xs=Subdivide[-2.5,10,999];

fx=Table[NIntegrate[(1/π)*Exp[-t]*Cos[t*x+(2*t*Log[t])/π],{t,0,∞},Method->"GlobalAdaptive",MaxRecursion->15,AccuracyGoal->6,PrecisionGoal->6],{x,xs}];

skewCauchyPlot=ListLinePlot[Transpose[{xs,fx}],Frame->True,FrameLabel->{Style["x",18],Style["Density",18]},FrameStyle->Black,FrameTicksStyle->Directive[Black,14],PlotStyle->Thick,PlotRange->All,GridLines->Automatic,Epilog->{{Dashed,Gray,Line[{{Min[xs],0},{Max[xs],0}}]},Inset[Style[Column[{"Max skew Cauchy distribution","with log characteristic function",Row[{"-","|t|"," - 2i t log","|t|","/",Style[π,Plain]}]},Alignment->Left],18],Scaled[{0.95,0.95}],(*top-right*){Right,Top}]},ImageSize->Large]