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 Cosh

 http://functions.wolfram.com/01.20.21.0781.01

 Input Form

 Integrate[Cos[b Sqrt[z] + e]^m Cosh[c z], z] == ((-(1/2))^m Binomial[m, m/2] (1 - Mod[m, 2]) Sinh[c z])/c + 2^(-1 - m) Sum[Binomial[m, s] ((1/((-I) c)^(3/2)) ((-b) Sqrt[2 Pi] (m - 2 s) (Cos[e m - 2 e s - (I (b m - 2 b s)^2)/(4 c)] FresnelC[(b (m - 2 s) - 2 I c Sqrt[z])/(Sqrt[(-I) c] Sqrt[ 2 Pi])] - FresnelS[(b (m - 2 s) - 2 I c Sqrt[z])/ (Sqrt[(-I) c] Sqrt[2 Pi])] Sin[e m - 2 e s - (I (b m - 2 b s)^2)/(4 c)]) + 2 Sqrt[(-I) c] Sin[e m - 2 e s + (b m - 2 b s) Sqrt[z] - I c z]) + (1/((-I) c)^(3/2)) (b Sqrt[2 Pi] (m - 2 s) (Cos[e m - 2 e s + (I (b m - 2 b s)^2)/(4 c)] FresnelC[((-b) (m - 2 s) - 2 I c Sqrt[z])/(Sqrt[(-I) c] Sqrt[ 2 Pi])] + FresnelS[((-b) (m - 2 s) - 2 I c Sqrt[z])/ (Sqrt[(-I) c] Sqrt[2 Pi])] Sin[e m - 2 e s + (I (b m - 2 b s)^2)/(4 c)]) - 2 Sqrt[(-I) c] Sin[e m - 2 e s - ((-b) m + 2 b s) Sqrt[z] + I c 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[RowBox[List["b", " ", SqrtBox["z"]]], "+", "e"]], "]"]], "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["-", "1"]], "-", "m"]]], " ", RowBox[List["Sum", "[", " ", RowBox[List[RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", 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 MathML Form

 cos m ( z b + e ) 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 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 ( - c ) 3 / 2 ( b 2 π ( m - 2 s ) ( cos ( ( b m - 2 b s ) 2 4 c + e m - 2 e s ) C TagBox["C", FresnelC] ( - b ( m - 2 s ) - 2 c z - c 2 π ) + S TagBox["S", FresnelS] ( - b ( m - 2 s ) - 2 c z - c 2 π ) sin ( ( b m - 2 b s ) 2 4 c + e m - 2 e s ) ) - 2 - c sin ( e m - 2 e s + c z - ( 2 b s - b m ) z ) ) + 1 ( - c ) 3 / 2 ( 2 - c sin ( e m - 2 e s - c z + ( b m - 2 b s ) z ) - b 2 π ( m - 2 s ) ( cos ( - ( b m - 2 b s ) 2 4 c + e m - 2 e s ) C TagBox["C", FresnelC] ( b ( m - 2 s ) - 2 c z - c 2 π ) - S TagBox["S", FresnelS] ( b ( m - 2 s ) - 2 c z - c 2 π ) sin ( - ( b m - 2 b s ) 2 4 c + e m - 2 e s ) ) ) ) /; m + Condition z z 1 2 b e 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 s 0 m -1 2 -1 Binomial m s 1 -1 c 3 2 -1 b 2 1 2 m -1 2 s b m -1 2 b s 2 4 c -1 e m -1 2 e s FresnelC -1 b m -1 2 s -1 2 c z 1 2 -1 c 1 2 2 1 2 -1 FresnelS -1 b m -1 2 s -1 2 c z 1 2 -1 c 1 2 2 1 2 -1 b m -1 2 b s 2 4 c -1 e m -1 2 e s -1 2 -1 c 1 2 e m -1 2 e s c z -1 2 b s -1 b m z 1 2 1 -1 c 3 2 -1 2 -1 c 1 2 e m -1 2 e s -1 c z b m -1 2 b s z 1 2 -1 b 2 1 2 m -1 2 s -1 b m -1 2 b s 2 4 c -1 e m -1 2 e s FresnelC b m -1 2 s -1 2 c z 1 2 -1 c 1 2 2 1 2 -1 -1 FresnelS b m -1 2 s -1 2 c z 1 2 -1 c 1 2 2 1 2 -1 -1 b m -1 2 b s 2 4 c -1 e m -1 2 e s m SuperPlus [/itex]

 Rule Form

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

 2002-12-18