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 Cosh

 http://functions.wolfram.com/01.20.21.0783.01

 Input Form

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

 Standard Form

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RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]], " ", "z"]]]], "]"]]]]]], ")"]], "/", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 cos m ( z b + d z ) cosh ( c z ) z 2 - m ( 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 ) c + 2 - m - 1 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]]]]] ( 1 ( - c + 2 d k - d m ) 3 / 2 ( ( b m - 2 b k ) 2 π cos ( ( 2 b k - b m ) 2 4 ( - c + 2 d k - d m ) ) C TagBox["C", FresnelC] ( 2 b k - b m + 2 ( - c + 2 d k - d m ) z - c + 2 d k - d m 2 π ) - ( 2 b k - b m ) 2 π S TagBox["S", FresnelS] ( 2 b k - b m + 2 ( - c + 2 d k - d m ) z - c + 2 d k - d m 2 π ) sin ( ( 2 b k - b m ) 2 4 ( - c + 2 d k - d m ) ) + 2 - c + 2 d k - d m sin ( ( - c + 2 d k - d m ) z + ( 2 b k - b m ) z ) ) + 1 ( c + 2 d k - d m ) 3 / 2 ( ( b m - 2 b k ) 2 π cos ( ( 2 b k - b m ) 2 4 ( c + 2 d k - d m ) ) C TagBox["C", FresnelC] ( 2 b k - b m + 2 ( c + 2 d k - d m ) z c + 2 d k - d m 2 π ) - ( 2 b k - b m ) 2 π S TagBox["S", FresnelS] ( 2 b k - b m + 2 ( c + 2 d k - d m ) z c + 2 d k - d m 2 π ) sin ( ( 2 b k - b m ) 2 4 ( c + 2 d k - d m ) ) + 2 c + 2 d k - d m sin ( ( c + 2 d k - d m ) z + ( 2 b k - b m ) z ) ) ) /; m + Condition z z 1 2 b d z m c z 2 -1 m Binomial m m 2 -1 1 -1 \$CellContext`m 2 c z c -1 2 -1 m -1 k 0 m -1 2 -1 Binomial m k 1 -1 c 2 d k -1 d m 3 2 -1 b m -1 2 b k 2 1 2 2 b k -1 b m 2 4 -1 c 2 d k -1 d m -1 FresnelC 2 b k -1 b m 2 -1 c 2 d k -1 d m z 1 2 -1 c 2 d k -1 d m 1 2 2 1 2 -1 -1 2 b k -1 b m 2 1 2 FresnelS 2 b k -1 b m 2 -1 c 2 d k -1 d m z 1 2 -1 c 2 d k -1 d m 1 2 2 1 2 -1 2 b k -1 b m 2 4 -1 c 2 d k -1 d m -1 2 -1 c 2 d k -1 d m 1 2 -1 c 2 d k -1 d m z 2 b k -1 b m z 1 2 1 c 2 d k -1 d m 3 2 -1 b m -1 2 b k 2 1 2 2 b k -1 b m 2 4 c 2 d k -1 d m -1 FresnelC 2 b k -1 b m 2 c 2 d k -1 d m z 1 2 c 2 d k -1 d m 1 2 2 1 2 -1 -1 2 b k -1 b m 2 1 2 FresnelS 2 b k -1 b m 2 c 2 d k -1 d m z 1 2 c 2 d k -1 d m 1 2 2 1 2 -1 2 b k -1 b m 2 4 c 2 d k -1 d m -1 2 c 2 d k -1 d m 1 2 c 2 d k -1 d m z 2 b k -1 b m z 1 2 m SuperPlus [/itex]

 Rule Form

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

 2002-12-18