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http://functions.wolfram.com/01.23.21.0416.01
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Integrate[Tanh[Pi/4 + z]^2 Csch[Pi/4 + z]^3, z] ==
-Log[Cosh[(1/2) (Pi/4 + z)]] + Log[Sinh[(1/2) (Pi/4 + z)]] + Sech[Pi/4 + z]
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", " ", RowBox[List[SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List[FractionBox["\[Pi]", "4"], "+", "z"]], "]"]], "2"], SuperscriptBox[RowBox[List["Csch", "[", RowBox[List[FractionBox["\[Pi]", "4"], "+", "z"]], "]"]], "3"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List["Cosh", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "4"], "+", "z"]], ")"]]]], "]"]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["Sinh", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "4"], "+", "z"]], ")"]]]], "]"]], "]"]], "+", RowBox[List["Sech", "[", RowBox[List[FractionBox["\[Pi]", "4"], "+", "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> <mrow> <msup> <mi> tanh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mi> π </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> csch </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mi> π </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mi> π </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mi> π </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> sech </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mi> π </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <apply> <tanh /> <apply> <plus /> <ci> z </ci> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <csch /> <apply> <plus /> <ci> z </ci> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <cosh /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> z </ci> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ln /> <apply> <sinh /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> z </ci> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <sech /> <apply> <plus /> <ci> z </ci> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </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[RowBox[List["Tanh", "[", RowBox[List[FractionBox["\[Pi]", "4"], "+", "z_"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["Csch", "[", RowBox[List[FractionBox["\[Pi]", "4"], "+", "z_"]], "]"]], "3"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List["Cosh", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "4"], "+", "z"]], ")"]]]], "]"]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["Sinh", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "4"], "+", "z"]], ")"]]]], "]"]], "]"]], "+", RowBox[List["Sech", "[", RowBox[List[FractionBox["\[Pi]", "4"], "+", "z"]], "]"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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