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 Cosh

 http://functions.wolfram.com/01.20.21.0748.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "+", "e"]], "]"]], RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["f", " ", "z"]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["(", RowBox[List[FractionBox["1", SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["e", "-", FractionBox[SuperscriptBox["f", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]]]]], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "f"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "f"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["e", "-", FractionBox[SuperscriptBox["f", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]]]]], "]"]]]]]], ")"]], "-", RowBox[List[FractionBox["1", SqrtBox[RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["e", "+", FractionBox[SuperscriptBox["f", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]]]]], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "f"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "f"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["e", "+", FractionBox[SuperscriptBox["f", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 cos ( b z 2 + e ) cosh ( c z 2 + f z ) z 1 2 π 2 ( cos ( - f 2 4 ( c - b ) + e ) C TagBox["C", FresnelC] ( f + 2 ( c - b ) z c - b 2 π ) + S TagBox["S", FresnelS] ( f + 2 ( c - b ) z c - b 2 π ) sin ( - f 2 4 ( c - b ) + e ) c - b - cos ( f 2 4 ( b + c ) + e ) C TagBox["C", FresnelC] ( f + 2 ( b + c ) z - b - c 2 π ) + S TagBox["S", FresnelS] ( f + 2 ( b + c ) z - b - c 2 π ) sin ( f 2 4 ( b + c ) + e ) - b - c ) z b z 2 e c z 2 f z 1 2 2 -1 1 2 -1 f 2 4 c -1 b -1 e FresnelC f 2 c -1 b z c -1 b 1 2 2 1 2 -1 FresnelS f 2 c -1 b z c -1 b 1 2 2 1 2 -1 -1 f 2 4 c -1 b -1 e c -1 b 1 2 -1 -1 f 2 4 b c -1 e FresnelC f 2 b c z -1 b -1 c 1 2 2 1 2 -1 FresnelS f 2 b c z -1 b -1 c 1 2 2 1 2 -1 f 2 4 b c -1 e -1 b -1 c 1 2 -1 [/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_", " ", SuperscriptBox["z_", "2"]]], "+", "e_"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "+", RowBox[List["f_", " ", "z_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["e", "-", FractionBox[SuperscriptBox["f", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]]]]], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "f"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "f"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["e", "-", FractionBox[SuperscriptBox["f", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]]]]], "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]], "-", FractionBox[RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["e", "+", FractionBox[SuperscriptBox["f", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]]]]], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "f"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "f"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["e", "+", FractionBox[SuperscriptBox["f", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]]]]], "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]]]], ")"]]]]]]]]

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

 2002-12-18