Carlson symmetric elliptic integral of the second kind R_G
CarlsonRG[x, y, z]
CarlsonRG[3., 5., 11.]
→ 2.4807804428774662 Sin[ϕ]CarlsonRG[Cos[ϕ]^2, 1 - m Sin[ϕ]^2, 1] /. {m -> 0.3, ϕ -> Pi / 5.}
→ 1.08238725326179Plot[{CarlsonRG[t, 2, 3], CarlsonRG[1, t, 3], CarlsonRG[1, 2, t]}, {t, 0, 6}, PlotLegends -> "Expressions"]
→ -Graphics-