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 ArcCoth

 http://functions.wolfram.com/01.28.16.0181.01

 Input Form

 ArcCoth[x] + ArcCosh[y] == ArcCoth[ ((-1)^Floor[1/2 - Arg[(y + (Sqrt[y - 1] Sqrt[y + 1])/x)/Sqrt[1 - 1/x^2]]/ Pi] (y/x + Sqrt[y - 1] Sqrt[y + 1]))/(Sqrt[1 - 1/x^2] Sqrt[1 + (y/x + Sqrt[y - 1] Sqrt[y + 1])^2/(1 - 1/x^2)])] - (1/2) Pi I (-1)^(Floor[1/2 - Arg[(y + (Sqrt[y - 1] Sqrt[y + 1])/x)/Sqrt[1 - 1/x^2]]/ Pi] + Floor[1/2 - Arg[1 - 1/x^2]/(2 Pi) + Arg[(I y)/x + I Sqrt[-1 + y] Sqrt[1 + y]]/Pi]) - (1/4) (1 + (-1)^(-Floor[-(Arg[1 - y]/(2 Pi))])) Pi I (2 (1 + (-1)^Floor[1/2 - Arg[(y + (I Sqrt[1 - y^2])/x)/Sqrt[1 - 1/x^2]]/ Pi]) Floor[(Arg[(I - I/x)/Sqrt[1 - 1/x^2]] + Arg[I y + Sqrt[1 - y^2]])/(2 Pi)] + (-1)^Floor[1/2 - Arg[(y + (I Sqrt[1 - y^2])/x)/Sqrt[1 - 1/x^2]]/Pi] - 2 (-1 + (-1)^Floor[1/2 - Arg[(y + (I Sqrt[1 - y^2])/x)/Sqrt[1 - 1/x^2]]/ Pi]) Floor[1/2 - (Arg[(I - I/x)/Sqrt[1 - 1/x^2]] + Arg[I y + Sqrt[1 - y^2]])/(2 Pi)] - 1) + (1/4) (1 - (-1)^(-Floor[-(Arg[1 - y]/(2 Pi))])) Pi I (2 (1 + (-1)^Floor[1/2 - Arg[(y - (I Sqrt[1 - y^2])/x)/Sqrt[1 - 1/x^2]]/ Pi]) Floor[(Arg[(I + I/x)/Sqrt[1 - 1/x^2]] + Arg[I y + Sqrt[1 - y^2]])/(2 Pi)] + (-1)^Floor[1/2 - Arg[(y - (I Sqrt[1 - y^2])/x)/Sqrt[1 - 1/x^2]]/Pi] - 2 (-1 + (-1)^Floor[1/2 - Arg[(y - (I Sqrt[1 - y^2])/x)/Sqrt[1 - 1/x^2]]/ Pi]) Floor[1/2 - (Arg[(I + I/x)/Sqrt[1 - 1/x^2]] + Arg[I y + Sqrt[1 - y^2]])/(2 Pi)] - 1)

 Standard Form

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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[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], "]"]]]], 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["y", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], "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["y", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], "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[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]], "-", "1"]], ")"]]]]]]]]]]

