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http://functions.wolfram.com/01.20.21.4289.01
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Integrate[1/(a Sinh[e z]^2 + b Cosh[e z]^2)^3, z] ==
(1/(8 a^(5/2) b^(5/2) e)) ((3 a^2 - 2 a b + 3 b^2)
ArcTan[(Sqrt[a] Tanh[e z])/Sqrt[b]] +
(Sqrt[a] Sqrt[b] (a + b) ((-(a - 3 b)) (3 a - b) +
3 (a - b) (a + b) Cosh[2 e z]) Sinh[2 e z])/
(-a + b + (a + b) Cosh[2 e z])^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> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> sinh </mi> <mn> 2 </mn> </msup> <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> <msup> <mi> cosh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mrow> <mn> 5 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> b </mi> <mrow> <mn> 5 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <mi> e </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> - 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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a_", " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["e_", " ", "z_"]], "]"]], "2"]]], "+", RowBox[List["b_", " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["e_", " ", "z_"]], "]"]], "2"]]]]], ")"]], "3"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["3", " ", SuperscriptBox["a", "2"]]], "-", RowBox[List["2", " ", "a", " ", "b"]], "+", RowBox[List["3", " ", SuperscriptBox["b", "2"]]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[SqrtBox["a"], " ", RowBox[List["Tanh", "[", RowBox[List["e", " ", "z"]], "]"]]]], SqrtBox["b"]], "]"]]]], "+", FractionBox[RowBox[List[SqrtBox["a"], " ", SqrtBox["b"], " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["a", "-", RowBox[List["3", " ", "b"]]]], ")"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "a"]], "-", "b"]], ")"]]]], "+", RowBox[List["3", " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]], ")"]], "2"]]]], RowBox[List["8", " ", SuperscriptBox["a", RowBox[List["5", "/", "2"]]], " ", SuperscriptBox["b", RowBox[List["5", "/", "2"]]], " ", "e"]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
