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 Cosh

 http://functions.wolfram.com/01.20.21.0154.01

 Input Form

 Integrate[Cosh[d z]/(a z^2 + b z + c), z] == (1/Sqrt[b^2 - 4 a c]) (Cosh[((b - Sqrt[b^2 - 4 a c]) d)/(2 a)] CosIntegral[(I d (b - Sqrt[b^2 - 4 a c] + 2 a z))/(2 a)] - Cosh[((b + Sqrt[b^2 - 4 a c]) d)/(2 a)] CosIntegral[(I d (b + Sqrt[b^2 - 4 a c] + 2 a z))/(2 a)] - Sinh[((b - Sqrt[b^2 - 4 a c]) d)/(2 a)] SinhIntegral[(d (b - Sqrt[b^2 - 4 a c] + 2 a z))/(2 a)] + Sinh[((b + Sqrt[b^2 - 4 a c]) d)/(2 a)] SinhIntegral[(d (b + Sqrt[b^2 - 4 a c] + 2 a z))/(2 a)])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Cosh", "[", RowBox[List["d", " ", "z"]], "]"]], RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["b", " ", "z"]], "+", "c"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[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["CosIntegral", "[", FractionBox[RowBox[List["\[ImaginaryI]", " ", "d", " ", RowBox[List["(", RowBox[List["b", "-", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]], "+", RowBox[List["2", " ", "a", " ", "z"]]]], ")"]]]], RowBox[List["2", " ", "a"]]], "]"]]]], "-", RowBox[List[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["CosIntegral", "[", FractionBox[RowBox[List["\[ImaginaryI]", " ", "d", " ", RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]], "+", RowBox[List["2", " ", "a", " ", "z"]]]], ")"]]]], RowBox[List["2", " ", "a"]]], "]"]]]], "-", RowBox[List[RowBox[List["Sinh", "[", 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["SinhIntegral", "[", FractionBox[RowBox[List["d", " ", RowBox[List["(", RowBox[List["b", "-", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]], "+", RowBox[List["2", " ", "a", " ", "z"]]]], ")"]]]], RowBox[List["2", " ", "a"]]], "]"]]]], "+", RowBox[List[RowBox[List["Sinh", "[", 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["SinhIntegral", "[", FractionBox[RowBox[List["d", " ", RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]], "+", RowBox[List["2", " ", "a", " ", "z"]]]], ")"]]]], RowBox[List["2", " ", "a"]]], "]"]]]]]], ")"]]]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Cosh", "[", RowBox[List["d_", " ", "z_"]], "]"]], RowBox[List[RowBox[List["a_", " ", SuperscriptBox["z_", "2"]]], "+", RowBox[List["b_", " ", "z_"]], "+", "c_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[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["CosIntegral", "[", FractionBox[RowBox[List["\[ImaginaryI]", " ", "d", " ", RowBox[List["(", RowBox[List["b", "-", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]], "+", RowBox[List["2", " ", "a", " ", "z"]]]], ")"]]]], RowBox[List["2", " ", "a"]]], "]"]]]], "-", RowBox[List[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["CosIntegral", "[", FractionBox[RowBox[List["\[ImaginaryI]", " ", "d", " ", RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]], "+", RowBox[List["2", " ", "a", " ", "z"]]]], ")"]]]], RowBox[List["2", " ", "a"]]], "]"]]]], "-", RowBox[List[RowBox[List["Sinh", "[", 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["SinhIntegral", "[", FractionBox[RowBox[List["d", " ", RowBox[List["(", RowBox[List["b", "-", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]], "+", RowBox[List["2", " ", "a", " ", "z"]]]], ")"]]]], RowBox[List["2", " ", "a"]]], "]"]]]], "+", RowBox[List[RowBox[List["Sinh", "[", 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["SinhIntegral", "[", FractionBox[RowBox[List["d", " ", RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]], "+", RowBox[List["2", " ", "a", " ", "z"]]]], ")"]]]], RowBox[List["2", " ", "a"]]], "]"]]]]]], SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]]]]]]]

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

 2002-12-18