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 Cosh

 http://functions.wolfram.com/01.20.21.1529.01

 Input Form

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

 Standard Form

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 MathML Form

 sinh ( z b + e + d z ) cosh ( z c + g + f z ) z 1 4 ( 1 ( ( f - d ) ) 3 / 2 ( - 2 ( f - d ) cosh ( - z ( c - b ) + e - g - ( f - d ) z ) - ( c - b ) 2 π ( cosh ( ( c - b ) 2 4 f - 4 d + e - g ) S TagBox["S", FresnelS] ( - b + c + 2 ( f - d ) z ( f - d ) 2 π ) - C TagBox["C", FresnelC] ( - b + c + 2 ( f - d ) z ( f - d ) 2 π ) sinh ( ( c - b ) 2 4 f - 4 d + e - g ) ) ) + 1 ( ( - d - f ) ) 3 / 2 ( - 2 ( - d - f ) cosh ( - z ( - b - c ) + e + g - ( - d - f ) z ) - ( - b - c ) 2 π ( cosh ( ( b + c ) 2 - 4 d - 4 f + e + g ) S TagBox["S", FresnelS] ( - b - c + 2 ( - d - f ) z ( - d - f ) 2 π ) - C TagBox["C", FresnelC] ( - b - c + 2 ( - d - f ) z ( - d - f ) 2 π ) sinh ( ( b + c ) 2 - 4 d - 4 f + e + g ) ) ) ) z z 1 2 b e d z z 1 2 c g f z 1 4 1 f -1 d 3 2 -1 -2 f -1 d 1 2 -1 z 1 2 c -1 b e -1 g -1 f -1 d z -1 c -1 b 2 1 2 c -1 b 2 4 f -1 4 d -1 e -1 g FresnelS -1 b c 2 f -1 d z 1 2 f -1 d 1 2 2 1 2 -1 -1 FresnelC -1 b c 2 f -1 d z 1 2 f -1 d 1 2 2 1 2 -1 c -1 b 2 4 f -1 4 d -1 e -1 g 1 -1 d -1 f 3 2 -1 -2 -1 d -1 f 1 2 -1 z 1 2 -1 b -1 c e g -1 -1 d -1 f z -1 -1 b -1 c 2 1 2 b c 2 -4 d -1 4 f -1 e g FresnelS -1 b -1 c 2 -1 d -1 f z 1 2 -1 d -1 f 1 2 2 1 2 -1 -1 FresnelC -1 b -1 c 2 -1 d -1 f z 1 2 -1 d -1 f 1 2 2 1 2 -1 b c 2 -4 d -1 4 f -1 e g [/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_", " ", SqrtBox["z_"]]], "+", RowBox[List["d_", " ", "z_"]], "+", "e_"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SqrtBox["z_"]]], "+", 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["-", "2"]], " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", "f"]], ")"]]]]], " ", RowBox[List["Cosh", "[", RowBox[List["e", "-", "g", "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], " ", SqrtBox["z"]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", "f"]], ")"]], " ", "z"]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Cosh", "[", RowBox[List["e", "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "d"]], "+", RowBox[List["4", " ", "f"]]]]], "-", "g"]], "]"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "c", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", "f"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", "f"]], ")"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "c", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", "f"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", "f"]], ")"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["e", "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "d"]], "+", RowBox[List["4", " ", "f"]]]]], "-", "g"]], "]"]]]]]], ")"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", "f"]], ")"]]]], ")"]], RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "-", "f"]], ")"]]]]], " ", RowBox[List["Cosh", "[", RowBox[List["e", "+", "g", "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", "c"]], ")"]], " ", SqrtBox["z"]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "-", "f"]], ")"]], " ", "z"]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", "c"]], ")"]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Cosh", "[", RowBox[List["e", "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "d"]], "-", RowBox[List["4", " ", "f"]]]]], "+", "g"]], "]"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "-", "c", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "-", "f"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "-", "f"]], ")"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "-", "c", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "-", "f"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "-", "f"]], ")"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["e", "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "d"]], "-", RowBox[List["4", " ", "f"]]]]], "+", "g"]], "]"]]]]]], ")"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "-", "f"]], ")"]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], ")"]]]]]]]]

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

 2002-12-18