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 Cosh

 http://functions.wolfram.com/01.20.21.0794.01

 Input Form

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

 Standard Form

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 MathML Form

 cos m ( b z ) cosh ( c z 2 ) z 2 - m - 1 2 π ( 1 - m mod 2 \$CellContext`m 2 ) c ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] C TagBox["C", FresnelC] ( c 2 π z c ) + 2 - m - 1 2 π k = 0 m - 1 2 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( cosh ( ( 2 b k - b m ) 2 4 c ) C TagBox["C", FresnelC] ( 2 b k - b m + 2 c z c 2 π ) - S TagBox["S", FresnelS] ( 2 b k - b m + 2 c z c 2 π ) sinh ( ( 2 b k - b m ) 2 4 c ) c + cosh ( ( 2 b k - b m ) 2 4 c ) C TagBox["C", FresnelC] ( 2 b k - b m - 2 c z - c 2 π ) + S TagBox["S", FresnelS] ( 2 b k - b m - 2 c z - c 2 π ) sinh ( ( 2 b k - b m ) 2 4 c ) - c ) /; m + Condition z b z m c z 2 2 -1 m -1 1 2 1 2 1 -1 \$CellContext`m 2 c 1 2 -1 Binomial m m 2 -1 FresnelC c 2 -1 1 2 z c 1 2 -1 2 -1 m -1 1 2 1 2 k 0 m -1 2 -1 Binomial m k 2 b k -1 b m 2 4 c -1 FresnelC 2 b k -1 b m 2 c z c 1 2 2 1 2 -1 -1 FresnelS 2 b k -1 b m 2 c z c 1 2 2 1 2 -1 2 b k -1 b m 2 4 c -1 c 1 2 -1 2 b k -1 b m 2 4 c -1 FresnelC 2 b k -1 b m -1 2 c z -1 c 1 2 2 1 2 -1 FresnelS 2 b k -1 b m -1 2 c z -1 c 1 2 2 1 2 -1 2 b k -1 b m 2 4 c -1 -1 c 1 2 -1 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_", " ", "z_"]], "]"]], "m_"], " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "m"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List["c", " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c"]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]]]], SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c"]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "m"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["Cosh", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], ")"]], "2"], RowBox[List["4", " ", "c"]]], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sinh", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], ")"]], "2"], RowBox[List["4", " ", "c"]]], "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]]]], "+", FractionBox[RowBox[List[RowBox[List[RowBox[List["Cosh", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], ")"]], "2"], RowBox[List["4", " ", "c"]]], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sinh", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], ")"]], "2"], RowBox[List["4", " ", "c"]]], "]"]]]]]], SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c"]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]

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

 2002-12-18