BUG: Fix Series.str.contains with compiled regex on Arrow string dtype (#61942) and add whatsnew note

Author: Aniketsy

pandas/core/arrays/string_arrow.py

@@ -344,8 +344,10 @@ def _str_contains(
         na=lib.no_default,
         regex: bool = True,
     ):
-        if (isinstance(pat, re.Pattern) and regex) or flags:
+        if flags:
             return super()._str_contains(pat, case, flags, na, regex)
+        if isinstance(pat, re.pattern):
+            pat = pat.pattern
 
         return ArrowStringArrayMixin._str_contains(self, pat, case, flags, na, regex)