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 Cosh

 http://functions.wolfram.com/01.20.21.0793.01

 Input Form

 Integrate[Cos[b Sqrt[z] + d z + e]^m Cosh[f z + g], z] == (Binomial[m, m/2] (1 - Mod[m, 2]) Sinh[g + f z])/(2^m f) + 2^(-1 - m) Sum[Binomial[m, k] (((-2 b k + b m) Sqrt[2 Pi] Cos[(-I) g + 2 e k - e m - (2 b k - b m)^2/(4 ((-I) f + 2 d k - d m))] FresnelC[(2 b k - b m + 2 ((-I) f + 2 d k - d m) Sqrt[z])/ (Sqrt[(-I) f + 2 d k - d m] Sqrt[2 Pi])] + (2 b k - b m) Sqrt[2 Pi] FresnelS[(2 b k - b m + 2 ((-I) f + 2 d k - d m) Sqrt[z])/(Sqrt[(-I) f + 2 d k - d m] Sqrt[2 Pi])] Sin[(-I) g + 2 e k - e m - (2 b k - b m)^2/ (4 ((-I) f + 2 d k - d m))] - 2 I Sqrt[(-I) f + 2 d k - d m] Sinh[g + 2 I e k - I e m + I (2 b k - b m) Sqrt[z] + I ((-I) f + 2 d k - d m) z])/((-I) f + 2 d k - d m)^(3/2) + ((-2 b k + b m) Sqrt[2 Pi] Cosh[g - 2 I e k + I e m + (I (2 b k - b m)^2)/(4 (I f + 2 d k - d m))] FresnelC[(2 b k - b m + 2 (I f + 2 d k - d m) Sqrt[z])/ (Sqrt[I f + 2 d k - d m] Sqrt[2 Pi])] + I (2 b k - b m) Sqrt[2 Pi] FresnelS[(2 b k - b m + 2 (I f + 2 d k - d m) Sqrt[z])/ (Sqrt[I f + 2 d k - d m] Sqrt[2 Pi])] Sinh[g - 2 I e k + I e m + (I (2 b k - b m)^2)/ (4 (I f + 2 d k - d m))] + 2 I Sqrt[I f + 2 d k - d m] Sinh[g - 2 I e k + I e m - I (2 b k - b m) Sqrt[z] - I (I f + 2 d k - d m) z])/(I f + 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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"b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], ")"]], " ", SqrtBox["z"]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "f"]], "+", RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]], " ", "z"]]]], "]"]]]]]], ")"]], "/", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "f"]], "+", 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 + e + d z ) cosh ( g + f z ) z 2 - m ( 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 ) f + 2 - m - 1 k = 0 m - 1 2 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 ( - f + 2 d k - d m ) 3 / 2 ( ( b m - 2 b k ) 2 π cos ( - ( 2 b k - b m ) 2 4 ( - f + 2 d k - d m ) - g + 2 e k - e m ) C TagBox["C", FresnelC] ( 2 b k - b m + 2 ( - f + 2 d k - d m ) z - f + 2 d k - d m 2 π ) + ( 2 b k - b m ) 2 π S TagBox["S", FresnelS] ( 2 b k - b m + 2 ( - f + 2 d k - d m ) z - f + 2 d k - d m 2 π ) sin ( - ( 2 b k - b m ) 2 4 ( - f + 2 d k - d m ) - g + 2 e k - e m ) - 2 - f + 2 d k - d m sinh ( g + 2 e k - e m + ( - f + 2 d k - d m ) z + ( 2 b k - b m ) z ) ) + 1 ( f + 2 d k - d m ) 3 / 2 ( ( b m - 2 b k ) 2 π cosh ( ( 2 b k - b m ) 2 4 ( f + 2 d k - d m ) + g - 2 e k + e m ) C TagBox["C", FresnelC] ( 2 b k - b m + 2 ( f + 2 d k - d m ) z f + 2 d k - d m 2 π ) + ( 2 b k - b m ) 2 π S TagBox["S", FresnelS] ( 2 b k - b m + 2 ( f + 2 d k - d m ) z f + 2 d k - d m 2 π ) sinh ( ( 2 b k - b m ) 2 4 ( f + 2 d k - d m ) + g - 2 e k + e m ) + 2 f + 2 d k - d m sinh ( g - 2 e k + e m - ( f + 2 d k - d m ) z - ( 2 b k - b m ) z ) ) ) /; m + Condition z z 1 2 b e d z m g f z 2 -1 m Binomial m m 2 -1 1 -1 \$CellContext`m 2 g f z f -1 2 -1 m -1 k 0 m -1 2 -1 Binomial m k 1 -1 f 2 d k -1 d m 3 2 -1 b m -1 2 b k 2 1 2 -1 2 b k -1 b m 2 4 -1 f 2 d k -1 d m -1 -1 g 2 e k -1 e m FresnelC 2 b k -1 b m 2 -1 f 2 d k -1 d m z 1 2 -1 f 2 d k -1 d m 1 2 2 1 2 -1 2 b k -1 b m 2 1 2 FresnelS 2 b k -1 b m 2 -1 f 2 d k -1 d m z 1 2 -1 f 2 d k -1 d m 1 2 2 1 2 -1 -1 2 b k -1 b m 2 4 -1 f 2 d k -1 d m -1 -1 g 2 e k -1 e m -1 2 -1 f 2 d k -1 d m 1 2 g 2 e k -1 e m -1 f 2 d k -1 d m z 2 b k -1 b m z 1 2 1 f 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 f 2 d k -1 d m -1 g -1 2 e k e m FresnelC 2 b k -1 b m 2 f 2 d k -1 d m z 1 2 f 2 d k -1 d m 1 2 2 1 2 -1 2 b k -1 b m 2 1 2 FresnelS 2 b k -1 b m 2 f 2 d k -1 d m z 1 2 f 2 d k -1 d m 1 2 2 1 2 -1 2 b k -1 b m 2 4 f 2 d k -1 d m -1 g -1 2 e k e m 2 f 2 d k -1 d m 1 2 g -1 2 e k e m -1 f 2 d k -1 d m z -1 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