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 Cosh

 http://functions.wolfram.com/01.20.21.1295.01

 Input Form

 Integrate[E^(p z) Cos[b z]^m Cosh[c z^2], z] == (-(1/c)) (2^(-2 - m) Sqrt[Pi] Binomial[m, m/2] (Sqrt[-c] E^(p^2/(4 c)) Erfi[(p - 2 c z)/(2 Sqrt[-c])] - (Sqrt[c] Erfi[(p + 2 c z)/(2 Sqrt[c])])/E^(p^2/(4 c))) (1 - Mod[m, 2])) - ((2^(-2 - m) Sqrt[Pi])/c) Sum[Binomial[m, s] ((Sqrt[-c] E^(((-I) b m + p + 2 I b s)^2/(4 c)) Erfi[((-I) b m + p + 2 I b s - 2 c z)/(2 Sqrt[-c])] - (Sqrt[c] Erfi[(I b m + p - 2 I b s + 2 c z)/(2 Sqrt[c])])/ E^((I b m + p - 2 I b s)^2/(4 c))) + (Sqrt[-c] E^((I b m + p - 2 I b s)^2/(4 c)) Erfi[(I b m + p - 2 I b s - 2 c z)/(2 Sqrt[-c])] - (Sqrt[c] Erfi[((-I) b m + p + 2 I b s + 2 c z)/(2 Sqrt[c])])/ E^(((-I) b m + p + 2 I b s)^2/(4 c)))), {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["\[ExponentialE]", RowBox[List["p", " ", "z"]]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["b", " ", "z"]], "]"]], "m"], " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "c"]]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "m"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", "c"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["p", "2"], RowBox[List["4", " ", "c"]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "-", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "c"]]]]]], "]"]]]], "-", RowBox[List[SqrtBox["c"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["p", "2"], RowBox[List["4", " ", "c"]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "+", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox["c"]]]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]]]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "m"]]], " ", SqrtBox["\[Pi]"], " "]], "c"], 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["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", "c"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", "m"]], "+", "p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "s"]]]], ")"]], "2"], RowBox[List["4", " ", "c"]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", "m"]], "+", "p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "s"]], "-", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "c"]]]]]], "]"]]]], "-", RowBox[List[SqrtBox["c"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]], "+", "p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "s"]]]], ")"]], "2"], RowBox[List["4", " ", "c"]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]], "+", "p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "s"]], "+", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox["c"]]]], "]"]]]]]], ")"]], "+", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", "c"]]], SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]], "+", "p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "s"]]]], ")"]], "2"], RowBox[List["4", " ", "c"]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]], "+", "p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "s"]], "-", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "c"]]]]]], "]"]]]], "-", RowBox[List[SqrtBox["c"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", "m"]], "+", "p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "s"]]]], ")"]], "2"], RowBox[List["4", " ", "c"]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", "m"]], "+", "p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "s"]], "+", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox["c"]]]], "]"]]]]]], ")"]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 p z cos m ( b z ) cosh ( c z 2 ) z - 2 - m - 2 π ( 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]]]]] ( - c p 2 4 c erfi ( p - 2 c z 2 - c ) - c - p 2 4 c erfi ( p + 2 c z 2 c ) ) ( 1 - m mod 2 \$CellContext`m 2 ) c - 1 c ( ( 2 - m - 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]]]]] ( ( - c ( b m + p - 2 b s ) 2 4 c erfi ( b m + p - 2 b s - 2 c z 2 - c ) - c - ( - b m + p + 2 b s ) 2 4 c erfi ( - b m + p + 2 b s + 2 c z 2 c ) ) + ( - c ( - b m + p + 2 b s ) 2 4 c erfi ( - b m + p + 2 b s - 2 c z 2 - c ) - c - ( b m + p - 2 b s ) 2 4 c erfi ( b m + p - 2 b s + 2 c z 2 c ) ) ) ) /; m TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] m > 0 Condition z p z b z m c z 2 -1 2 -1 m -2 1 2 Binomial m m 2 -1 -1 c 1 2 p 2 4 c -1 Erfi p -1 2 c z 2 -1 c 1 2 -1 -1 c 1 2 -1 p 2 4 c -1 Erfi p 2 c z 2 c 1 2 -1 1 -1 \$CellContext`m 2 c -1 -1 1 c -1 2 -1 m -2 1 2 s 0 m -1 2 -1 Binomial m s -1 c 1 2 b m p -1 2 b s 2 4 c -1 Erfi b m p -1 2 b s -1 2 c z 2 -1 c 1 2 -1 -1 c 1 2 -1 -1 b m p 2 b s 2 4 c -1 Erfi -1 b m p 2 b s 2 c z 2 c 1 2 -1 -1 c 1 2 -1 b m p 2 b s 2 4 c -1 Erfi -1 b m p 2 b s -1 2 c z 2 -1 c 1 2 -1 -1 c 1 2 -1 b m p -1 2 b s 2 4 c -1 Erfi b m p -1 2 b s 2 c z 2 c 1 2 -1 m m 0 [/itex]

 Rule Form

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

 2002-12-18