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 Cosh

 http://functions.wolfram.com/01.20.21.0638.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18