restart;
a := Pi*(1+(1/4)*sinh(u)^2)*cosh(u)^2;

a2 := subs(sinh(u) = (exp(u)-exp(-u))*(1/2), a);

a3 := subs(cosh(u) = (exp(u)+exp(-u))*(1/2), a2);

a4 := expand(a3);

a5 := int(a4, u);

g := unapply(a5, u);

a6 := g(log(2+sqrt(5)))-g(0);

a7 := int(2*Pi*(1+x^2)*sqrt(1+4*x^2), x = 0 .. 1);

simplify(a6-a7);

evalf(a6);