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http://functions.wolfram.com/01.20.21.2121.01
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Integrate[Cosh[a z] Sqrt[Cosh[2 a z]], z] ==
(Sqrt[2] ArcSinh[Sqrt[2] Sinh[a z]] + 2 Sqrt[Cosh[2 a z]] Sinh[a z])/(4 a)
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List["Cosh", "[", RowBox[List["2", "a", " ", "z"]], "]"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", FractionBox[RowBox[List[RowBox[List[SqrtBox["2"], " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox["2"], " ", RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]]]], "]"]]]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "a", " ", "z"]], "]"]]], " ", RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], RowBox[List["4", " ", "a"]]]]]]]
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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> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> cosh </mi> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> cosh </mi> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <cosh /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arcsinh /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <sinh /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <sinh /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </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[RowBox[List["Cosh", "[", RowBox[List["a_", " ", "z_"]], "]"]], " ", SqrtBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "a_", " ", "z_"]], "]"]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[SqrtBox["2"], " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox["2"], " ", RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]]]], "]"]]]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", "a", " ", "z"]], "]"]]], " ", RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], RowBox[List["4", " ", "a"]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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