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 Cosh

 http://functions.wolfram.com/01.20.21.0152.01

 Input Form

 Integrate[Cosh[c z]/(a z^2 + b)^2, z] == (1/(4 b^(3/2))) ((2 Sqrt[b] z Cosh[c z])/(b + a z^2) + (1/a) (I CosIntegral[-((Sqrt[b] c)/Sqrt[a]) + I c z] (Sqrt[a] Cos[(Sqrt[b] c)/Sqrt[a]] + Sqrt[b] c Sin[(Sqrt[b] c)/Sqrt[a]])) - (1/a) (I CosIntegral[(Sqrt[b] c)/Sqrt[a] + I c z] (Sqrt[a] Cos[(Sqrt[b] c)/Sqrt[a]] + Sqrt[b] c Sin[(Sqrt[b] c)/Sqrt[a]])) - (I Sqrt[b] c Cos[(Sqrt[b] c)/Sqrt[a]] SinIntegral[(Sqrt[b] c)/Sqrt[a] - I c z])/a + (I Sin[(Sqrt[b] c)/Sqrt[a]] SinIntegral[(Sqrt[b] c)/Sqrt[a] - I c z])/ Sqrt[a] + (I Sqrt[b] c Cos[(Sqrt[b] c)/Sqrt[a]] SinIntegral[(Sqrt[b] c)/Sqrt[a] + I c z])/a - (I Sin[(Sqrt[b] c)/Sqrt[a]] SinIntegral[(Sqrt[b] c)/Sqrt[a] + I c z])/ Sqrt[a])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "+", "b"]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["4", " ", SuperscriptBox["b", RowBox[List["3", "/", "2"]]]]]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", SqrtBox["b"], " ", "z", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List["b", "+", RowBox[List["a", " ", SuperscriptBox["z", "2"]]]]]], "+", RowBox[List[FractionBox["1", "a"], RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["CosIntegral", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["a"], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "]"]]]], "+", RowBox[List[SqrtBox["b"], " ", "c", " ", RowBox[List["Sin", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "]"]]]]]], ")"]]]], ")"]]]], "-", RowBox[List[FractionBox["1", "a"], RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["CosIntegral", "[", RowBox[List[FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["a"], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "]"]]]], "+", RowBox[List[SqrtBox["b"], " ", "c", " ", RowBox[List["Sin", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "]"]]]]]], ")"]]]], ")"]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"], " ", "c", " ", RowBox[List["Cos", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List[FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]]], "]"]]]], "a"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List[FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]]], "]"]]]], SqrtBox["a"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"], " ", "c", " ", RowBox[List["Cos", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List[FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]]], "]"]]]], "a"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List[FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]]], "]"]]]], SqrtBox["a"]]]], ")"]]]]]]]]

 MathML Form

 cosh ( c z ) ( a z 2 + b ) 2 z 1 4 b 3 / 2 ( 2 b z cosh ( c z ) a z 2 + b - Ci ( z c + b c a ) ( a cos ( b c a ) + b c sin ( b c a ) ) a + Ci ( c z - b c a ) ( a cos ( b c a ) + b c sin ( b c a ) ) a + b c cos ( b c a ) Si ( z c + b c a ) a + sin ( b c a ) Si ( b c a - c z ) a - sin ( b c a ) Si ( z c + b c a ) a - b c cos ( b c a ) Si ( b c a - c z ) a ) z c z a z 2 b 2 -1 1 4 b 3 2 -1 2 b 1 2 z c z a z 2 b -1 -1 CosIntegral z c b 1 2 c a 1 2 -1 a 1 2 b 1 2 c a 1 2 -1 b 1 2 c b 1 2 c a 1 2 -1 a -1 CosIntegral c z -1 b 1 2 c a 1 2 -1 a 1 2 b 1 2 c a 1 2 -1 b 1 2 c b 1 2 c a 1 2 -1 a -1 b 1 2 c b 1 2 c a 1 2 -1 SinIntegral z c b 1 2 c a 1 2 -1 a -1 b 1 2 c a 1 2 -1 SinIntegral b 1 2 c a 1 2 -1 -1 c z a 1 2 -1 -1 b 1 2 c a 1 2 -1 SinIntegral z c b 1 2 c a 1 2 -1 a 1 2 -1 -1 b 1 2 c b 1 2 c a 1 2 -1 SinIntegral b 1 2 c a 1 2 -1 -1 c z a -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a_", " ", SuperscriptBox["z_", "2"]]], "+", "b_"]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[FractionBox[RowBox[List["2", " ", SqrtBox["b"], " ", "z", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List["b", "+", RowBox[List["a", " ", SuperscriptBox["z", "2"]]]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["CosIntegral", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["a"], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "]"]]]], "+", RowBox[List[SqrtBox["b"], " ", "c", " ", RowBox[List["Sin", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "]"]]]]]], ")"]]]], "a"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["CosIntegral", "[", RowBox[List[FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["a"], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "]"]]]], "+", RowBox[List[SqrtBox["b"], " ", "c", " ", RowBox[List["Sin", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "]"]]]]]], ")"]]]], "a"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"], " ", "c", " ", RowBox[List["Cos", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List[FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]]], "]"]]]], "a"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List[FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]]], "]"]]]], SqrtBox["a"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"], " ", "c", " ", RowBox[List["Cos", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List[FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]]], "]"]]]], "a"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List[FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]]], "]"]]]], SqrtBox["a"]]]], RowBox[List["4", " ", SuperscriptBox["b", RowBox[List["3", "/", "2"]]]]]]]]]]

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

 2002-12-18