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

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