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 Cosh

 http://functions.wolfram.com/01.20.21.1258.01

 Input Form

 Integrate[E^(p Sqrt[z]) Cos[b Sqrt[z]] Cosh[c z], z] == (1/8) ((8 E^(p Sqrt[z]) Cos[b Sqrt[z]] Sinh[c z])/c + (c ((-I) b + p) Sqrt[Pi] Erfi[((-I) b + p - 2 c Sqrt[z])/(2 Sqrt[-c])])/ (E^((b + I p)^2/(4 c)) (-c)^(5/2)) + (c E^((I b + p)^2/(4 c)) (I b + p) Sqrt[Pi] Erfi[(I b + p - 2 c Sqrt[z])/(2 Sqrt[-c])])/(-c)^(5/2) + (I E^((b + I p)^2/(4 c)) (b + I p) Sqrt[Pi] Erfi[((-I) b + p + 2 c Sqrt[z])/(2 Sqrt[c])])/c^(3/2) - (E^((b - I p)^2/(4 c)) (I b + p) Sqrt[Pi] Erfi[(I b + p + 2 c Sqrt[z])/(2 Sqrt[c])])/c^(3/2))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", SqrtBox["z"]]]], " ", RowBox[List["Cos", "[", RowBox[List["b", " ", SqrtBox["z"]]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["8", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", SqrtBox["z"]]]], " ", RowBox[List["Cos", "[", RowBox[List["b", " ", SqrtBox["z"]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]], "c"], "+", FractionBox[RowBox[List["c", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "p"]]]], ")"]], "2"], RowBox[List["4", " ", "c"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "p"]], ")"]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "p", "-", RowBox[List["2", " ", "c", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "c"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "c"]], ")"]], RowBox[List["5", "/", "2"]]]], "+", FractionBox[RowBox[List["c", " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p"]], ")"]], "2"], RowBox[List["4", " ", "c"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p"]], ")"]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p", "-", RowBox[List["2", " ", "c", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "c"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "c"]], ")"]], RowBox[List["5", "/", "2"]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "p"]]]], ")"]], "2"], RowBox[List["4", " ", "c"]]]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "p"]]]], ")"]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "p", "+", RowBox[List["2", " ", "c", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["c"]]]], "]"]]]], SuperscriptBox["c", RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], ")"]], "2"], RowBox[List["4", " ", "c"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p"]], ")"]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p", "+", RowBox[List["2", " ", "c", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["c"]]]], "]"]]]], SuperscriptBox["c", RowBox[List["3", "/", "2"]]]]]], ")"]]]]]]]]

 MathML Form

 p z cos ( b z ) cosh ( c z ) z 1 8 ( 8 p z cos ( b z ) sinh ( c z ) c + c - ( b + p ) 2 4 c ( - b + p ) π erfi ( - 2 z c - b + p 2 - c ) ( - c ) 5 / 2 + c ( b + p ) 2 4 c ( b + p ) π erfi ( - 2 z c + b + p 2 - c ) ( - c ) 5 / 2 + ( b + p ) 2 4 c ( b + p ) π erfi ( 2 z c - b + p 2 c ) c 3 / 2 - ( b - p ) 2 4 c ( b + p ) π erfi ( 2 z c + b + p 2 c ) c 3 / 2 ) z p z 1 2 b z 1 2 c z 1 8 8 p z 1 2 b z 1 2 c z c -1 c -1 b p 2 4 c -1 -1 b p 1 2 Erfi -2 z 1 2 c -1 b p 2 -1 c 1 2 -1 -1 c 5 2 -1 c b p 2 4 c -1 b p 1 2 Erfi -2 z 1 2 c b p 2 -1 c 1 2 -1 -1 c 5 2 -1 b p 2 4 c -1 b p 1 2 Erfi 2 z 1 2 c -1 b p 2 c 1 2 -1 c 3 2 -1 -1 b -1 p 2 4 c -1 b p 1 2 Erfi 2 z 1 2 c b p 2 c 1 2 -1 c 3 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", SqrtBox["z_"]]]], " ", RowBox[List["Cos", "[", RowBox[List["b_", " ", SqrtBox["z_"]]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["8", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", SqrtBox["z"]]]], " ", RowBox[List["Cos", "[", RowBox[List["b", " ", SqrtBox["z"]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]], "c"], "+", FractionBox[RowBox[List["c", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "p"]]]], ")"]], "2"], RowBox[List["4", " ", "c"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "p"]], ")"]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "p", "-", RowBox[List["2", " ", "c", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "c"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "c"]], ")"]], RowBox[List["5", "/", "2"]]]], "+", FractionBox[RowBox[List["c", " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p"]], ")"]], "2"], RowBox[List["4", " ", "c"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p"]], ")"]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p", "-", RowBox[List["2", " ", "c", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "c"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "c"]], ")"]], RowBox[List["5", "/", "2"]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "p"]]]], ")"]], "2"], RowBox[List["4", " ", "c"]]]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "p"]]]], ")"]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "p", "+", RowBox[List["2", " ", "c", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["c"]]]], "]"]]]], SuperscriptBox["c", RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], ")"]], "2"], RowBox[List["4", " ", "c"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p"]], ")"]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p", "+", RowBox[List["2", " ", "c", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["c"]]]], "]"]]]], SuperscriptBox["c", RowBox[List["3", "/", "2"]]]]]], ")"]]]]]]]]

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

 2002-12-18