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The complex pieces cancel, I believe.

sage: f=integrate(integrate(integrate(1,z,sqrt(x^2+y^2),H+sqrt(3)*y/3),y,-sqrt(H^2*(3-sqrt(3))^2/4-x^2),sqrt(H^2*(3-sqrt(3))^2/4-x^2)),x,-H*(3-sqrt(3))/2,H*(3-sqrt(3))/2)
sage: f.subs(H=1)
1/4*sqrt(2)*(((3*sqrt(3)) - 6)*sqrt(-3*sqrt(3) + 6)*arcsin(1/2*sqrt(2)*sqrt(-3*sqrt(3) + 6)*(5*sqrt(3) - 9)/(12*sqrt(3) - 21)) + I*((3*sqrt(3)) - 6)*sqrt(sqrt(3) - 2)*arcsin(1/2*sqrt(2)*sqrt(-3*sqrt(3) + 6)*(5*sqrt(3) - 9)/(12*sqrt(3) - 21)) - ((3*sqrt(3))*(2*sqrt(2)) - 12*sqrt(2))*arcsin(1/2*sqrt(2)*sqrt(-3*sqrt(3) + 6)*(5*sqrt(3) - 9)/(12*sqrt(3) - 21)))
sage: f.subs(H=1).n()
1.09351366847695 - 1.41499750148729e-17*I
sage: f.subs(H=2).n()
8.74810934781556 - 1.13199800118983e-16*I

As you can see, the problem is that Sage gets something that isn't simplifying enough to get rid of vanishingly small imaginary parts when numerically evaluated, though

sage: plot(lambda x: f.subs(H=x).n().real(),(x,0,5))

probably gives what you want.

However, in general simplification is a computationally hard problem. You might have to 'unsimplify' before a good simplification is possible.

sage: f.expand().canonicalize_radical()
-3/4*(sqrt(3)*sqrt(2) - sqrt(2))*H^3*sqrt(sqrt(3) - 2)*arcsinh((3*sqrt(3) - 5)*sqrt(sqrt(3) - 2)/(4*sqrt(3)*sqrt(2) - 7*sqrt(2))) - 1/4*H^3*(-12*I*sqrt(3) + 24*I)*arcsinh((3*sqrt(3) - 5)*sqrt(sqrt(3) - 2)/(4*sqrt(3)*sqrt(2) - 7*sqrt(2)))

Unfortunately, due to the 'canonical' nature of this simplification, this gives quite different answers, presumably wrong, which the documentation for canonicalize_radical points out.

This all doesn't answer the question of what the right answer is, but I can't see your picture in any case, and hopefully it does help some.