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 Cosh

 http://functions.wolfram.com/01.20.21.0787.01

 Input Form

 Integrate[Cos[b Sqrt[z]]^m Cosh[f z + g], z] == ((-(1/2))^m Binomial[m, m/2] (1 - Mod[m, 2]) Sinh[g + f z])/f + 2^(-1 - m) Sum[Binomial[m, s] ((1/((-I) f)^(3/2)) (Sqrt[2 Pi] ((-b) m + 2 b s) Cosh[g + (b m - 2 b s)^2/(4 f)] FresnelC[(b m - 2 b s - 2 I f Sqrt[z])/(Sqrt[(-I) f] Sqrt[2 Pi])] - I Sqrt[2 Pi] (b m - 2 b s) FresnelS[(b m - 2 b s - 2 I f Sqrt[z])/ (Sqrt[(-I) f] Sqrt[2 Pi])] Sinh[g + (b m - 2 b s)^2/(4 f)] - 2 I Sqrt[(-I) f] Sinh[g + I (b m - 2 b s) Sqrt[z] + f z]) + (1/((-I) f)^(3/2)) (Sqrt[2 Pi] (b m - 2 b s) Cosh[g + ((-b) m + 2 b s)^2/(4 f)] FresnelC[ ((-b) m + 2 b s - 2 I f Sqrt[z])/(Sqrt[(-I) f] Sqrt[2 Pi])] - I Sqrt[2 Pi] ((-b) m + 2 b s) FresnelS[((-b) m + 2 b s - 2 I f Sqrt[z])/(Sqrt[(-I) f] Sqrt[2 Pi])] Sinh[g + ((-b) m + 2 b s)^2/(4 f)] - 2 I Sqrt[(-I) f] Sinh[g + I ((-b) m + 2 b s) Sqrt[z] + f z])), {s, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["b", " ", SqrtBox["z"]]], "]"]], "m"], RowBox[List["Cosh", "[", RowBox[List[RowBox[List["f", " ", "z"]], "+", "g"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "m"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["g", "+", RowBox[List["f", " ", "z"]]]], "]"]]]], "f"], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", 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SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]]], " ", RowBox[List["Sinh", "[", RowBox[List["g", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "b", " ", "s"]]]], ")"]], " ", SqrtBox["z"]]], "+", RowBox[List["f", " ", "z"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]], ")"]], RowBox[List["3", "/", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "b", " ", "s"]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["g", "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]], ")"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["g", "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]]], " ", RowBox[List["Sinh", "[", RowBox[List["g", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]], ")"]], " ", SqrtBox["z"]]], "+", RowBox[List["f", " ", "z"]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 cos m ( b z ) cosh ( g + f z ) z ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["m", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 - m mod 2 \$CellContext`m 2 ) sinh ( g + f z ) ( - 1 2 ) m f + 2 - m - 1 s = 0 m - 1 2 ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 ( - f ) 3 / 2 ( 2 π ( b m - 2 b s ) cosh ( ( 2 b s - b m ) 2 4 f + g ) C TagBox["C", FresnelC] ( - b m + 2 b s - 2 f z - f 2 π ) - 2 π ( 2 b s - b m ) S TagBox["S", FresnelS] ( - b m + 2 b s - 2 f z - f 2 π ) sinh ( ( 2 b s - b m ) 2 4 f + g ) - 2 - f sinh ( g + f z + ( 2 b s - b m ) z ) ) + 1 ( - f ) 3 / 2 ( 2 π ( 2 b s - b m ) cosh ( ( b m - 2 b s ) 2 4 f + g ) C TagBox["C", FresnelC] ( b m - 2 b s - 2 f z - f 2 π ) - 2 π ( b m - 2 b s ) S TagBox["S", FresnelS] ( b m - 2 b s - 2 f z - f 2 π ) sinh ( ( b m - 2 b s ) 2 4 f + g ) - 2 - f sinh ( g + f z + ( b m - 2 b s ) z ) ) ) /; m + Condition z b z 1 2 m g f z Binomial m m 2 -1 1 -1 \$CellContext`m 2 g f z -1 1 2 m f -1 2 -1 m -1 s 0 m -1 2 -1 Binomial m s 1 -1 f 3 2 -1 2 1 2 b m -1 2 b s 2 b s -1 b m 2 4 f -1 g FresnelC -1 b m 2 b s -1 2 f z 1 2 -1 f 1 2 2 1 2 -1 -1 2 1 2 2 b s -1 b m FresnelS -1 b m 2 b s -1 2 f z 1 2 -1 f 1 2 2 1 2 -1 2 b s -1 b m 2 4 f -1 g -1 2 -1 f 1 2 g f z 2 b s -1 b m z 1 2 1 -1 f 3 2 -1 2 1 2 2 b s -1 b m b m -1 2 b s 2 4 f -1 g FresnelC b m -1 2 b s -1 2 f z 1 2 -1 f 1 2 2 1 2 -1 -1 2 1 2 b m -1 2 b s FresnelS b m -1 2 b s -1 2 f z 1 2 -1 f 1 2 2 1 2 -1 b m -1 2 b s 2 4 f -1 g -1 2 -1 f 1 2 g f z b m -1 2 b s z 1 2 m SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["b_", " ", SqrtBox["z_"]]], "]"]], "m_"], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["f_", " ", "z_"]], "+", "g_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "m"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["g", "+", RowBox[List["f", " ", "z"]]]], "]"]]]], "f"], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", 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 Date Added to functions.wolfram.com (modification date)

 2002-12-18