Using Rubi ( see https://community.wolfram.com/groups/-/m/t/1421180 ), one obtains
Needs@"Rubi`"
int2 = Int[1/Sqrt[x^3 + 1], x]
(*
(2 Sqrt[2 + Sqrt[3]] (1 + x) Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]
EllipticF[
ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4 Sqrt[3]])/(3^(
1/4) Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2] Sqrt[1 + x^3])
*)
For x > -1, the hypergeometric solution is equal to int2 - (int2 /. x -> 0). However, the hypergeometric solution has simpler branch cuts and differs from this (by a locally constant function) on the whole complex plane. Consequently, the two solutions cannot be simplified into each other. The elliptic function solution can be simplified with the assumption x > -1:
FullSimplify[int2, x > -1]
(*
2 Sqrt[1 + 2/Sqrt[3]]
EllipticF[ArcSin[1 - (2 Sqrt[3])/(1 + Sqrt[3] + x)], -7 - 4 Sqrt[3]]
*)
The hypergeometric solution cannot be simplified into this form under the assumption, since, as an expression tree, it is simpler than the elliptic function expression. I don't know how to get an elliptic function solution using just the built-in functionality.
