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 ArcCoth

 http://functions.wolfram.com/01.28.16.0179.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2007-05-02