Integrate[Sqrt[2]/(-8 + Exp[y] + 2 Exp[-y]), {y, 0, t}]

Solve[-((ArcTanh[3/Sqrt[14]] + ArcTanh[(-4 + E^t)/Sqrt[14]])/Sqrt[7]) == y, t]

t[y_] = Log[4 - Sqrt[14] Tanh[Sqrt[7] y + ArcTanh[3/Sqrt[14]]]];
Plot[t[y], {y, -1, 1}]

t[0]

t[1]

t'[0]