cSpherical[t_] := {t, Pi(((1+Sin[16t+4t^2])/2)), 1+2t+3t^2}

cCartesian[t_] = CoordinateTransform["Spherical"->"Cartesian",cSpherical[t]]

Show[Graphics3D[{Opacity[.25],Sphere[]}],ParametricPlot3D[cCartesian[t],{t,0,1},PlotStyle->Thick,ColorFunction->(Hue[.8#4]&)]]

differentialLength[t_] = Simplify@Sqrt[ cSpherical'[t]. CoordinateChartData["Spherical","Metric",cSpherical[t]]. cSpherical'[t]]

NIntegrate[differentialLength[t],{t,0,1}]