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 Cosh

 http://functions.wolfram.com/01.20.21.4474.01

 Input Form

 Integrate[z^(\[Alpha] - 1) Sinh[b z^r + e] Cosh[c z^r]^v, z] == 2^(-1 - v) z^\[Alpha] ((1/r) (Binomial[v, v/2] ((E^e Gamma[\[Alpha]/r, (-b) z^r])/ ((-b) z^r)^(\[Alpha]/r) - Gamma[\[Alpha]/r, b z^r]/ (E^e (b z^r)^(\[Alpha]/r))) (-1 + Mod[v, 2])) - ((1/r) Sum[Binomial[v, s] ((E^(2 e) Gamma[\[Alpha]/r, (-b + 2 c s - c v) z^r])/((-b + 2 c s - c v) z^r)^(\[Alpha]/r) - Gamma[\[Alpha]/r, (b + 2 c s - c v) z^r]/((b + 2 c s - c v) z^r)^ (\[Alpha]/r) + (E^(2 e) Gamma[\[Alpha]/r, (-b - 2 c s + c v) z^r])/ ((-b - 2 c s + c v) z^r)^(\[Alpha]/r) - Gamma[\[Alpha]/r, (b - 2 c s + c v) z^r]/((b - 2 c s + c v) z^r)^ (\[Alpha]/r)), {s, 0, Floor[(1/2) (-1 + v)]}])/E^e) /; Element[v, Integers] && v > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["Sinh", "[", RowBox[List[RowBox[List["b", " ", SuperscriptBox["z", "r"]]], "+", "e"]], "]"]], SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", SuperscriptBox["z", "r"]]], "]"]], "v"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "v"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "r"], RowBox[List["(", RowBox[List[RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", "e"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "r"], ",", RowBox[List[RowBox[List["-", "b"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "e"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "r"], ",", RowBox[List["b", " ", SuperscriptBox["z", "r"]]]]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], ")"]]]], "-", RowBox[List[FractionBox["1", "r"], SuperscriptBox["\[ExponentialE]", RowBox[List["-", "e"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "r"], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "-", " ", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "r"], ",", RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "r"], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["b", "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "r"], ",", RowBox[List[RowBox[List["(", RowBox[List["b", "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]]]], ")"]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

 z α - 1 sinh ( b z r + e ) cosh v ( c z r ) z 2 - v - 1 z α ( ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( e ( - b z r ) - α r Γ ( α r , - b z r ) - - e ( b z r ) - α r Γ ( α r , b z r ) ) ( v mod 2 \$CellContext`v 2 - 1 ) r - 1 r - e s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 e Γ ( α r , ( - b - 2 c s + c v ) z r ) ( ( - b - 2 c s + c v ) z r ) - α r - ( ( b - 2 c s + c v ) z r ) - α r Γ ( α r , ( b - 2 c s + c v ) z r ) + 2 e ( ( - b + 2 c s - c v ) z r ) - α r Γ ( α r , ( - b + 2 c s - c v ) z r ) - ( ( b + 2 c s - c v ) z r ) - α r Γ ( α r , ( b + 2 c s - c v ) z r ) ) ) /; v + Condition z z α -1 b z r e c z r v 2 -1 v -1 z α Binomial v v 2 -1 e -1 b z r -1 α r -1 Gamma α r -1 -1 b z r -1 -1 e b z r -1 α r -1 Gamma α r -1 b z r \$CellContext`v 2 -1 r -1 -1 1 r -1 -1 e s 0 v -1 2 -1 Binomial v s 2 e Gamma α r -1 -1 b -1 2 c s c v z r -1 b -1 2 c s c v z r -1 α r -1 -1 b -1 2 c s c v z r -1 α r -1 Gamma α r -1 b -1 2 c s c v z r 2 e -1 b 2 c s -1 c v z r -1 α r -1 Gamma α r -1 -1 b 2 c s -1 c v z r -1 b 2 c s -1 c v z r -1 α r -1 Gamma α r -1 b 2 c s -1 c v z r v SuperPlus [/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["Sinh", "[", RowBox[List[RowBox[List["b_", " ", SuperscriptBox["z_", "r_"]]], "+", "e_"]], "]"]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c_", " ", SuperscriptBox["z_", "r_"]]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "v"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", "e"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "r"], ",", RowBox[List[RowBox[List["-", "b"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "e"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "r"], ",", RowBox[List["b", " ", SuperscriptBox["z", "r"]]]]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], "r"], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "e"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "r"], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "r"], ",", RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "r"], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["b", "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "r"], ",", RowBox[List[RowBox[List["(", RowBox[List["b", "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SuperscriptBox["z", "r"]]]]], "]"]]]]]], ")"]]]]]]]], "r"]]], ")"]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "&&", RowBox[List["v", ">", "0"]]]]]]]]]]

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

 2002-12-18