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 ArcCoth

 http://functions.wolfram.com/01.28.16.0180.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcCoth", "[", "x", "]"]], "+", RowBox[List["ArcSinh", "[", "y", "]"]]]], "\[Equal]", RowBox[List[RowBox[List["ArcCoth", "[", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], "x"]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]]], "]"]], "\[Pi]"]]], "]"]]], " ", RowBox[List["(", RowBox[List[FractionBox["y", "x"], "+", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], ")"]]]], RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["y", "x"], "+", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], ")"]], "2"], RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]]]]]]]], "]"]], "-", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], "x"]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]]], "]"]], "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "+", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", FractionBox["y", "x"]]], "-", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]], "]"]], "\[Pi]"]]], "]"]]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], "x"]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]]], "]"]], "\[Pi]"]]], "]"]]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Arg", "[", FractionBox[RowBox[List["\[ImaginaryI]", "-", FractionBox["\[ImaginaryI]", "x"]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]]], "]"]], "+", RowBox[List["Arg", "[", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]], "-", "y"]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], "x"]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]]], "]"]], "\[Pi]"]]], "]"]]], "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], "x"]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]]], "]"]], "\[Pi]"]]], "]"]]]]], ")"]], " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List[RowBox[List["Arg", "[", FractionBox[RowBox[List["\[ImaginaryI]", "-", FractionBox["\[ImaginaryI]", "x"]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]]], "]"]], "+", RowBox[List["Arg", "[", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]], "-", "y"]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]]], ")"]]]]]]]]]]

 MathML Form

 coth - 1 ( x ) + sinh - 1 ( y ) coth - 1 ( ( - 1 ) 1 2 - arg ( y + y 2 + 1 x 1 - 1 x 2 ) π ( y x + y 2 + 1 ) 1 - 1 x 2 1 - ( y x + y 2 + 1 ) 2 1 - 1 x 2 ) - 1 2 π ( 2 ( 1 + ( - 1 ) 1 2 - arg ( y + y 2 + 1 x 1 - 1 x 2 ) π ) arg ( - x 1 - 1 x 2 ) + arg ( y 2 + 1 - y ) 2 π + ( - 1 ) - arg ( 1 - 1 x 2 ) 2 π + arg ( - y x - y 2 + 1 ) π + 1 2 + 1 2 - arg ( y + y 2 + 1 x 1 - 1 x 2 ) π + ( - 1 ) 1 2 - arg ( y + y 2 + 1 x 1 - 1 x 2 ) π - 2 ( - 1 + ( - 1 ) 1 2 - arg ( y + y 2 + 1 x 1 - 1 x 2 ) π ) 1 2 - arg ( - x 1 - 1 x 2 ) + arg ( y 2 + 1 - y ) 2 π ) x y -1 1 2 -1 y y 2 1 1 2 x -1 1 -1 1 x 2 -1 1 2 -1 -1 y x -1 y 2 1 1 2 1 -1 1 x 2 -1 1 2 1 -1 y x -1 y 2 1 1 2 2 1 -1 1 x 2 -1 -1 1 2 -1 -1 1 2 2 1 -1 1 2 -1 y y 2 1 1 2 x -1 1 -1 1 x 2 -1 1 2 -1 -1 -1 x -1 1 -1 1 x 2 -1 1 2 -1 y 2 1 1 2 -1 y 2 -1 -1 -1 1 -1 1 x 2 -1 2 -1 -1 y x -1 -1 y 2 1 1 2 -1 1 2 1 2 -1 y y 2 1 1 2 x -1 1 -1 1 x 2 -1 1 2 -1 -1 -1 1 2 -1 y y 2 1 1 2 x -1 1 -1 1 x 2 -1 1 2 -1 -1 -1 2 -1 -1 1 2 -1 y y 2 1 1 2 x -1 1 -1 1 x 2 -1 1 2 -1 -1 1 2 -1 -1 x -1 1 -1 1 x 2 -1 1 2 -1 y 2 1 1 2 -1 y 2 -1 [/itex]

 Rule Form

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

 Date Added to functions.wolfram.com (modification date)

 2007-05-02