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http://functions.wolfram.com/01.14.21.0004.01
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Integrate[z^n ArcTan[z], z] == (-(z^(n + 1)/(2 (n + 1))))
(z LerchPhi[-z^2, 1, (n + 2)/2] - 2 ArcTan[z])
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z", "n"], " ", RowBox[List["ArcTan", "[", "z", "]"]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["z", RowBox[List["n", "+", "1"]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]]]]], RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["LerchPhi", "[", RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], ",", "1", ",", FractionBox[RowBox[List["n", "+", "2"]], "2"]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["ArcTan", "[", "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> <mo> ∫ </mo> <mrow> <mrow> <msup> <mi> z </mi> <mi> n </mi> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> z </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ⁢ </mo> <semantics> <mrow> <mi> Φ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> , </mo> <mn> 1 </mn> <mo> , </mo> <mfrac> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[CapitalPhi]", "(", RowBox[List[TagBox[RowBox[List["-", SuperscriptBox["z", "2"]]], Rule[Editable, True]], ",", TagBox["1", Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["n", "+", "2"]], "2"], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[LerchPhi[Slot[1], Slot[2], Slot[3]]]]] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> <apply> <arctan /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <ci> LerchPhi </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arctan /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", RowBox[List["ArcTan", "[", "z_", "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["n", "+", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["LerchPhi", "[", RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], ",", "1", ",", FractionBox[RowBox[List["n", "+", "2"]], "2"]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["ArcTan", "[", "z", "]"]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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