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http://functions.wolfram.com/01.20.21.4274.01
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Integrate[(A + B Sinh[e z] + C Cosh[e z])/(a Sinh[e z] + b Cosh[e z]), z] ==
-(2 A (a - b) Sqrt[a + b] ArcTan[(a + b Tanh[(e z)/2])/
(Sqrt[-a + b] Sqrt[a + b])] + Sqrt[-a + b] ((a B - b C) e z +
((-b) B + a C) Log[b Cosh[e z] + a Sinh[e z]]))/
((-a + b)^(3/2) (a + b) e)
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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> <mrow> <mi> a </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> <mrow> <mo> - 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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[RowBox[List["a_", " ", RowBox[List["Sinh", "[", RowBox[List["e_", " ", "z_"]], "]"]]]], "+", RowBox[List["b_", " ", RowBox[List["Cosh", "[", RowBox[List["e_", " ", "z_"]], "]"]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["2", " ", "A", " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SqrtBox[RowBox[List["a", "+", "b"]]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Tanh", "[", FractionBox[RowBox[List["e", " ", "z"]], "2"], "]"]]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b"]]], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", "b"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["a", " ", "B"]], "-", RowBox[List["b", " ", "C"]]]], ")"]], " ", "e", " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", "B"]], "+", RowBox[List["a", " ", "C"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["a", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], "]"]]]]]], ")"]]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "e"]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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