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 Cosh

 http://functions.wolfram.com/01.20.21.0153.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2002-12-18