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http://functions.wolfram.com/01.14.16.0248.01
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ArcTan[x] - ArcTanh[x] == Log[((I + x)/(I - x))^(I/2)
(Sqrt[1 - x]/Sqrt[1 + x])] /;
!IntervalMemberQ[Interval[{-Infinity, -1}], I z]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["ArcTan", "[", "x", "]"]], "-", RowBox[List["ArcTanh", "[", "x", "]"]]]], "\[Equal]", RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", "+", "x"]], RowBox[List["\[ImaginaryI]", "-", "x"]]], ")"]], FractionBox["\[ImaginaryI]", "2"]], FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "-", "x"]]], " "]], SqrtBox[RowBox[List["1", "+", "x"]]]]]], "]"]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", " ", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]], "]"]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo>  </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> + </mo> <mi> x </mi> </mrow> <mrow> <mi> ⅈ </mi> <mo> - </mo> <mi> x </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> ⅈ </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <mfrac> <mrow> <mtext> </mtext> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> x </mi> </mrow> </msqrt> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> x </mi> </mrow> </msqrt> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mtext> ​​ </mtext> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo>  </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> + </mo> <mi> x </mi> </mrow> <mrow> <mi> ⅈ </mi> <mo> - </mo> <mi> x </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> ⅈ </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <mfrac> <mrow> <mtext> </mtext> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> x </mi> </mrow> </msqrt> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> x </mi> </mrow> </msqrt> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mtext> ​​ </mtext> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcTan", "[", "x_", "]"]], "-", RowBox[List["ArcTanh", "[", "x_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Log", "[", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", "+", "x"]], RowBox[List["\[ImaginaryI]", "-", "x"]]], ")"]], RowBox[List["\[ImaginaryI]", "/", "2"]]], " ", SqrtBox[RowBox[List["1", "-", "x"]]]]], SqrtBox[RowBox[List["1", "+", "x"]]]], "]"]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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