (((3+s))^2)))+((1408 (s^3))/((((1+s))^2) (((2+s))^2) (((3+s))^2)))+((288 (s^4))/((((1+s))^2) (((2+s))^2) (((3+s))^2)))+((24 (s^5))/((((1+s))^2) (((2+s))^2) (((3+s))^2)))+4 (4+s) Log[2]-2 (4+s) PolyGamma[0,1+(s/2)]+2 (4+s) PolyGamma[0,((1+s)/2)]+(((4+s))^2) PolyGamma[1,1+(s/2)]-(((4+s))^2) PolyGamma[1,((1+s)/2)])-(1/(((5+s))^3))1.3242` ((57328/((((1+s))^2) (((2+s))^2) (((3+s))^2) (((4+s))^2)))+((146144 s)/((((1+s))^2) (((2+s))^2) (((3+s))^2) (((4+s))^2)))+((162160 (s^2))/((((1+s))^2) (((2+s))^2) (((3+s))^2) (((4+s))^2)))+((103728 (s^3))/((((1+s))^2) (((2+s))^2) (((3+s))^2) (((4+s))^2)))+((42144 (s^4))/((((1+s))^2) (((2+s))^2) (((3+s))^2) (((4+s))^2)))+((11160 (s^5))/((((1+s))^2) (((2+s))^2) (((3+s))^2) (((4+s))^2)))+((1880 (s^6))/((((1+s))^2) (((2+s))^2) (((3+s))^2) (((4+s))^2)))+((184 (s^7))/((((1+s))^2) (((2+s))^2) (((3+s))^2) (((4+s))^2)))+((8 (s^8))/((((1+s))^2) (((2+s))^2) (((3+s))^2) (((4+s))^2)))-4 (5+s) Log[2]-2 (5+s) PolyGamma[0,1+(s/2)]+2 (5+s) PolyGamma[0,((1+s)/2)]+(((5+s))^2) PolyGamma[1,1+(s/2)]-(((5+s))^2) PolyGamma[1,((1+s)/2)])-(1/(((6+s))^2))0.6348` (Log[16]-2 PolyGamma[0,4+(s/2)]+2 PolyGamma[0,((7+s)/2)]+(6+s) PolyGamma[1,4+(s/2)]-(6+s) PolyGamma[1,((7+s)/2)])+(1/(((7+s))^2))0.1398` (Log[16]+2 PolyGamma[0,4+(s/2)]-2 PolyGamma[0,((9+s)/2)]-(7+s) PolyGamma[1,4+(s/2)]+(7+s) PolyGamma[1,((9+s)/2)])+(1/2) (-Zeta[3,1+(s/2)]+Zeta[3,((1+s)/2)]));
%/.s->1//N

pqq1p=ppsqq1+pnsp1;

c3nlo=cf(- (9+((2 (π^2))/3))-((2 )/(((1+s))^2))+((7 )/(1+s))-((2 )/(((2+s))^2))+((5 )/(2+s))+3  HarmonicNumber[s]+((2 HarmonicNumber[1+s])/(1+s))+((2  HarmonicNumber[2+s])/(2+s))+(1/3)  ((π^2)+6 ((HarmonicNumber[s])^2)-6 PolyGamma[1,1+s])+4  Zeta[2,1+s]);

