maxima先生の解答

r(s):=sin(s)/(cos(s)-1/2);
a(s,t):=r(s)*cos(t)+r(s)*sin(s)+cos(s);
b(s,t):=r(s)*sin(t)+sin(s)-r(s)*cos(s);
define(c(s,t),diff(a(s,t),t));
define(f(s),ratsimp(integrate(b(s,t)*c(s,t),t,5/6*%pi,%pi/2+s)+integrate(-(sin(t))^2,t,s,0)));
g(x):=f(x) - %pi/8;
h(x):=if g((x[1]+x[2])/2)>0 then [x[1],(x[1]+x[2])/2] else [(x[1]+x[2])/2,x[2]];
k(n,x):=if n = 1 then x else k(n-1,h(x));
s0:k(50,[0,1])[1],numer;
r(s0);
2 *a(s0,5/6*%pi)+4*r(s0)*(5/6*%pi - (%pi/2+s0)),numer;

(%o9) .8486751477323029
(%o10) 4.661542714220053
(%o11) 3.945702967267186