Incorrectly calculated integral.

[parisse]

If you want an approx answer, it is always best to introduce an approx boundary because there is no attempt to find an antiderivative.
Without dot, the CAS tries to compute a symbolic antiderivative. It transforms
f:=integrate(1/sqrt(1-(2+sqrt(3))/4*sin(x)^2),x,0,80*pi/180)
to integration(2/(sqrt(sin(x)^2+4*sqrt(3)-8)*(i*sqrt(2)/2-i*sqrt(6)/2)),x,0,80*pi/180), with a rewrite of the sqrt that unfortunately introduces complex numbers. After that when you press shift-Enter, the new integral is evaluated numerically, but the choice of sqrt branch is not the same as for the initial integrand.