Comparing two equivalent definite integrals

Reading this question on Math.SE, I tried the following Mathematica instructions

In[1]:= i1 = Integrate[1/Sqrt[1 + t^4], {t, 0, 1}];
        i2 = Integrate[Sec[u]^2/Sqrt[1 + Tan[u]^4], {u, 0, [Pi]/4}];
        FullSimplify[i1 == i2]
        FullSimplify[i1 == i2] // N
Out[3]= Sqrt[[Pi]] EllipticK[1/2] == 4 Gamma[5/4]^2
Out[4]= True

so, given that the two integrals are the same, apart from a change of variable, I ask: why Mathematica does not know?

$Version

(* "12.1.1 for Mac OS X x86 (64-bit) (June 19, 2020)" *)

Clear["Global`*"]

i1 = Integrate[1/Sqrt[1 + t^4], {t, 0, 1}]

(* (2 Gamma[5/4]^2)/Sqrt[π] *)

i2 = Integrate[Sec[u]^2/Sqrt[1 + Tan[u]^4], {u, 0, π/4}]

(* 1/2 EllipticK[1/2] *)

i1 == i2 // FunctionExpand // FullSimplify

(* True *)