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 ArcTanh

 http://functions.wolfram.com/01.27.16.0172.01

 Input Form

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

 Standard Form

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

 tanh - 1 ( x ) + cosh - 1 ( y ) tanh - 1 ( ( - 1 ) - 1 2 - arg ( y - 1 y + 1 x + y 1 - x 2 ) π 1 - x 2 1 - ( x y + y - 1 y + 1 ) 2 1 - x 2 x y + y - 1 y + 1 ) - 1 2 ( - 1 ) - arg ( 1 - x 2 ) 2 π + arg ( x y + y - 1 y + 1 ) π + 1 2 + 1 2 - arg ( y - 1 y + 1 x + y 1 - x 2 ) π π - 1 4 ( 1 + ( - 1 ) - - arg ( 1 - y ) 2 π ) π ( 2 ( 1 + ( - 1 ) 1 2 - arg ( 1 - y 2 x + y 1 - x 2 ) π ) arg ( - x 1 - x 2 ) + arg ( y + 1 - y 2 ) 2 π + ( - 1 ) 1 2 - arg ( 1 - y 2 x + y 1 - x 2 ) π - 2 ( - 1 + ( - 1 ) 1 2 - arg ( 1 - y 2 x + y 1 - x 2 ) π ) 1 2 - arg ( - x 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 - x 1 - y 2 1 - x 2 ) π ) arg ( x + 1 - x 2 ) + arg ( y + 1 - y 2 ) 2 π + ( - 1 ) 1 2 - arg ( y - x 1 - y 2 1 - x 2 ) π - 2 ( - 1 + ( - 1 ) 1 2 - arg ( y - x 1 - y 2 1 - x 2 ) π ) 1 2 - arg ( x + 1 - x 2 ) + arg ( y + 1 - y 2 ) 2 π - 1 ) x y -1 -1 1 2 -1 y -1 1 2 y 1 1 2 x y 1 -1 x 2 1 2 -1 -1 1 -1 x 2 1 2 1 -1 x y y -1 1 2 y 1 1 2 2 1 -1 x 2 -1 1 2 x y y -1 1 2 y 1 1 2 -1 -1 1 2 -1 -1 1 -1 x 2 2 -1 x y y -1 1 2 y 1 1 2 -1 1 2 1 2 -1 y -1 1 2 y 1 1 2 x y 1 -1 x 2 1 2 -1 -1 -1 1 4 1 -1 -1 -1 1 -1 y 2 -1 2 1 -1 1 2 -1 1 -1 y 2 1 2 x y 1 -1 x 2 1 2 -1 -1 -1 x 1 -1 x 2 1 2 -1 y 1 -1 y 2 1 2 2 -1 -1 1 2 -1 1 -1 y 2 1 2 x y 1 -1 x 2 1 2 -1 -1 -1 2 -1 -1 1 2 -1 1 -1 y 2 1 2 x y 1 -1 x 2 1 2 -1 -1 1 2 -1 -1 x 1 -1 x 2 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 x 1 -1 y 2 1 2 1 -1 x 2 1 2 -1 -1 x 1 -1 x 2 1 2 -1 y 1 -1 y 2 1 2 2 -1 -1 1 2 -1 y -1 x 1 -1 y 2 1 2 1 -1 x 2 1 2 -1 -1 -1 2 -1 -1 1 2 -1 y -1 x 1 -1 y 2 1 2 1 -1 x 2 1 2 -1 -1 1 2 -1 x 1 -1 x 2 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["ArcTanh", "[", "x_", "]"]], "+", RowBox[List["ArcCosh", "[", "y_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["ArcTanh", "[", FractionBox[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[SqrtBox[RowBox[List["y", "-", "1"]]], " ", SqrtBox[RowBox[List["y", "+", "1"]]], " ", "x"]], "+", "y"]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]], "]"]], "\[Pi]"]]], "]"]]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["y", "-", "1"]]], " ", SqrtBox[RowBox[List["y", "+", "1"]]]]]]], ")"]], 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02