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 Cosh

 http://functions.wolfram.com/01.20.21.0778.01

 Input Form

 Integrate[Cos[b z^2]^m Cosh[c z], z] == ((-(1/2))^m Binomial[m, m/2] (1 - Mod[m, 2]) Sinh[c z])/c + 2^(-(1/2) - m) Sqrt[Pi] Sum[Binomial[m, s] ((1/Sqrt[b m - 2 b s]) (Cos[c^2/(4 (b m - 2 b s))] FresnelC[((-I) c + 2 (b m - 2 b s) z)/(Sqrt[2 Pi] Sqrt[b m - 2 b s])] - FresnelS[((-I) c + 2 (b m - 2 b s) z)/ (Sqrt[2 Pi] Sqrt[b m - 2 b s])] Sin[c^2/(4 (b m - 2 b s))]) + (1/Sqrt[(-b) m + 2 b s]) (Cos[c^2/(4 ((-b) m + 2 b s))] FresnelC[((-I) c + 2 ((-b) m + 2 b s) z)/(Sqrt[2 Pi] Sqrt[(-b) m + 2 b s])] - FresnelS[((-I) c + 2 ((-b) m + 2 b s) z)/(Sqrt[2 Pi] Sqrt[(-b) m + 2 b s])] Sin[c^2/(4 ((-b) m + 2 b s))])), {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", " ", SuperscriptBox["z", "2"]]], "]"]], "m"], RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], 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["c", " ", "z"]], "]"]]]], "c"], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "m"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "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", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", SqrtBox[RowBox[List[RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "b", " ", "s"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", FractionBox[SuperscriptBox["c", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "b", " ", "s"]]]], ")"]]]]], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "b", " ", "s"]]]], ")"]], " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SqrtBox[RowBox[List[RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "b", " ", "s"]]]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "b", " ", "s"]]]], ")"]], " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SqrtBox[RowBox[List[RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "b", " ", "s"]]]]]]]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[SuperscriptBox["c", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "b", " ", "s"]]]], ")"]]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", FractionBox[SuperscriptBox["c", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]], ")"]]]]], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]], ")"]], " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]], ")"]], " ", "z"]]]], RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]]]]]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[SuperscriptBox["c", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]], ")"]]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2002-12-18