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 Cosh

 http://functions.wolfram.com/01.20.21.1419.01

 Input Form

 Integrate[z^n E^(p z^2) Cos[b z^2] Cosh[c z^2], z] == (-(1/8)) z^(1 + n) ((((-I) b - c - p) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, ((-I) b - c - p) z^2] + ((I b - c - p) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (I b - c - p) z^2] + (((-I) b + c - p) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, ((-I) b + c - p) z^2] + ((I b + c - p) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (I b + c - p) z^2]) /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", SuperscriptBox["z", "2"]]]], RowBox[List["Cos", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]], RowBox[List["Cosh", "[", RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "8"]]], " ", SuperscriptBox["z", RowBox[List["1", "+", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c", "-", "p"]], ")"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "n"]], "2"], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c", "-", "p"]], ")"]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c", "-", "p"]], ")"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "n"]], "2"], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c", "-", "p"]], ")"]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c", "-", "p"]], ")"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "n"]], "2"], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c", "-", "p"]], ")"]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c", "-", "p"]], ")"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "n"]], "2"], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c", "-", "p"]], ")"]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n p z 2 cos ( b z 2 ) cosh ( c z 2 ) z - 1 8 z n + 1 ( Γ ( n + 1 2 , ( - c - b - p ) z 2 ) ( ( - c - b - p ) z 2 ) 1 2 ( - n - 1 ) + ( ( - c + b - p ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , ( - c + b - p ) z 2 ) + ( ( c - b - p ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , ( c - b - p ) z 2 ) + ( ( c + b - p ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , ( c + b - p ) z 2 ) ) /; n Condition z z n p z 2 b z 2 c z 2 -1 1 8 z n 1 Gamma n 1 2 -1 -1 c -1 b -1 p z 2 -1 c -1 b -1 p z 2 1 2 -1 n -1 -1 c b -1 p z 2 1 2 -1 n -1 Gamma n 1 2 -1 -1 c b -1 p z 2 c -1 b -1 p z 2 1 2 -1 n -1 Gamma n 1 2 -1 c -1 b -1 p z 2 c b -1 p z 2 1 2 -1 n -1 Gamma n 1 2 -1 c b -1 p z 2 n [/itex]

 Rule Form

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

 2002-12-18