c3nnlo=-338.635`-((188.64`)/s)+((23.532`)/(((1+s))^4))-((66.62`)/(((1+s))^3))+((67.6`)/(((1+s))^2))-((206.1`)/(1+s))-((576.8`)/(2+s))-((31.105` (EulerGamma+PolyGamma[1+s]))/s)+((409.6` (EulerGamma+PolyGamma[2+s]))/(1+s))+(10.2222/s) ((π^2)+6 (((EulerGamma+PolyGamma[1+s]))^2)-6 PolyGamma[1,1+s])+(1/(6+6 s))94.61` ((π^2)+6(((EulerGamma+PolyGamma[2+s]))^2)-6 PolyGamma[1,2+s])+((24.65`)/(((1+s))^2)) (6 (EulerGamma^2)+(π^2)+((12 EulerGamma)/(1+s))+12 EulerGamma PolyGamma[0,1+s]-6 (1+s) PolyGamma[0,1+s] ((PolyGamma[0,2+s])^2)+6 (1+s) ((PolyGamma[0,2+s])^3)-6 (3+2 EulerGamma (1+s)) PolyGamma[1,2+s]-12 (1+s) PolyGamma[0,1+s] PolyGamma[1,2+s]+6 (1+s) PolyGamma[2,2+s])+((7.1111`)/s) (2 (EulerGamma^3)+EulerGamma (π^2)+6 EulerGamma ((PolyGamma[0,1+s])^2)+2 ((PolyGamma[0,1+s])^3)+PolyGamma[0,1+s] (6 (EulerGamma^2)+(π^2)-6 PolyGamma[1,1+s])-6 EulerGamma PolyGamma[1,1+s]+2 PolyGamma[2,1+s]+4 Zeta[3])+((7.6`)/(1+s)) (2 (EulerGamma^3)+EulerGamma (π^2)+6 EulerGamma ((PolyGamma[0,2+s])^2)+2 ((PolyGamma[0,2+s])^3)+PolyGamma[0,2+s] (6 (EulerGamma^2)+(π^2)-6 PolyGamma[1,2+s])-6 EulerGamma PolyGamma[1,2+s]+2 PolyGamma[2,2+s]+4 Zeta[3])+f (46.857` -((6.3489`)/s)+((4.414`)/(((1+s))^3))-((8.683`)/(((1+s))^2))-((6.337`)/(1+s))-((14.97`)/(2+s))-((8.5926` (EulerGamma+PolyGamma[1+s]))/s)-((25 (EulerGamma+PolyGamma[2+s]))/(1+s))-(0.29629/s) ((π^2)+6 (((EulerGamma+PolyGamma[1+s]))^2)-6 PolyGamma[1,1+s])-(1/(6+6 s))0.808` ((π^2)+6 (((EulerGamma+PolyGamma[2+s]))^2)-6 PolyGamma[1,2+s])+(1/(((1+s))^2))9.684` (EulerGamma+(1/(1+s))+PolyGamma[0,1+s]-(1+s) PolyGamma[1,2+s])-((0.021`)/(1+s)) (2 (EulerGamma^3)+EulerGamma (π^2)+6 EulerGamma ((PolyGamma[0,2+s])^2)+2 ((PolyGamma[0,2+s])^3)+PolyGamma[0,2+s] (6 (EulerGamma^2)+(π^2)-6 PolyGamma[1,2+s])-6 EulerGamma PolyGamma[1,2+s]+2 PolyGamma[2,2+s]+4 Zeta[3]));

pnspnnlo=1295.384` +(1024/(27 (((1+s))^5)))-(1600/(9 (((1+s))^4)))+((589.8`)/(((1+s))^3))-(1258/(((1+s))^2))+((1641.1`)/(1+s))-(3135/(2+s))+((243.6`)/(3+s))-((522.1`)/(4+s))+1174.898` (HarmonicNumber[-1+s]-HarmonicNumber[s])-((714.1` HarmonicNumber[1+s])/(1+s))+(1/(((1+s))^2))563.9` (EulerGamma+(1/(1+s))+PolyGamma[0,1+s]-(1+s) PolyGamma[1,2+s])+f (173.927` +(128/(9 (((1+s))^4)))-(5216/(81 (((1+s))^3)))+((152.6`)/(((1+s))^2))-(197/(1+s))+((8.982000000000001`)/(((2+s))^4))+((381.1`)/(2+s))+((72.94`)/(3+s))+((44.79`)/(4+s))-183.187` (HarmonicNumber[-1+s]-HarmonicNumber[s])+((5120 HarmonicNumber[1+s])/(81 (1+s)))-(1/(((1+s))^2))56.66` (EulerGamma+(1/(1+s))+PolyGamma[0,1+s]-(1+s) PolyGamma[1,2+s]))-(1/(((1+s))^4))256.8` (3+2 EulerGamma (1+s)+2 EulerGamma (((1+s))^2) PolyGamma[0,1+s]-(1+s) (-1+2 EulerGamma (1+s)) PolyGamma[0,1+s]+(((1+s))^3) ((PolyGamma[0,1+s])^2) PolyGamma[0,2+s]-2 (((1+s))^3) PolyGamma[0,1+s] ((PolyGamma[0,2+s])^2)+(((1+s))^3) ((PolyGamma[0,2+s])^3)-2 (((1+s))^2) PolyGamma[1,1+s]+(((1+s))^3) PolyGamma[2,2+s])+(f^2) (-(64/81) (HarmonicNumber[-1+s]-HarmonicNumber[s])+(64/81) (-(51/16)+((5 (π^2))/6)+(3/(2 (((1+s))^3)))-(11/(2 (((1+s))^2)))+(7/(1+s))-(3/(2 (((2+s))^3)))+(11/(2 (((2+s))^2)))-(6/(2+s))-3 Zeta[3]-5 Zeta[2,2+s]+3 Zeta[3,2+s]));

