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Well...

sage: R=vector([3*cos(t),3*sin(t),4*t]);R
(3*cos(t), 3*sin(t), 4*t)
sage: with assuming(t,"real"): (R.diff(t).norm()^2).trig_reduce().sqrt()
5

Is that acceptable to you ?

Well...

sage: R=vector([3*cos(t),3*sin(t),4*t]);R
(3*cos(t), 3*sin(t), 4*t)
sage: with assuming(t,"real"): (R.diff(t).norm()^2).trig_reduce().sqrt()
5

or even :

sage: with assuming(t,"real"): R.diff(t).norm().trig_reduce()
5

Is that acceptable to you ?