Epicycloid curve calculating arch length using integral method

Im trying to calculate the length of the Epicycloid curve.

i have the x and y coordinates right and i get the correct looking plot, but cant figure out how to calculate the length of the epicycloid curve.

I already tried to calculate, but didnt get results that make sense.

Any help would be greatly appreciated.

Here is a picture of what im calculating:

clear all

R=8;

L=6;

Alfa=2*pi;

Bertta=4*pi;

T=2*pi/Alfa;

syms t

alfa0=Alfa*t;

beta0=Bertta*t;

%%%%%%%%%%%%%%%%%%% %R Coordinates

x0(t)=R*cos(alfa0);

y0(t)=R*sin(alfa0);

%%%%%%%%%%%%%%%%%% %L Coordinates

x(t)=x0(t)+L*cos(alfa0+beta0);

y(t)=y0(t)+L*sin(alfa0+beta0);

%

%x1(t)=diff(x,t)

%y1(t)=diff(y,t)

%t0=2;

%dx=x1(t0)

%dy=y1(t0)

%s0 = @(t) sqrt( ( R.*cosd(t)+L.*cosd(3.*t) ).^2 + (R.*sind(t)+L.*sind(3.*t)).^2);

%s = integral(s0,0,2*pi)

%pit = sqrt(1+(dy/dx).^2*dx)

%lenght = integral(pit,0,2*pi)

fplot(x,y,[0,T],'linewidth',2)

title(['R = ',num2str(R),', L = ',num2str(L),', s = ',num2str(5)])

hold off

grid

axis equal

xlabel('x')

ylabel('y')

Answer 1

[Torsten]

curvelength = vpaintegral(sqrt(diff(x,t)^2+diff(y,t)^2),t,0,T)

[pauli relanderi]

Thank you so much !