mnsNNLO=Exp[   ta  (pqq0+ta2n/ta pnsp1+ta3n/ta pnspnnlo )];

uvn=mnsNNLO Un;
dvn=mnsNNLO Dn;

F3=(uvn+dvn)(1+ta/(4 Pi) c3nlo +(((ta/(4 Pi)))^2)c3nnlo);

x=.
Nmax=.
Nmax=9;
α=3;
β=0.5;
θ=(Sqrt[((  k!(2  k+α+β+1) Gamma[k+α+β+1])/(Gamma[k+α+1]Gamma[k+β+1]))  ])((((-1))^k)/(k!(x^β)(((1-x))^α)))(∂_({x,k}))((x^(β+k))(((1-x))^(α+k)));
ckj=(Sqrt[(( k! (2 k+α+β+1) Gamma[k+α+β+1])/(Gamma[k+α+1]Gamma[k+β+1]))  ])((((-1))^k)/(k!))(1/(j!))(Pochhammer[-k,j]Pochhammer[α+β+k+1,j]Pochhammer[β+j+1,k-j]);Momentg1NLOxF3=(F3)/.s->j+1;

aαβ[k_]:=(∑_(j=0)^k)ckj*Momentg1NLOxF3;

xF3xQ=(x^β)(((1-x))^α)*((∑_(k=0)^Nmax)θ*aαβ[k]);

Needs["ErrorBarLogPlots`"]
xf3Q20={{0.0075,0.1821,0.04039},{0.0125,0.2239,0.03512},{0.0175,0.2767,0.03549},{0.025,0.3517,0.0267},{0.035,0.4121,0.035},{0.05,0.5068,0.036},{0.07,0.5093,0.05839},{0.09,0.7605,0.09046}};
ErrorListLogLinearPlot[xf3Q20];
LogLinearPlot[xF3xQ/.f->4/.Q->20/.lan->0.41,{x,0.001,0.999}];
Show[%,%%]

xf3Q7p9={{0.0175`,0.3513`,0.06743`},{0.025`,0.3555`,0.03521`},{0.035`,0.4316`,0.03064`},{0.05`,0.58`,0.01979`},{0.07`,0.6377`,0.01946`},{0.09`,0.734`,0.02048`},{0.11`,0.7624`,0.022`},{0.14`,0.7931`,0.01804`},{0.18`,0.8026`,0.02349`},{0.225`,0.7854`,0.0271`},{0.275`,0.7803`,0.03387`},{0.35`,0.6352`,0.03202`}};
ErrorListLogLinearPlot[xf3Q7p9];
LogLinearPlot[xF3xQ/.f->4/.Q->7.9/.lan->0.41,{x,0.001,0.999}];
Show[%,%%]

NIntegrate[ xF3xQ/x/.f->4/.Q->8/.lan->0.41,{x,0,1}]