 MathML Form

 coth - 1 ( x ) + cosh - 1 ( y ) coth - 1 ( ( - 1 ) 1 2 - arg ( y + y - 1 y + 1 x 1 - 1 x 2 ) π ( y x + y - 1 y + 1 ) 1 - 1 x 2 ( y x + y - 1 y + 1 ) 2 1 - 1 x 2 + 1 ) - 1 2 π ( - 1 ) - arg ( 1 - 1 x 2 ) 2 π + arg ( y x + y - 1 y + 1 ) π + 1 2 + 1 2 - arg ( y + y - 1 y + 1 x 1 - 1 x 2 ) π - 1 4 ( 1 + ( - 1 ) - - arg ( 1 - y ) 2 π ) π ( 2 ( 1 + ( - 1 ) 1 2 - arg ( y + 1 - y 2 x 1 - 1 x 2 ) π ) arg ( - x 1 - 1 x 2 ) + arg ( y + 1 - y 2 ) 2 π + ( - 1 ) 1 2 - arg ( y + 1 - y 2 x 1 - 1 x 2 ) π - 2 ( - 1 + ( - 1 ) 1 2 - arg ( y + 1 - y 2 x 1 - 1 x 2 ) π ) 1 2 - arg ( - x 1 - 1 x 2 ) + arg ( y + 1 - y 2 ) 2 π - 1 ) + 1 4 ( 1 - ( - 1 ) - - arg ( 1 - y ) 2 π ) π ( 2 ( 1 + ( - 1 ) 1 2 - arg ( y - 1 - y 2 x 1 - 1 x 2 ) π ) arg ( + x 1 - 1 x 2 ) + arg ( y + 1 - y 2 ) 2 π + ( - 1 ) 1 2 - arg ( y - 1 - y 2 x 1 - 1 x 2 ) π - 2 ( - 1 + ( - 1 ) 1 2 - arg ( y - 1 - y 2 x 1 - 1 x 2 ) π ) 1 2 - arg ( + x 1 - 1 x 2 ) + arg ( y + 1 - y 2 ) 2 π - 1 ) x y -1 1 2 -1 y y -1 1 2 y 1 1 2 x -1 1 -1 1 x 2 -1 1 2 -1 -1 y x -1 y -1 1 2 y 1 1 2 1 -1 1 x 2 -1 1 2 y x -1 y -1 1 2 y 1 1 2 2 1 -1 1 x 2 -1 -1 1 1 2 -1 -1 1 2 -1 -1 1 -1 1 x 2 -1 2 -1 y x -1 y -1 1 2 y 1 1 2 -1 1 2 1 2 -1 y y -1 1 2 y 1 1 2 x -1 1 -1 1 x 2 -1 1 2 -1 -1 -1 1 4 1 -1 -1 -1 1 -1 y 2 -1 2 1 -1 1 2 -1 y 1 -1 y 2 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 1 -1 y 2 1 2 2 -1 -1 1 2 -1 y 1 -1 y 2 1 2 x -1 1 -1 1 x 2 -1 1 2 -1 -1 -1 2 -1 -1 1 2 -1 y 1 -1 y 2 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 1 -1 y 2 1 2 2 -1 -1 1 4 1 -1 -1 -1 -1 1 -1 y 2 -1 2 1 -1 1 2 -1 y -1 1 -1 y 2 1 2 x -1 1 -1 1 x 2 -1 1 2 -1 -1 x -1 1 -1 1 x 2 -1 1 2 -1 y 1 -1 y 2 1 2 2 -1 -1 1 2 -1 y -1 1 -1 y 2 1 2 x -1 1 -1 1 x 2 -1 1 2 -1 -1 -1 2 -1 -1 1 2 -1 y -1 1 -1 y 2 1 2 x -1 1 -1 1 x 2 -1 1 2 -1 -1 1 2 -1 x -1 1 -1 1 x 2 -1 1 2 -1 y 1 -1 y 2 1 2 2 -1 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcCoth", "[", "x_", "]"]], "+", RowBox[List["ArcCosh", "[", "y_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["ArcCoth", "[", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List["y", "+", FractionBox[RowBox[List[SqrtBox[RowBox[List["y", "-", "1"]]], " ", SqrtBox[RowBox[List["y", "+", "1"]]]]], "x"]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]]], "]"]], "\[Pi]"]]], "]"]]], " ", RowBox[List["(", RowBox[List[FractionBox["y", "x"], "+", RowBox[List[SqrtBox[RowBox[List["y", "-", "1"]]], " ", SqrtBox[RowBox[List["y", "+", "1"]]]]]]], ")"]]]], RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", SqrtBox[RowBox[List["1", "+", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02