I am trying to determine the coefficient q in the parametrization of a epicycloid which gives me the arc length of 4.25. The parametrization can be glimpsed in my attempt of a solution in the following Matlab code.

R=0.5;

r=R/3;

c=(R+r)/r;

t = 0:0.01:2*pi;

fun = @(t,q) sqrt((c.^2).*(r.^2).*sin(t).^2+(c.^2).*(q.^2).*(r.^2).*sin(c.*t).^2+(c.^2).*(r.^2).*cos(t).^2+(c.^2).*(q.^2).*(r.^2).*cos(c.*t).^2+1);

fun2 = @(q) integral(@(t) fun(t,q),0,2*pi)

qsolve=fsolve(@(q) fun2(q)-4.25, 0)

The problem is that solve can not find any solution. I am very much grateful if someone can help me with this one.

Cheers!