ArcTan[x] + I ArcSinh[y] == -ArcTan[(Sqrt[1 + x^2] Sqrt[1 - (I x y - Sqrt[1 + y^2])^2/(1 + x^2)])/ ((-1)^Floor[1/2 - Arg[(I y + x Sqrt[1 + y^2])/Sqrt[1 + x^2]]/Pi] (I x y - Sqrt[1 + y^2]))] + (1/2) Pi ((-1)^(Floor[1/2 - Arg[1 + x^2]/(2 Pi) + Arg[I x y - Sqrt[1 + y^2]]/Pi] + Floor[1/2 + Arg[1 + x^2]/(2 Pi) - Arg[I y + x Sqrt[1 + y^2]]/Pi] - 2 Floor[1/2 + Arg[1 + x^2]/(4 Pi) - Arg[I y + x Sqrt[1 + y^2]]/ (2 Pi)]) + (-1)^Floor[1/2 - Arg[(I y + x Sqrt[1 + y^2])/Sqrt[1 + x^2]]/Pi] + 2 (1 + (-1)^Floor[1/2 - Arg[(I y + x Sqrt[1 + y^2])/Sqrt[1 + x^2]]/Pi]) Floor[(Arg[(I - x)/Sqrt[1 + x^2]] + Arg[-y + Sqrt[1 + y^2]])/(2 Pi)] - 2 (-1 + (-1)^Floor[1/2 - Arg[(I y + x Sqrt[1 + y^2])/Sqrt[1 + x^2]]/Pi]) Floor[1/2 - (Arg[(I - x)/Sqrt[1 + x^2]] + Arg[-y + Sqrt[1 + y^2]])/ (2 Pi)])

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