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 Cosh

 http://functions.wolfram.com/01.20.21.0156.01

 Input Form

 Integrate[Cosh[d z]/(a z^2 + b z + c)^2, z] == ((d Cosh[((b + Sqrt[b^2 - 4 a c]) d)/(2 a)])/(b^2 - 4 a c) - (2 a Sinh[((b + Sqrt[b^2 - 4 a c]) d)/(2 a)])/(b^2 - 4 a c)^(3/2)) SinhIntegral[(d (b + Sqrt[b^2 - 4 a c] + 2 a z))/(2 a)] + ((d Cosh[((b - Sqrt[b^2 - 4 a c]) d)/(2 a)])/(b^2 - 4 a c) + (2 a Sinh[((b - Sqrt[b^2 - 4 a c]) d)/(2 a)])/(b^2 - 4 a c)^(3/2)) SinhIntegral[(d (b - Sqrt[b^2 - 4 a c] + 2 a z))/(2 a)] - ((2 a Cosh[((b - Sqrt[b^2 - 4 a c]) d)/(2 a)])/(b^2 - 4 a c)^(3/2) + (d Sinh[((b - Sqrt[b^2 - 4 a c]) d)/(2 a)])/(b^2 - 4 a c)) CosIntegral[(I d (b - Sqrt[b^2 - 4 a c] + 2 a z))/(2 a)] + ((2 a Cosh[((b + Sqrt[b^2 - 4 a c]) d)/(2 a)])/(b^2 - 4 a c)^(3/2) - (d Sinh[((b + Sqrt[b^2 - 4 a c]) d)/(2 a)])/(b^2 - 4 a c)) CosIntegral[(I d (b + Sqrt[b^2 - 4 a c] + 2 a z))/(2 a)] - ((b + 2 a z) Cosh[d z])/((b^2 - 4 a c) (c + z (b + a z)))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Cosh", "[", RowBox[List["d", " ", "z"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["b", " ", "z"]], "+", "c"]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["d", " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]]]], ")"]], " ", "d"]], RowBox[List["2", " ", "a"]]], "]"]]]], RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]], "-", FractionBox[RowBox[List["2", " ", "a", " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]]]], ")"]], " 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"z"]]]], ")"]]]], RowBox[List["2", " ", "a"]]], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "a", " ", "z"]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["d", " ", "z"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]]]], ")"]]]]]]]]]]]

 MathML Form

 cosh ( d z ) ( a z 2 + b z + c ) 2 z - ( b + 2 a z ) cosh ( d z ) ( b 2 - 4 a c ) ( c + z ( b + a z ) ) - ( 2 a cosh ( ( b - b 2 - 4 a c ) d 2 a ) ( b 2 - 4 a c ) 3 / 2 + d sinh ( ( b - b 2 - 4 a c ) d 2 a ) b 2 - 4 a c ) Ci ( d ( b + 2 a z - b 2 - 4 a c ) 2 a ) + ( 2 a cosh ( ( b + b 2 - 4 a c ) d 2 a ) ( b 2 - 4 a c ) 3 / 2 - d sinh ( ( b + b 2 - 4 a c ) d 2 a ) b 2 - 4 a c ) Ci ( d ( b + 2 a z + b 2 - 4 a c ) 2 a ) + ( d cosh ( ( b - b 2 - 4 a c ) d 2 a ) b 2 - 4 a c + 2 a sinh ( ( b - b 2 - 4 a c ) d 2 a ) ( b 2 - 4 a c ) 3 / 2 ) Shi ( d ( b + 2 a z - b 2 - 4 a c ) 2 a ) + ( d cosh ( ( b + b 2 - 4 a c ) d 2 a ) b 2 - 4 a c - 2 a sinh ( ( b + b 2 - 4 a c ) d 2 a ) ( b 2 - 4 a c ) 3 / 2 ) Shi ( d ( b + 2 a z + b 2 - 4 a c ) 2 a ) z d z a z 2 b z c 2 -1 -1 b 2 a z d z b 2 -1 4 a c c z b a z -1 -1 2 a b -1 b 2 -1 4 a c 1 2 d 2 a -1 b 2 -1 4 a c 3 2 -1 d b -1 b 2 -1 4 a c 1 2 d 2 a -1 b 2 -1 4 a c -1 CosIntegral d b 2 a z -1 b 2 -1 4 a c 1 2 2 a -1 2 a b b 2 -1 4 a c 1 2 d 2 a -1 b 2 -1 4 a c 3 2 -1 -1 d b b 2 -1 4 a c 1 2 d 2 a -1 b 2 -1 4 a c -1 CosIntegral d b 2 a z b 2 -1 4 a c 1 2 2 a -1 d b -1 b 2 -1 4 a c 1 2 d 2 a -1 b 2 -1 4 a c -1 2 a b -1 b 2 -1 4 a c 1 2 d 2 a -1 b 2 -1 4 a c 3 2 -1 SinhIntegral d b 2 a z -1 b 2 -1 4 a c 1 2 2 a -1 d b b 2 -1 4 a c 1 2 d 2 a -1 b 2 -1 4 a c -1 -1 2 a b b 2 -1 4 a c 1 2 d 2 a -1 b 2 -1 4 a c 3 2 -1 SinhIntegral d b 2 a z b 2 -1 4 a c 1 2 2 a -1 [/itex]

 Rule Form

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 Date Added to functions.wolfram.com (modification date)

 2002-12-18