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 Cosh

 http://functions.wolfram.com/01.20.21.0709.01

 Input Form

 Integrate[Cos[b Sqrt[z] + e] Cosh[f z + g], z] == (1/4) ((1/((-I) f)^(3/2)) ((-b) Sqrt[2 Pi] Cos[e - (I b^2)/(4 f) - I g] FresnelC[(b - 2 I f Sqrt[z])/(Sqrt[(-I) f] Sqrt[2 Pi])] + b Sqrt[2 Pi] FresnelS[(b - 2 I f Sqrt[z])/(Sqrt[(-I) f] Sqrt[2 Pi])] Sin[e - (I b^2)/(4 f) - I g] + 2 Sqrt[(-I) f] Sin[e - I g + b Sqrt[z] - I f z]) - (1/((-I) f)^(3/2)) (b Sqrt[2 Pi] Cos[e + (I b^2)/(4 f) + I g] FresnelC[(b + 2 I f Sqrt[z])/(Sqrt[(-I) f] Sqrt[2 Pi])] + b Sqrt[2 Pi] FresnelS[(b + 2 I f Sqrt[z])/(Sqrt[(-I) f] Sqrt[2 Pi])] Sin[e + (I b^2)/(4 f) + I g] + 2 Sqrt[(-I) f] Sin[e + I g + b Sqrt[z] + I f z]))

 Standard Form

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 MathML Form

 cos ( z b + e ) cosh ( g + f z ) z 1 4 ( 1 ( - f ) 3 / 2 ( - b 2 π cos ( - b 2 4 f + e - g ) C TagBox["C", FresnelC] ( b - 2 f z - f 2 π ) + b 2 π S TagBox["S", FresnelS] ( b - 2 f z - f 2 π ) sin ( - b 2 4 f + e - g ) + 2 - f sin ( z b + e - g - f z ) ) - 1 ( - f ) 3 / 2 ( b 2 π cos ( b 2 4 f + e + g ) C TagBox["C", FresnelC] ( b + 2 f z - f 2 π ) + b 2 π S TagBox["S", FresnelS] ( b + 2 f z - f 2 π ) sin ( b 2 4 f + e + g ) + 2 - f sin ( z b + e + g + f z ) ) ) z z 1 2 b e g f z 1 4 1 -1 f 3 2 -1 -1 b 2 1 2 -1 b 2 4 f -1 e -1 g FresnelC b -1 2 f z 1 2 -1 f 1 2 2 1 2 -1 b 2 1 2 FresnelS b -1 2 f z 1 2 -1 f 1 2 2 1 2 -1 -1 b 2 4 f -1 e -1 g 2 -1 f 1 2 z 1 2 b e -1 g -1 f z -1 1 -1 f 3 2 -1 b 2 1 2 b 2 4 f -1 e g FresnelC b 2 f z 1 2 -1 f 1 2 2 1 2 -1 b 2 1 2 FresnelS b 2 f z 1 2 -1 f 1 2 2 1 2 -1 b 2 4 f -1 e g 2 -1 f 1 2 z 1 2 b e g f z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List["b_", " ", SqrtBox["z_"]]], "+", "e_"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["f_", " ", "z_"]], "+", "g_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["Cos", "[", RowBox[List["e", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], RowBox[List["4", " ", "f"]]], "-", RowBox[List["\[ImaginaryI]", " ", "g"]]]], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List["b", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "+", RowBox[List["b", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List["b", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["e", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], RowBox[List["4", " ", "f"]]], "-", RowBox[List["\[ImaginaryI]", " ", "g"]]]], "]"]]]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]]], " ", RowBox[List["Sin", "[", RowBox[List["e", "-", RowBox[List["\[ImaginaryI]", " ", "g"]], "+", RowBox[List["b", " ", SqrtBox["z"]]], "-", RowBox[List["\[ImaginaryI]", " ", "f", " ", "z"]]]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]], ")"]], RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List[RowBox[List["b", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["Cos", "[", RowBox[List["e", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], RowBox[List["4", " ", "f"]]], "+", RowBox[List["\[ImaginaryI]", " ", "g"]]]], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "+", RowBox[List["b", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["e", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], RowBox[List["4", " ", "f"]]], "+", RowBox[List["\[ImaginaryI]", " ", "g"]]]], "]"]]]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]]], " ", RowBox[List["Sin", "[", RowBox[List["e", "+", RowBox[List["\[ImaginaryI]", " ", "g"]], "+", RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["\[ImaginaryI]", " ", "f", " ", "z"]]]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]], ")"]], RowBox[List["3", "/", "2"]]]]]], ")"]]]]]]]]

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

 2002-12-18