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 ArcTanh

 http://functions.wolfram.com/01.27.16.0176.01

 Input Form

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

 Standard Form

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 MathML Form

 tanh - 1 ( x ) + sech - 1 ( y ) tanh - 1 ( ( - 1 ) - 1 2 - arg ( 1 y - 1 1 + 1 y x + 1 y 1 - x 2 ) π 1 - x 2 ( x y + 1 y - 1 1 + 1 y ) 2 1 - x 2 + 1 x y + 1 y - 1 1 + 1 y ) - 1 2 ( - 1 ) - arg ( 1 - x 2 ) 2 π + arg ( 1 y - 1 1 + 1 y + x y ) π + 1 2 + 1 2 - arg ( 1 y - 1 1 + 1 y x + 1 y 1 - x 2 ) π π - 1 4 ( 1 + ( - 1 ) - - arg ( 1 - 1 y ) 2 π ) π ( 2 ( 1 + ( - 1 ) 1 2 - arg ( 1 - 1 y 2 x + 1 y 1 - x 2 ) π ) arg ( - x 1 - x 2 ) + arg ( 1 - 1 y 2 + y ) 2 π + ( - 1 ) 1 2 - arg ( 1 - 1 y 2 x + 1 y 1 - x 2 ) π - 2 ( - 1 + ( - 1 ) 1 2 - arg ( 1 - 1 y 2 x + 1 y 1 - x 2 ) π ) 1 2 - arg ( - x 1 - x 2 ) + arg ( 1 - 1 y 2 + y ) 2 π - 1 ) + 1 4 ( 1 - ( - 1 ) - - arg ( 1 - 1 y ) 2 π ) π ( 2 ( 1 + ( - 1 ) 1 2 - arg ( 1 y - x 1 - 1 y 2 1 - x 2 ) π ) arg ( x + 1 - x 2 ) + arg ( 1 - 1 y 2 + y ) 2 π + ( - 1 ) 1 2 - arg ( 1 y - x 1 - 1 y 2 1 - x 2 ) π - 2 ( - 1 + ( - 1 ) 1 2 - arg ( 1 y - x 1 - 1 y 2 1 - x 2 ) π ) 1 2 - arg ( x + 1 - x 2 ) + arg ( 1 - 1 y 2 + y ) 2 π - 1 ) x y -1 -1 1 2 -1 1 y -1 -1 1 2 1 1 y -1 1 2 x 1 y -1 1 -1 x 2 1 2 -1 -1 1 -1 x 2 1 2 x y -1 1 y -1 -1 1 2 1 1 y -1 1 2 2 1 -1 x 2 -1 1 1 2 x y -1 1 y -1 -1 1 2 1 1 y -1 1 2 -1 -1 1 2 -1 -1 1 -1 x 2 2 -1 1 y -1 -1 1 2 1 1 y -1 1 2 x y -1 -1 1 2 1 2 -1 1 y -1 -1 1 2 1 1 y -1 1 2 x 1 y -1 1 -1 x 2 1 2 -1 -1 -1 1 4 1 -1 -1 -1 1 -1 1 y -1 2 -1 2 1 -1 1 2 -1 1 -1 1 y 2 -1 1 2 x 1 y -1 1 -1 x 2 1 2 -1 -1 -1 x 1 -1 x 2 1 2 -1 1 -1 1 y 2 -1 1 2 y -1 2 -1 -1 1 2 -1 1 -1 1 y 2 -1 1 2 x 1 y -1 1 -1 x 2 1 2 -1 -1 -1 2 -1 -1 1 2 -1 1 -1 1 y 2 -1 1 2 x 1 y -1 1 -1 x 2 1 2 -1 -1 1 2 -1 -1 x 1 -1 x 2 1 2 -1 1 -1 1 y 2 -1 1 2 y -1 2 -1 -1 1 4 1 -1 -1 -1 -1 1 -1 1 y -1 2 -1 2 1 -1 1 2 -1 1 y -1 -1 x 1 -1 1 y 2 -1 1 2 1 -1 x 2 1 2 -1 -1 x 1 -1 x 2 1 2 -1 1 -1 1 y 2 -1 1 2 y -1 2 -1 -1 1 2 -1 1 y -1 -1 x 1 -1 1 y 2 -1 1 2 1 -1 x 2 1 2 -1 -1 -1 2 -1 -1 1 2 -1 1 y -1 -1 x 1 -1 1 y 2 -1 1 2 1 -1 x 2 1 2 -1 -1 1 2 -1 x 1 -1 x 2 1 2 -1 1 -1 1 y 2 -1 1 2 y -1 2 -1 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcTanh", "[", "x_", "]"]], "+", RowBox[List["ArcSech", "[", "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[FractionBox["1", "y"], "-", "1"]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "y"]]]], " ", "x"]], "+", FractionBox["1", "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[FractionBox["x", "y"], "+", RowBox[List[SqrtBox[RowBox[List[FractionBox["1", "y"], "-", "1"]]], " ", SqrtBox[RowBox[List["1", "+", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02