ArcTanh[x] + I ArcCsc[y] == ArcTanh[(Sqrt[1 - x^2] Sqrt[1 - (-((I x)/y) - Sqrt[1 - 1/y^2])^2/(1 - x^2)])/ ((-1)^Floor[1/2 - Arg[(1/y - I x Sqrt[1 - 1/y^2])/Sqrt[1 - x^2]]/Pi] (x/y - I Sqrt[1 - 1/y^2]))] + (1/2) Pi I ((-1)^(Floor[1/2 - Arg[(1/y - I x Sqrt[1 - 1/y^2])/Sqrt[1 - x^2]]/Pi] + Floor[1/2 - Arg[1 - x^2]/(2 Pi) + Arg[-Sqrt[1 - 1/y^2] - (I x)/y]/ Pi]) + 2 (1 + (-1)^Floor[1/2 - Arg[(1/y - I x Sqrt[1 - 1/y^2])/Sqrt[1 - x^2]]/ Pi]) Floor[(Arg[(I x + I)/Sqrt[1 - x^2]] + Arg[Sqrt[1 - 1/y^2] + I/y])/(2 Pi)] + (-1)^Floor[1/2 - Arg[(1/y - I x Sqrt[1 - 1/y^2])/Sqrt[1 - x^2]]/Pi] - 2 (-1 + (-1)^Floor[1/2 - Arg[(1/y - I x Sqrt[1 - 1/y^2])/Sqrt[1 - x^2]]/ Pi]) Floor[1/2 - (Arg[(I x + I)/Sqrt[1 - x^2]] + Arg[Sqrt[1 - 1/y^2] + I/y])/(2 Pi)])

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