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 Cosh

 http://functions.wolfram.com/01.20.21.4817.01

 Input Form

 Integrate[z^n E^(p Sqrt[z]) Sinh[b z] Cosh[c Sqrt[z]]^v, z] == (2^(-2 - 2 n - v) Binomial[v, v/2] (1 - Mod[v, 2]) ((-E^(p^2/(4 b))) Sum[(-1)^(-h + i) 4^i p^(-h - i + 2 n) (p - 2 b Sqrt[z])^(h + i) ((p - 2 b Sqrt[z])^2/b)^ ((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] (p (p - 2 b Sqrt[z]) Gamma[(1/2) (1 + h + i), (p - 2 b Sqrt[z])^2/ (4 b)] - 2 b Sqrt[(p - 2 b Sqrt[z])^2/b] Gamma[(1/2) (2 + h + i), (p - 2 b Sqrt[z])^2/(4 b)]), {i, 0, n}, {h, 0, i}] + Sum[(-1)^(-h + i) 4^i p^(-h - i + 2 n) (p + 2 b Sqrt[z])^(h + i) (-((p + 2 b Sqrt[z])^2/b))^ ((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] (p (p + 2 b Sqrt[z]) Gamma[(1/2) (1 + h + i), -((p + 2 b Sqrt[z])^2/(4 b))] + 2 b Sqrt[-((p + 2 b Sqrt[z])^2/b)] Gamma[(1/2) (2 + h + i), -((p + 2 b Sqrt[z])^2/(4 b))]), {i, 0, n}, {h, 0, i}]/ E^(p^2/(4 b))))/b^(2 (1 + n)) + (2^(-2 - 2 n - v) Sum[Binomial[v, s] ((-E^((p - c (-2 s + v))^2/(4 b))) Sum[(-1)^(-h + i) 4^i (p - c (-2 s + v))^(-h - i + 2 n) (p - c (-2 s + v) - 2 b Sqrt[z])^(h + i) ((p - c (-2 s + v) - 2 b Sqrt[z])^2/b)^((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] ((p - c (-2 s + v)) (p - c (-2 s + v) - 2 b Sqrt[z]) Gamma[(1/2) (1 + h + i), (p - c (-2 s + v) - 2 b Sqrt[z])^2/(4 b)] - 2 b Sqrt[(p - c (-2 s + v) - 2 b Sqrt[z])^2/b] Gamma[(1/2) (2 + h + i), (p - c (-2 s + v) - 2 b Sqrt[z])^2/(4 b)]), {i, 0, n}, {h, 0, i}] - E^((p + c (-2 s + v))^2/(4 b)) Sum[(-1)^(-h + i) 4^i (p + c (-2 s + v))^(-h - i + 2 n) (p + c (-2 s + v) - 2 b Sqrt[z])^(h + i) ((p + c (-2 s + v) - 2 b Sqrt[z])^2/b)^((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] ((p + c (-2 s + v)) (p + c (-2 s + v) - 2 b Sqrt[z]) Gamma[(1/2) (1 + h + i), (p + c (-2 s + v) - 2 b Sqrt[z])^2/(4 b)] - 2 b Sqrt[(p + c (-2 s + v) - 2 b Sqrt[z])^2/b] Gamma[(1/2) (2 + h + i), (p + c (-2 s + v) - 2 b Sqrt[z])^2/(4 b)]), {i, 0, n}, {h, 0, i}] + Sum[(-1)^(-h + i) 4^i (p - c (-2 s + v))^(-h - i + 2 n) (p - c (-2 s + v) + 2 b Sqrt[z])^(h + i) (-((p - c (-2 s + v) + 2 b Sqrt[z])^2/b))^((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] ((p - c (-2 s + v)) (p - c (-2 s + v) + 2 b Sqrt[z]) Gamma[(1/2) (1 + h + i), -((p - c (-2 s + v) + 2 b Sqrt[z])^2/(4 b))] + 2 b Sqrt[-((p - c (-2 s + v) + 2 b Sqrt[z])^2/b)] Gamma[(1/2) (2 + h + i), -((p - c (-2 s + v) + 2 b Sqrt[z])^2/ (4 b))]), {i, 0, n}, {h, 0, i}]/ E^((p - c (-2 s + v))^2/(4 b)) + Sum[(-1)^(-h + i) 4^i (p + c (-2 s + v))^(-h - i + 2 n) (p + c (-2 s + v) + 2 b Sqrt[z])^(h + i) (-((p + c (-2 s + v) + 2 b Sqrt[z])^2/b))^((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] ((p + c (-2 s + v)) (p + c (-2 s + v) + 2 b Sqrt[z]) Gamma[(1/2) (1 + h + i), -((p + c (-2 s + v) + 2 b Sqrt[z])^2/(4 b))] + 2 b Sqrt[-((p + c (-2 s + v) + 2 b Sqrt[z])^2/b)] Gamma[(1/2) (2 + h + i), -((p + c (-2 s + v) + 2 b Sqrt[z])^2/ (4 b))]), {i, 0, n}, {h, 0, i}]/ E^((p + c (-2 s + v))^2/(4 b))), {s, 0, Floor[(1/2) (-1 + v)]}])/ b^(2 (1 + n)) /; Element[v, Integers] && v > 0 && Element[n, Integers] && n >= 0

