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Curve in Spherical Coordinates

The following is a curve expressed in spherical coordinates. It starts at the origin and moves to the surface of the unit sphere.

    
Wolfram Language code: cSpherical[t_] := {t, Pi((1 + Sin[16t + 4t ^ 2]/2)), 1 + 2t + 3t ^ 2}
Wolfram Language code: cCartesian[t_] = CoordinateTransform["Spherical" -> "Cartesian", cSpherical[t]]
A plot of the curve inside the unit sphere. Notice the coloring, which indicates it starts quite slowly and then speeds up as it gets closer to the surface:
Wolfram Language code: Show[Graphics3D[{Opacity[.25], Sphere[]}], ParametricPlot3D[cCartesian[t], {t, 0, 1}, PlotStyle -> Thick, ColorFunction -> (Hue[.8#4]&)]]
Compute the length of the curve. It is about 11.2 radii:
Wolfram Language code: differentialLength[t_] = Simplify@Sqrt[ cSpherical'[t]. CoordinateChartData["Spherical", "Metric", cSpherical[t]]. cSpherical'[t]]
Wolfram Language code: NIntegrate[differentialLength[t], {t, 0, 1}]