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 Cosh

 http://functions.wolfram.com/01.20.21.0582.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]], "+", RowBox[List["d_", " ", "z_"]]]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "+", RowBox[List["f_", " ", "z_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["Cos", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]]], "]"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["-", "d"]], "+", 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["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["-", "d"]], "+", 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", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", RowBox[List["\[ImaginaryI]", " ", "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", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "-", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]]], "]"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["-", "d"]], "-", 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["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["-", "d"]], "-", 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", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "-", RowBox[List["\[ImaginaryI]", " ", "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"]]]]]]]], ")"]]]]]]]]

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

 2002-12-18