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 Cosh

 http://functions.wolfram.com/01.20.21.1421.01

 Input Form

 Integrate[z^(\[Alpha] - 1) E^(b z^r + e) Cos[a z^r + q] Cosh[c z^r + g], z] == (-(z^\[Alpha]/(4 r))) ((E^(e + g + I q) Gamma[\[Alpha]/r, ((-I) a - b - c) z^r])/ (((-I) a - b - c) z^r)^(\[Alpha]/r) + (E^(e + g - I q) Gamma[\[Alpha]/r, (I a - b - c) z^r])/ ((I a - b - c) z^r)^(\[Alpha]/r) + (E^(e - g + I q) Gamma[\[Alpha]/r, ((-I) a - b + c) z^r])/ (((-I) a - b + c) z^r)^(\[Alpha]/r) + (E^(e - g - I q) Gamma[\[Alpha]/r, (I a - b + c) z^r])/ ((I a - b + c) z^r)^(\[Alpha]/r))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["b", " ", SuperscriptBox["z", "r"]]], "+", "e"]]], RowBox[List["Cos", "[", RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "+", "q"]], "]"]], RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c", " ", SuperscriptBox["z", "r"]]], "+", "g"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " "]], RowBox[List["4", " ", "r"]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["e", "+", "g", "+", RowBox[List["\[ImaginaryI]", " ", "q"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "-", "b", "-", "c"]], ")"]], " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "r"], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "-", "b", "-", "c"]], ")"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["e", "+", "g", "-", RowBox[List["\[ImaginaryI]", " ", "q"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "b", "-", "c"]], ")"]], " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "r"], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "b", "-", "c"]], ")"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["e", "-", "g", "+", RowBox[List["\[ImaginaryI]", " ", "q"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "-", "b", "+", "c"]], ")"]], " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "r"], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "-", "b", "+", "c"]], ")"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["e", "-", "g", "-", RowBox[List["\[ImaginaryI]", " ", "q"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "b", "+", "c"]], ")"]], " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "r"], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "b", "+", "c"]], ")"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 b z r + e cos ( a z r + q ) cosh ( c z r + g ) z - z α 4 r ( e + g + q Γ ( α r , ( - b - c - a ) z r ) ( ( - b - c - a ) z r ) - α r + e + g - q ( ( - b - c + a ) z r ) - α r Γ ( α r , ( - b - c + a ) z r ) + e - g + q ( ( - b + c - a ) z r ) - α r Γ ( α r , ( - b + c - a ) z r ) + e - g - q ( ( - b + c + a ) z r ) - α r Γ ( α r , ( - b + c + a ) z r ) ) z z α -1 b z r e a z r q c z r g -1 z α 4 r -1 e g q Gamma α r -1 -1 b -1 c -1 a z r -1 b -1 c -1 a z r -1 α r -1 e g -1 q -1 b -1 c a z r -1 α r -1 Gamma α r -1 -1 b -1 c a z r e -1 g q -1 b c -1 a z r -1 α r -1 Gamma α r -1 -1 b c -1 a z r e -1 g -1 q -1 b c a z r -1 α r -1 Gamma α r -1 -1 b c a z r [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["b_", " ", SuperscriptBox["z_", "r_"]]], "+", "e_"]]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]], "+", "q_"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SuperscriptBox["z_", "r_"]]], "+", "g_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["e", "+", "g", "+", RowBox[List["\[ImaginaryI]", " ", "q"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "-", "b", "-", "c"]], ")"]], " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "r"], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "-", "b", "-", "c"]], ")"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["e", "+", "g", "-", RowBox[List["\[ImaginaryI]", " ", "q"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "b", "-", "c"]], ")"]], " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "r"], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "b", "-", "c"]], ")"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["e", "-", "g", "+", RowBox[List["\[ImaginaryI]", " ", "q"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "-", "b", "+", "c"]], ")"]], " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "r"], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "-", "b", "+", "c"]], ")"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["e", "-", "g", "-", RowBox[List["\[ImaginaryI]", " ", "q"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "b", "+", "c"]], ")"]], " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "r"], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "b", "+", "c"]], ")"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]]]], ")"]]]], RowBox[List["4", " ", "r"]]]]]]]]]

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

 2002-12-18