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http://functions.wolfram.com/01.30.21.0033.01
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Integrate[z ArcSech[a z]^2, z] ==
-((Sqrt[(1 - a z)/(1 + a z)] (1 + a z) ArcSech[a z])/a^2) +
(1/2) z^2 ArcSech[a z]^2 - Log[z]/a^2
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List["z", " ", SuperscriptBox[RowBox[List["ArcSech", "[", RowBox[List["a", " ", "z"]], "]"]], "2"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SqrtBox[FractionBox[RowBox[List["1", "-", RowBox[List["a", " ", "z"]]]], RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]], ")"]], " ", RowBox[List["ArcSech", "[", RowBox[List["a", " ", "z"]], "]"]]]], SuperscriptBox["a", "2"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["z", "2"], " ", SuperscriptBox[RowBox[List["ArcSech", "[", RowBox[List["a", " ", "z"]], "]"]], "2"]]], "-", FractionBox[RowBox[List["Log", "[", "z", "]"]], SuperscriptBox["a", "2"]]]]]]]]
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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> <mi> z </mi> <mo> ⁢ </mo> <msup> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mfrac> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </mfrac> </msqrt> </mrow> <mo> - </mo> <mfrac> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <arcsech /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <arcsech /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <arcsech /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ln /> <ci> z </ci> </apply> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </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["z_", " ", SuperscriptBox[RowBox[List["ArcSech", "[", RowBox[List["a_", " ", "z_"]], "]"]], "2"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SqrtBox[FractionBox[RowBox[List["1", "-", RowBox[List["a", " ", "z"]]]], RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]], ")"]], " ", RowBox[List["ArcSech", "[", RowBox[List["a", " ", "z"]], "]"]]]], SuperscriptBox["a", "2"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["z", "2"], " ", SuperscriptBox[RowBox[List["ArcSech", "[", RowBox[List["a", " ", "z"]], "]"]], "2"]]], "-", FractionBox[RowBox[List["Log", "[", "z", "]"]], SuperscriptBox["a", "2"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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