Add PlotRange -> All or reduce the x range
Plot[Piecewise[
         {{5.66048 E^(0.76` x), x < -5}, 
          {0.398447 Cos[0.25 x], -5 <= x <= 5}, 
          {5.66048 E^(-0.76` x), x > 5}}, 0],
   {x, -10, 10}]

LogPlot[Piecewise[
         {{5.66048 E^(0.76` x), x < -5}, 
          {0.398447 Cos[0.25 x], -5 <= x <= 5}, 
          {5.66048 E^(-0.76` x), x > 5}}, 0],
   {x, -10, 10}]
