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 ArcTanh

 http://functions.wolfram.com/01.27.27.2146.01

 Input Form

 ArcTanh[Sqrt[z/(z + 1)]] == ArcSinh[Sqrt[z]] + ((Pi I)/2) (Sqrt[z + 1] Sqrt[1/(z + 1)] - 1)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["ArcTanh", "[", SqrtBox[FractionBox["z", RowBox[List["z", "+", "1"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["ArcSinh", "[", SqrtBox["z"], "]"]], "+", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"], RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["z", "+", "1"]]], SqrtBox[FractionBox["1", RowBox[List["z", "+", "1"]]]]]], "-", "1"]], ")"]]]]]]]]]]

 MathML Form

 tanh - 1 ( z z + 1 ) sinh - 1 ( z ) + π 2 ( z + 1 1 z + 1 - 1 ) z z 1 -1 1 2 z 1 2 2 -1 z 1 1 2 1 z 1 -1 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcTanh", "[", SqrtBox[FractionBox["z_", RowBox[List["z_", "+", "1"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["ArcSinh", "[", SqrtBox["z"], "]"]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["z", "+", "1"]]], " ", SqrtBox[FractionBox["1", RowBox[List["z", "+", "1"]]]]]], "-", "1"]], ")"]]]]]]]]]]

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

 2003-08-21