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 ArcSech

 http://functions.wolfram.com/01.30.16.0180.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

 2007-05-02