Does $c^*(b \cup (ac)^*)^*$ define all strings over $\{a,b,c\}$ that don't contain the substring $bc$
I checked an older edition of Sudkamp, and it had the same example. Actually the expression is $c^*(b\cup ac^*)^*$, which has a different bracketing.
Your $(ac)^*$ specifies strings of the form $acac\dots ac$, whereas Sudkamp has $ac^*$ which means strings like $acc\dots c$.

It follows the arguments of the answer by Raphael, but backwards. Any sequence of $c$'s is either at the beginning of the string, or else it cannot be preceeded by $b$ because there is an explicit $a$ in the expressing.