How can I check if a function remains unevaluated?

check if a function remains unevaluated.

Why not just check the head?

func = Log[3 x]^5*Sec[3 x];
result = Integrate[func, x]

If[Head[result] === Integrate,
 Print["Opps"],
 Print["it worked"]
 ]

Update for comment

Use this to check if result contains Integrate anywhere

func = 20* Log[3 x]^5*Sec[3 x];
result = Integrate[func, x]

If[FreeQ[result, Integrate, Infinity],
 Print["it worked"],
 Print["Opps"]
 ]

ValueQ should do exactly what you want. From the documentation: ValueQ[expr] gives True if a value has been defined for expr, and gives False otherwise. ValueQ gives False only if expr would not change if it were to be entered as Wolfram Language input.

So

ValueQ[Integrate[Log[3 x]^5*Sec[3 x], x]]
(* False *)

and

ValueQ[Integrate[Sec[3 x], x]]
(* True *)