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http://functions.wolfram.com/01.20.21.4279.01
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Integrate[(A + B Sinh[e z] + C Cosh[e z])/(a + b Sinh[e z] + b Cosh[e z]),
z] == (1/(4 a^2 b e))
((2 a b (B - C) + E^(e z) ((2 a A b + b^2 (B - C) + a^2 (B + C)) e z +
2 (-2 a A b + b^2 (-B + C) + a^2 (B + C))
Log[(a + b E^(e z))/E^((e z)/2)]))/E^(e 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> <mfrac> <mrow> <mi> A </mi> <mo> + </mo> <mrow> <mi> B </mi> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> C </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> e </mi> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> e </mi> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> B </mi> <mo> - </mo> <mi> C </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> B </mi> <mo> + </mo> <mi> C </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> A </mi> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> B </mi> <mo> - </mo> <mi> C </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> B </mi> <mo> + </mo> <mi> C </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> A </mi> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> C </mi> <mo> - </mo> <mi> B </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <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> <plus /> <ci> A </ci> <apply> <times /> <ci> B </ci> <apply> <sinh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> C </ci> <apply> <cosh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <sinh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <ci> b </ci> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> b </ci> <apply> <plus /> <ci> B </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> C </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> B </ci> <ci> C </ci> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> A </ci> <ci> b </ci> <ci> a </ci> </apply> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> B </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> C </ci> </apply> </apply> </apply> </apply> <ci> e </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> B </ci> <ci> C </ci> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> A </ci> <ci> b </ci> <ci> a </ci> </apply> </apply> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> C </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> B </ci> </apply> </apply> </apply> </apply> <apply> <ln /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> e </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </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[FractionBox[RowBox[List["A_", "+", RowBox[List["B_", " ", RowBox[List["Sinh", "[", RowBox[List["e_", " ", "z_"]], "]"]]]], "+", RowBox[List["C", " ", RowBox[List["Cosh", "[", RowBox[List["e_", " ", "z_"]], "]"]]]]]], RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Sinh", "[", RowBox[List["e_", " ", "z_"]], "]"]]]], "+", RowBox[List["b_", " ", RowBox[List["Cosh", "[", RowBox[List["e_", " ", "z_"]], "]"]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "e"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a", " ", "b", " ", RowBox[List["(", RowBox[List["B", "-", "C"]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a", " ", "A", " ", "b"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List["B", "-", "C"]], ")"]]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", RowBox[List["(", RowBox[List["B", "+", "C"]], ")"]]]]]], ")"]], " ", "e", " ", "z"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a", " ", "A", " ", "b"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "B"]], "+", "C"]], ")"]]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", RowBox[List["(", RowBox[List["B", "+", "C"]], ")"]]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["e", " ", "z"]], ")"]]]]], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]], ")"]]]], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["4", " ", SuperscriptBox["a", "2"], " ", "b", " ", "e"]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
