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

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