N@Integrate[Sqrt[16 E^(4 I*t) - 2 E^(I*t)]*2 I*E^(I*t), {t, 0, 2 Pi}]
6.79746 - 6.40204 I

NIntegrate[Sqrt[16 E^(4 I*t) - 2 E^(I*t)]*2 I*E^(I*t), {t, 0, 2 Pi}]
7.60503*10^-15 + 1.37562 I

f[z_] = Sqrt[z^4 - z]

integrand[t_] =
   f[2 E^(I t)]*2 I*E^(I*t) //
        ComplexExpand[#, TargetFunctions -> {Re, Im}] & //
        FullSimplify[#, 0 < t < 2 Pi] &

(*   2 I Sqrt[2] E^(I t) Sqrt[E^(I t) (-1 + 8 E^(3 I t))]

2 I Sqrt[2] (65 - 16 Cos[3 t])^(1/4) (Cos[
   t + 1/2 ArcTan[-Cos[t] + 8 Cos[4 t], -Sin[t] + 8 Sin[4 t]]] +
   I Sin[t + 1/2 ArcTan[-Cos[t] + 8 Cos[4 t], -Sin[t] + 8 Sin[4 t]]])   *)

{sol = Solve[-Sin[t] + 8 Sin[4 t] == 0 && -Cos[t] + 8 Cos[4 t] < 0 &&
    0 < t < 2 Pi, t], t /. sol // N}

(*   {{{t -> 2 [Pi] -
 2 ArcTan[Sqrt[
   Root[-31 + 227 #1 - 221 #1^2 + 33 #1^3 &, 1]]]}, {t ->
2 ArcTan[Sqrt[
  Root[-31 + 227 #1 - 221 #1^2 + 33 #1^3 &, 1]]]}, {t ->
2 [Pi] -
 2 ArcTan[Sqrt[
   Root[-31 + 227 #1 - 221 #1^2 + 33 #1^3 &, 3]]]}, {t ->
2 ArcTan[Sqrt[
  Root[-31 + 227 #1 - 221 #1^2 + 33 #1^3 &, 3]]]}}, {5.51943,
0.763757, 3.94961, 2.33357}}   *)

integrand2[t_] =
  Piecewise[{{integrand[t],
     0 <= t <= (t /. sol[[2]])}, {-integrand[t], (t /. sol[[2]]) <
     t <= (t /. sol[[4]])}, {integrand[t], (t /. sol[[4]]) <
     t <= (t /. sol[[3]])}, {-integrand[t], (t /. sol[[3]]) <
     t <= (t /. sol[[1]])}, {integrand[t], (t /. sol[[1]]) < t <=
     2 Pi}}, 0]

NIntegrate[integrand2[t], {t, 0, 2 Pi}]

(*   1.77636*10^-15 - 3.14159 I   *)

l1 = Int[integrand2[t][[1, All, 1]], t];

l2 = integrand2[t][[1, All, 2]];

tp = Transpose[{l1, l2}];

rint[t_] = Piecewise[tp, 0]

Plot[{rint[t] // Re, rint[t] // Im}, {t, 0, 2 Pi}, PlotPoints -> 200,
 PlotStyle -> {Blue, Red}, GridLines -> {{2/3 Pi, 4/3 Pi}, Automatic},
 Ticks -> {{2/3 Pi, 4/3 Pi}, Automatic}]

lim = Plus @@ {Limit[rint[t], t -> 2 Pi,
 Direction -> 1], -Limit[rint[t], t -> 4/3 Pi, Direction -> -1],
Limit[rint[t], t -> 4/3 Pi,
 Direction -> 1], -Limit[rint[t], t -> 2/3 Pi, Direction -> -1],
Limit[rint[t], t -> 2/3 Pi,
 Direction -> 1], -Limit[rint[t], t -> 0, Direction -> -1]} //
 FullSimplify

(*  - I Pi   *)