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 Cosh

 http://functions.wolfram.com/01.20.21.0559.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]], "+", "e"]], "]"]], RowBox[List["Cosh", "[", RowBox[List["c", " ", SqrtBox["z"]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["-", "d"]], ")"]], RowBox[List["3", "/", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["-", "d"]]], " ", RowBox[List["Cos", "[", RowBox[List["e", "+", RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"], RowBox[List["4", " ", "d"]]], "-", "e"]], "]"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List["-", "d"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["FresnelC", "[", FractionBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List["-", "d"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"], RowBox[List["4", " ", "d"]]], "-", "e"]], "]"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", SuperscriptBox["d", RowBox[List["3", "/", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SqrtBox["d"], " ", RowBox[List["Cos", "[", RowBox[List["e", "+", RowBox[List[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"], RowBox[List["4", " ", "d"]]], "-", "e"]], "]"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox["d"], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox["d"], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"], RowBox[List["4", " ", "d"]]], "-", "e"]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 sin ( z b + e + d z ) cosh ( z c ) z 1 4 ( 1 d 3 / 2 ( - 2 d cos ( z ( b - c ) + e + d z ) - ( b - c ) 2 π cos ( ( b - c ) 2 4 d - e ) S TagBox["S", FresnelS] ( b - c + 2 d z d 2 π ) - ( c - b ) 2 π C TagBox["C", FresnelC] ( b - c + 2 d z d 2 π ) sin ( ( b - c ) 2 4 d - e ) ) + 1 ( - d ) 3 / 2 ( 2 - d cos ( z ( b + c ) + e + d z ) + ( b + c ) 2 π ( cos ( ( - b - c ) 2 4 d - e ) S TagBox["S", FresnelS] ( b + c + 2 d z - d 2 π ) + C TagBox["C", FresnelC] ( b + c + 2 d z - d 2 π ) sin ( ( - b - c ) 2 4 d - e ) ) ) ) z z 1 2 b e d z z 1 2 c 1 4 1 d 3 2 -1 -2 d 1 2 z 1 2 b -1 c e d z -1 b -1 c 2 1 2 b -1 c 2 4 d -1 -1 e FresnelS b -1 c 2 d z 1 2 d 1 2 2 1 2 -1 -1 c -1 b 2 1 2 FresnelC b -1 c 2 d z 1 2 d 1 2 2 1 2 -1 b -1 c 2 4 d -1 -1 e 1 -1 d 3 2 -1 2 -1 d 1 2 z 1 2 b c e d z b c 2 1 2 -1 b -1 c 2 4 d -1 -1 e FresnelS b c 2 d z 1 2 -1 d 1 2 2 1 2 -1 FresnelC b c 2 d z 1 2 -1 d 1 2 2 1 2 -1 -1 b -1 c 2 4 d -1 -1 e [/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_", " ", SqrtBox["z_"]]], "+", RowBox[List["d_", " ", "z_"]], "+", "e_"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", SqrtBox["z_"]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["-", "d"]]], " ", RowBox[List["Cos", "[", RowBox[List["e", "+", RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"], RowBox[List["4", " ", "d"]]], "-", "e"]], "]"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List["-", "d"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["FresnelC", "[", FractionBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List["-", "d"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"], RowBox[List["4", " ", "d"]]], "-", "e"]], "]"]]]]]], ")"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "d"]], ")"]], RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SqrtBox["d"], " ", RowBox[List["Cos", "[", RowBox[List["e", "+", RowBox[List[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"], RowBox[List["4", " ", "d"]]], "-", "e"]], "]"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox["d"], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox["d"], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"], RowBox[List["4", " ", "d"]]], "-", "e"]], "]"]]]]]], SuperscriptBox["d", RowBox[List["3", "/", "2"]]]]]], ")"]]]]]]]]

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

 2002-12-18