When to trust Integrate?

Question

I'm trying to get an analytical expression for the following integral:

f = 16 k Abs[Cos[t] + Sqrt[2] Sin[t] Cos[k z]]^5/Pi;
Integrate[f, {z, 0, 2 Pi/k}, Assumptions -> k > 0 && t > 0]

The answer Mathematica gives is not correct for t>ArcTan[1/Sqrt[2]]:

(* ConditionalExpression[-((-90 Cos[t] + 55 Cos[3 t] + 3 Cos[5 t])/Sign[Cos[t] + Sqrt[2] Sin[t]]^5), Cos[t] + Sqrt[2] Sin[t] != 0] *)

N@% /. t -> 1

(* 102.226 *)

NIntegrate[f /. {t -> 1, k -> 2}, {z, 0, 2 Pi/2}]

(* 103.146 *)

(these two numerical results should be equal.) The answer is independent of k (e.g. on change of variables to k z). If I set k=1 for Integrate, it gives me the correct (rather complicated) expression. Also, I know I can get a simpler expression by splitting the integral into two regions using ArcCos[-Cot[t]/Sqrt[2]]. However, I want to know when I can't trust Integrate.

Answer 1

Making use of RealAbs introduced in 2017 instead of Abs, we obtain the same results symbolically and numerically.

f = 16 k RealAbs[Cos[t] + Sqrt[2] Sin[t] Cos[k z]]^5/Pi;
Integrate[f /. t -> 1, {z, 0, 2 Pi/k}, Assumptions -> k > 0 ]

A long exppression

N[%]

103.146 + 4.45408*10^-15 I