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 ArcTanh

 http://functions.wolfram.com/01.27.16.0210.01

 Input Form

 a ArcCsc[x] + b Log[y] == I Pi (1 - (-1)^(Floor[-(Arg[y^b/(Sqrt[1 - 1/x^2] + I/x)^(I a) + 1]/ (2 Pi))] - Floor[-(Arg[y^b/(Sqrt[1 - 1/x^2] + I/x)^(I a)]/ (2 Pi))])) + I (-1)^(Floor[-(Arg[y^b/(Sqrt[1 - 1/x^2] + I/x)^(I a)]/Pi)] + Floor[Arg[y^b/(Sqrt[1 - 1/x^2] + I/x)^(I a) - 1]/(2 Pi) - Arg[y^b/(Sqrt[1 - 1/x^2] + I/x)^(I a) + 1]/(2 Pi) + 1/2]) (ArcCsc[(2 y^b)/((Sqrt[1 - 1/x^2] + I/x)^(I a) (y^(2 b)/(Sqrt[1 - 1/x^2] + I/x)^(2 I a) + 1))] - Pi/2) - 2 I Pi (Floor[(-Arg[(Sqrt[1 - 1/x^2] + I/x)^((-I) a)] - Arg[y^b] + Pi)/ (2 Pi)] + Floor[(Pi - Im[b Log[y]])/(2 Pi)] + Floor[(Re[a Log[Sqrt[1 - 1/x^2] + I/x]] + Pi)/(2 Pi)])

 Standard Form

 Cell[BoxData[RowBox[List[" ", RowBox[List[RowBox[List[RowBox[List["a", " ", RowBox[List["ArcCsc", "[", "x", "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Log", "[", "y", "]"]]]]]], "\[Equal]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List[SuperscriptBox["y", "b"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "x"]]], ")"]], RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]]]]], "+", "1"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]], "-", RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List["Arg", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "x"]]], ")"]], RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]]], " ", SuperscriptBox["y", "b"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List["Arg", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "x"]]], ")"]], RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]]], " ", SuperscriptBox["y", "b"]]], "]"]], "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "x"]]], ")"]], RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]]], " ", SuperscriptBox["y", "b"]]], "-", "1"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List[SuperscriptBox["y", "b"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "x"]]], ")"]], RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]]]]], "+", "1"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "+", FractionBox["1", "2"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["ArcCsc", "[", FractionBox[RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "x"]]], ")"]], RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]]], " ", SuperscriptBox["y", "b"]]], RowBox[List[RowBox[List[SuperscriptBox["y", RowBox[List["2", " ", "b"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "x"]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "a"]]]]], "+", "1"]]], "]"]], "-", FractionBox["\[Pi]", "2"]]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["-", RowBox[List["Arg", "[", SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "x"]]], ")"]], RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]]], "]"]]]], "-", RowBox[List["Arg", "[", SuperscriptBox["y", "b"], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Im", "[", RowBox[List["b", " ", RowBox[List["Log", "[", "y", "]"]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Re", "[", RowBox[List["a", " ", RowBox[List["Log", "[", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["x", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "x"]]], "]"]]]], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]]]]]]]]]]

 MathML Form

 a csc - 1 ( x ) + b log ( y ) π ( 1 - ( - 1 ) - arg ( y b ( 1 - 1 x 2 + x ) - a + 1 ) 2 π - - arg ( ( 1 - 1 x 2 + x ) - a y b ) 2 π ) + ( - 1 ) arg ( ( 1 - 1 x 2 + x ) - a y b - 1 ) 2 π + 1 2 - arg ( y b ( 1 - 1 x 2 + x ) - a + 1 ) 2 π + - arg ( ( 1 - 1 x 2 + x ) - a y b ) π ( csc - 1 ( 2 ( 1 - 1 x 2 + x ) - a y b y 2 b ( 1 - 1 x 2 + x ) - 2 a + 1 ) - π 2 ) - 2 π ( - arg ( ( 1 - 1 x 2 + x ) - a ) - arg ( y b ) + π 2 π + Re ( a log ( 1 - 1 x 2 + x ) ) + π 2 π + π - Im ( b log ( y ) ) 2 π ) a x b y 1 -1 -1 -1 y b 1 -1 1 x 2 -1 1 2 x -1 -1 a 1 2 -1 -1 -1 1 -1 1 x 2 -1 1 2 x -1 -1 a y b 2 -1 -1 1 -1 1 x 2 -1 1 2 x -1 -1 a y b -1 2 -1 1 2 -1 y b 1 -1 1 x 2 -1 1 2 x -1 -1 a 1 2 -1 -1 1 -1 1 x 2 -1 1 2 x -1 -1 a y b -1 2 1 -1 1 x 2 -1 1 2 x -1 -1 a y b y 2 b 1 -1 1 x 2 -1 1 2 x -1 -2 a 1 -1 -1 2 -1 -1 2 -1 1 -1 1 x 2 -1 1 2 x -1 -1 a -1 y b 2 -1 a 1 -1 1 x 2 -1 1 2 x -1 2 -1 -1 b y 2 -1 [/itex]

 Rule Form

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

 2007-05-02