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

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Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcCoth", "[", "x_", "]"]], "+", RowBox[List["ArcSin", "[", "y_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["-", RowBox[List["Arg", "[", RowBox[List["1", "-", FractionBox["1", "x"]]], "]"]]]], "+", RowBox[List["Arg", "[", RowBox[List["1", "+", FractionBox["1", "x"]]], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Arg", "[", FractionBox[RowBox[List["x", "-", "1"]], RowBox[List["x", "+", "1"]]], "]"]]]], "-", RowBox[List["Arg", "[", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], ")"]], RowBox[List["-", "\[ImaginaryI]"]]], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Im", "[", RowBox[List["Log", "[", FractionBox[RowBox[List["x", "-", "1"]], RowBox[List["x", "+", "1"]]], "]"]], "]"]]]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Re", "[", RowBox[List["Log", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], "]"]], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]], "+", RowBox[List["Log", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], ")"]], RowBox[List["-", "\[ImaginaryI]"]]], SqrtBox[FractionBox[RowBox[List["x", "-", "1"]], RowBox[List["x", "+", "1"]]]]], "]"]]]]]]]]

2007-05-02