What the output of SecondInt is telling you is that MATLAB is not able to find a closed form solution for the integral, except in the case where x = -1 or +1 in which case it can be expressed as a hypergeom.
There is actually a closed form solution for the second integral. And MATLAB can even compute one under reasonable circumstances:
syms x y real
syms z r h nonnegative
FirstInt = int(1,z,h/r*sqrt(x^2+y^2),h);
SecondInt = int(FirstInt,y,-sqrt(r^2-x^2),sqrt(r^2-x^2));
ThirdInt = int(SecondInt,x,-r,r);
This will give you a closed form SecondInt involving log. You would get a numeric error if you evaluate this at x = 0 because one term would involve log(0), but the limit as x = 0 is not a problem.
Unfortunately if there is a closed form solution for ThirdInt then it is difficult to find.