Question: Cannot calculate the curve length of the complicated 3d curve expression?

Hi everyone, I wrote a complex 3D curve expression and wanted to calculate its length (0..Pi), but Maple was always calculating and could not get the answer. The code and worksheet are attached.
The idea came from: https://www.mapleprimes.com/questions/239065-Curve-Length-2d-3d

with(plots); with(VectorCalculus)

p1 := spacecurve([t*sin(t/cos(exp(t))), t*cos(t), (2*sqrt(2)*(1/3))*t^(3/2)], t = 0 .. Pi)

 

curve := `<,>`(t*sin(t/cos(exp(t))), t*cos(t), (2*sqrt(2)*(1/3))*t^(3/2))

Vector(3, {(1) = t*sin(t/cos(exp(t))), (2) = t*cos(t), (3) = (2/3)*sqrt(2)*t^(3/2)})

(1)

NULL

x := proc (t) options operator, arrow; t*sin(t/cos(exp(t))) end proc; y := proc (t) options operator, arrow; t*cos(t) end proc; z := proc (t) options operator, arrow; (2/3)*sqrt(2)*t^(3/2) end proc; arcLengthIntegrand := sqrt((diff(x(t), t))^2+(diff(y(t), t))^2+(diff(z(t), t))^2); AL := evalf(Int(arcLengthIntegrand, t = 0 .. Pi)); AL

proc (t) options operator, arrow; VectorCalculus:-`*`(t, sin(VectorCalculus:-`*`(t, cos(exp(t))^VectorCalculus:-`-`(1)))) end proc

 

proc (t) options operator, arrow; VectorCalculus:-`*`(t, cos(t)) end proc

 

proc (t) options operator, arrow; VectorCalculus:-`*`(VectorCalculus:-`*`(VectorCalculus:-`*`(2, sqrt(2)), 3^VectorCalculus:-`-`(1)), t^VectorCalculus:-`*`(3, 2^VectorCalculus:-`-`(1))) end proc

 

((sin(t/cos(exp(t)))+t*(1/cos(exp(t))+t*exp(t)*sin(exp(t))/cos(exp(t))^2)*cos(t/cos(exp(t))))^2+(cos(t)-t*sin(t))^2+2*t)^(1/2)

 

AL

(2)

NULL

Download length_of_3d_curve.mw

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