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

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

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> y </mi> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> </mrow> <mrow> <mrow> <msqrt> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <mi> y </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> + </mo> <msqrt> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> y </mi> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> </mrow> <mrow> <mrow> <msqrt> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <mi> y </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msqrt> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> - </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> y </mi> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> </mrow> <mrow> <mrow> <msqrt> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <mi> y </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msqrt> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> - </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mi> &#960; </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> y </mi> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <arctanh /> <ci> x </ci> </apply> <apply> <arcsinh /> <ci> y </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> y </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> x </ci> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> x </ci> </apply> <ci> y </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arcsin /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <pi /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> y </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> x </ci> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> x </ci> </apply> <ci> y </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <arg /> <apply> <times /> <apply> <plus /> <imaginaryi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> x </ci> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <arg /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <pi /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> y </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> x </ci> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> x </ci> </apply> <ci> y </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <arg /> <apply> <times /> <apply> <plus /> <imaginaryi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> x </ci> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <arg /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> y </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> y </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> x </ci> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> x </ci> </apply> <ci> y </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcTanh", "[", "x_", "]"]], "+", RowBox[List["ArcSinh", "[", "y_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", SqrtBox[FractionBox[RowBox[List[SuperscriptBox["y", "2"], "+", RowBox[List["2", " ", "x", " ", "y", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]], "+", RowBox[List[SuperscriptBox["x", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["y", "2"]]], ")"]]]]]], RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["x", "2"]]]]]]], ")"]], " ", RowBox[List["ArcSin", "[", FractionBox[RowBox[List[RowBox[List["x", " ", "y"]], "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]], "]"]]]], RowBox[List["y", "+", RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]]]], "-", FractionBox[RowBox[List["\[Pi]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", SqrtBox[FractionBox[RowBox[List[SuperscriptBox["y", "2"], "+", RowBox[List["2", " ", "x", " ", "y", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]], "+", RowBox[List[SuperscriptBox["x", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["y", "2"]]], ")"]]]]]], RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["x", "2"]]]]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["y", "+", RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]]], ")"]]]]], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", SqrtBox[FractionBox[RowBox[List[SuperscriptBox["y", "2"], "+", RowBox[List["2", " ", "x", " ", "y", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]], "+", RowBox[List[SuperscriptBox["x", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["y", "2"]]], ")"]]]]]], RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["x", "2"]]]]]]], RowBox[List["y", "+", RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]]]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Arg", "[", FractionBox[RowBox[List["\[ImaginaryI]", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]], "]"]], "+", RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "y"]], "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", SqrtBox[FractionBox[RowBox[List[SuperscriptBox["y", "2"], "+", RowBox[List["2", " ", "x", " ", "y", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]], "+", RowBox[List[SuperscriptBox["x", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["y", "2"]]], ")"]]]]]], RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["x", "2"]]]]]]], RowBox[List["y", "+", RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]]]]]], ")"]], " ", RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["-", "\[Pi]"]], "+", RowBox[List["Arg", "[", FractionBox[RowBox[List["\[ImaginaryI]", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]], "]"]], "+", RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "y"]], "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]]]]]]]

2007-05-02