c1=Sqrt[(1/2)^2+1/2*((1)^2-(1/2)^2)*(1+(2/Pi)*ArcTan[(x+(8*Tan[(Pi/((1)^2-(1/2)^2))*((3/4)^2-1/2*((1/2)^2+1^2))]))/(8)])];
dc1=D[c1, x];
ddc1=D[dc1,x];

Plot[c1,  {x,-800, 800}]

v0=3/4;
xinitial1=N[10, 60];
x1=xinitial1;
x3=x1;
x2=-x1;
x4=x2;

h=FullSimplify[c1/(c1^2-v0^2)];
xstar[y_]:=HeavisideTheta[y]*NIntegrate[(h), {x,  x1,y}, WorkingPrecision->30, PrecisionGoal->20, MaxRecursion->50]+HeavisideTheta[-y]*NIntegrate[(h), {x,  -x1,y}, WorkingPrecision->30, PrecisionGoal->20, MaxRecursion->50]
xfinal=N[10^-18,60];
xfinal2=N[10^-18,60];

Plot[xstar[x],{x, -20, 20},PlotStyle->{Blue, Thick}, ImageSize-> Full, PlotRange->{-150, 30}, Ticks->{{-15,{ -10,Subscript["x","l"]},-5, 5,{ 10,Subscript["x","r"]}, 15}, Automatic},AxesLabel->{ Style[x, 20],Style[SuperscriptBox["x","*"]//DisplayForm, 20]},TicksStyle->Directive["Label", 14],GridLines->{{{-10,{Red,Thick}},{ 10,{Red,Thick}}},None}]

OutputPlotxstar=Plot[xstar[x],{x, -20, 20},PlotStyle->{Blue, Thick}, ImageSize-> Full, PlotRange->{-150, 30}, Ticks->{{-15,{ -10,Subscript["x","l"]},-5,5,{ 10,Subscript["x","r"]}, 15}, Automatic},AxesLabel->{ Style[x, 20],Style[SuperscriptBox["x","*"]//DisplayForm, 20]},TicksStyle->Directive["Label", 14]]

Export["D:\\Dropbox\\ParticleProductionDraft\\Figures\\OutputPlotxstar.pdf",OutputPlotxstar]