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 Cosh

 http://functions.wolfram.com/01.20.21.1073.01

 Input Form

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

 Standard Form

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

 MathML Form

 z α - 1 cos ( b z r + e ) cosh ( c z r + g ) z - 1 4 r ( z α ( g + e Γ ( α r , ( - c - b ) z r ) ( ( - c - b ) z r ) - α r + g - e ( ( b - c ) z r ) - α r Γ ( α r , ( b - c ) z r ) + e - g ( ( c - b ) z r ) - α r Γ ( α r , ( c - b ) z r ) + - g - e ( ( c + b ) z r ) - α r Γ ( α r , ( c + b ) z r ) ) ) z z α -1 b z r e c z r g -1 1 4 r -1 z α g e Gamma α r -1 -1 c -1 b z r -1 c -1 b z r -1 α r -1 g -1 e b -1 c z r -1 α r -1 Gamma α r -1 b -1 c z r e -1 g c -1 b z r -1 α r -1 Gamma α r -1 c -1 b z r -1 g -1 e c b z r -1 α r -1 Gamma α r -1 c b 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"]]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["b_", " ", SuperscriptBox["z_", "r_"]]], "+", "e_"]], "]"]], " ", 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[RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "g"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "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]"]], " ", "b"]], "-", "c"]], ")"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "e"]], "+", "g"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "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]", " ", "b"]], "-", "c"]], ")"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "e"]], "-", "g"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "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]"]], " ", "b"]], "+", "c"]], ")"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "e"]], "-", "g"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "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]", " ", "b"]], "+", "c"]], ")"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]]]], ")"]]]], RowBox[List["4", " ", "r"]]]]]]]]]

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

 2002-12-18