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 Cosh

 http://functions.wolfram.com/01.20.21.1527.01

 Input Form

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

 Standard Form

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

 MathML Form

 sinh ( b z 2 + d z + e ) cosh ( c z 2 + f z + g ) z - 1 2 π 2 ( cosh ( ( f - d ) 2 4 c - 4 b + e - g ) 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 ) sinh ( ( f - d ) 2 4 c - 4 b + e - g ) ( c - b ) + cosh ( ( d + f ) 2 - 4 b - 4 c + e + g ) S TagBox["S", FresnelS] ( - d - f - 2 b z - 2 c z ( - b - c ) 2 π ) - C TagBox["C", FresnelC] ( - d - f - 2 b z - 2 c z ( - b - c ) 2 π ) sinh ( ( d + f ) 2 - 4 b - 4 c + e + g ) - b - c ) z b z 2 d z e c z 2 f z g -1 1 2 2 -1 1 2 f -1 d 2 4 c -1 4 b -1 e -1 g 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 4 b -1 e -1 g c -1 b 1 2 -1 d f 2 -4 b -1 4 c -1 e g FresnelS -1 d -1 f -1 2 b z -1 2 c z -1 b -1 c 1 2 2 1 2 -1 -1 FresnelC -1 d -1 f -1 2 b z -1 2 c z -1 b -1 c 1 2 2 1 2 -1 d f 2 -4 b -1 4 c -1 e g -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["Sinh", "[", RowBox[List[RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]], "+", RowBox[List["d_", " ", "z_"]], "+", "e_"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "+", RowBox[List["f_", " ", "z_"]], "+", "g_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Cosh", "[", RowBox[List["e", "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", "f"]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "+", RowBox[List["4", " ", "c"]]]]], "-", "g"]], "]"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["-", "d"]], "+", "f", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["-", "d"]], "+", "f", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["e", "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", "f"]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "+", RowBox[List["4", " ", "c"]]]]], "-", "g"]], "]"]]]]]], SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Cosh", "[", RowBox[List["e", "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["d", "+", "f"]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "-", RowBox[List["4", " ", "c"]]]]], "+", "g"]], "]"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["-", "d"]], "-", "f", "-", RowBox[List["2", " ", "b", " ", "z"]], "-", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", "c"]], ")"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["-", "d"]], "-", "f", "-", RowBox[List["2", " ", "b", " ", "z"]], "-", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", "c"]], ")"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["e", "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["d", "+", "f"]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "-", RowBox[List["4", " ", "c"]]]]], "+", "g"]], "]"]]]]]], SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]]]], ")"]]]]]]]]

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

 2002-12-18