 Standard Form

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RowBox[List["1", "+", "h", "+", "i"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "b"]]]]]]], "]"]]]], "+", RowBox[List["2", " ", "b", " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"], "b"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "i"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["c", " ", RowBox[List["(", 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RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "i"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "b"]]]]]]], "]"]]]], "+", RowBox[List["2", " ", "b", " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", 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 MathML Form

 z n p z sinh ( b z ) cosh v ( c z ) z 2 - 2 n - v - 2 ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["v", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 - v mod 2 \$CellContext`v 2 ) ( - p 2 4 b i = 0 n h = 0 i ( - 1 ) i - h 4 i p - h - i + 2 n ( 2 z b + p ) h + i ( - ( 2 z b + p ) 2 b ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity, Rule[Editable, True]]], List[TagBox["h", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["i", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( p ( 2 z b + p ) Γ ( 1 2 ( h + i + 1 ) , - ( 2 z b + p ) 2 4 b ) + 2 - ( 2 z b + p ) 2 b b Γ ( 1 2 ( h + i + 2 ) , - ( 2 z b + p ) 2 4 b ) ) - p 2 4 b i = 0 n h = 0 i ( - 1 ) i - h 4 i p - h - i + 2 n ( p - 2 b z ) h + i ( ( p - 2 b z ) 2 b ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity, Rule[Editable, True]]], List[TagBox["h", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["i", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( p ( p - 2 b z ) Γ ( 1 2 ( h + i + 1 ) , ( p - 2 b z ) 2 4 b ) - 2 b ( p - 2 b z ) 2 b Γ ( 1 2 ( h + i + 2 ) , ( p - 2 b z ) 2 4 b ) ) ) b - 2 ( n + 1 ) + 2 - 2 n - v - 2 ( s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - ( p + c ( v - 2 s ) ) 2 4 b i = 0 n h = 0 i ( - 1 ) i - h 4 i ( p + c ( v - 2 s ) ) - h - i + 2 n ( - 2 z b + p + c ( v - 2 s ) ) h + i ( ( - 2 z b + p + c ( v - 2 s ) ) 2 b ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity, Rule[Editable, True]]], List[TagBox["h", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["i", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( p + c ( v - 2 s ) ) ( - 2 z b + p + c ( v - 2 s ) ) Γ ( 1 2 ( h + i + 1 ) , ( - 2 z b + p + c ( v - 2 s ) ) 2 4 b ) - 2 b ( - 2 z b + p + c ( v - 2 s ) ) 2 b Γ ( 1 2 ( h + i + 2 ) , ( - 2 z b + p + c ( v - 2 s ) ) 2 4 b ) ) - ( p - c ( v - 2 s ) ) 2 4 b i = 0 n h = 0 i ( - 1 ) i - h 4 i ( p - c ( v - 2 s ) ) - h - i + 2 n ( - 2 z b + p - c ( v - 2 s ) ) h + i ( ( - 2 z b + p - c ( v - 2 s ) ) 2 b ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity, Rule[Editable, True]]], List[TagBox["h", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["i", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( p - c ( v - 2 s ) ) ( - 2 z b + p - c ( v - 2 s ) ) Γ ( 1 2 ( h + i + 1 ) , ( - 2 z b + p - c ( v - 2 s ) ) 2 4 b ) - 2 b ( - 2 z b + p - c ( v - 2 s ) ) 2 b Γ ( 1 2 ( h + i + 2 ) , ( - 2 z b + p - c ( v - 2 s ) ) 2 4 b ) ) + - ( p + c ( v - 2 s ) ) 2 4 b i = 0 n h = 0 i ( - 1 ) i - h 4 i ( p + c ( v - 2 s ) ) - h - i + 2 n ( 2 z b + p + c ( v - 2 s ) ) h + i ( - ( 2 z b + p + c ( v - 2 s ) ) 2 b ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity, Rule[Editable, True]]], List[TagBox["h", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["i", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( p + c ( v - 2 s ) ) ( 2 z b + p + c ( v - 2 s ) ) Γ ( 1 2 ( h + i + 1 ) , - ( 2 z b + p + c ( v - 2 s ) ) 2 4 b ) + 2 - ( 2 z b + p + c ( v - 2 s ) ) 2 b b Γ ( 1 2 ( h + i + 2 ) , - ( 2 z b + p + c ( v - 2 s ) ) 2 4 b ) ) + - ( p - c ( v - 2 s ) ) 2 4 b i = 0 n h = 0 i ( - 1 ) i - h 4 i ( p - c ( v - 2 s ) ) - h - i + 2 n ( 2 z b + p - c ( v - 2 s ) ) h + i ( - ( 2 z b + p - c ( v - 2 s ) ) 2 b ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity, Rule[Editable, True]]], List[TagBox["h", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["i", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( p - c ( v - 2 s ) ) ( 2 z b + p - c ( v - 2 s ) ) Γ ( 1 2 ( h + i + 1 ) , - ( 2 z b + p - c ( v - 2 s ) ) 2 4 b ) + 2 - ( 2 z b + p - c ( v - 2 s ) ) 2 b b Γ ( 1 2 ( h + i + 2 ) , - ( 2 z b + p - c ( v - 2 s ) ) 2 4 b ) ) ) ) b - 2 ( n + 1 ) /; v + n Condition z z n p z 1 2 b z c z 1 2 v 2 -2 n -1 v -2 Binomial v v 2 -1 1 -1 \$CellContext`v 2 -1 p 2 4 b -1 h 0 i i 0 n -1 i -1 h 4 i p -1 h -1 i 2 n 2 z 1 2 b p h i -1 2 z 1 2 b p 2 b -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i p 2 z 1 2 b p Gamma 1 2 h i 1 -1 2 z 1 2 b p 2 4 b -1 2 -1 2 z 1 2 b p 2 b -1 1 2 b Gamma 1 2 h i 2 -1 2 z 1 2 b p 2 4 b -1 -1 p 2 4 b -1 h 0 i i 0 n -1 i -1 h 4 i p -1 h -1 i 2 n p -1 2 b z 1 2 h i p -1 2 b z 1 2 2 b -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i p p -1 2 b z 1 2 Gamma 1 2 h i 1 p -1 2 b z 1 2 2 4 b -1 -1 2 b p -1 2 b z 1 2 2 b -1 1 2 Gamma 1 2 h i 2 p -1 2 b z 1 2 2 4 b -1 b -2 n 1 2 -2 n -1 v -2 s 0 v -1 2 -1 Binomial v s -1 p c v -1 2 s 2 4 b -1 h 0 i i 0 n -1 i -1 h 4 i p c v -1 2 s -1 h -1 i 2 n -2 z 1 2 b p c v -1 2 s h i -2 z 1 2 b p c v -1 2 s 2 b -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i p c v -1 2 s -2 z 1 2 b p c v -1 2 s Gamma 1 2 h i 1 -2 z 1 2 b p c v -1 2 s 2 4 b -1 -1 2 b -2 z 1 2 b p c v -1 2 s 2 b -1 1 2 Gamma 1 2 h i 2 -2 z 1 2 b p c v -1 2 s 2 4 b -1 -1 p -1 c v -1 2 s 2 4 b -1 h 0 i i 0 n -1 i -1 h 4 i p -1 c v -1 2 s -1 h -1 i 2 n -2 z 1 2 b p -1 c v -1 2 s h i -2 z 1 2 b p -1 c v -1 2 s 2 b -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i p -1 c v -1 2 s -2 z 1 2 b p -1 c v -1 2 s Gamma 1 2 h i 1 -2 z 1 2 b p -1 c v -1 2 s 2 4 b -1 -1 2 b -2 z 1 2 b p -1 c v -1 2 s 2 b -1 1 2 Gamma 1 2 h i 2 -2 z 1 2 b p -1 c v -1 2 s 2 4 b -1 -1 p c v -1 2 s 2 4 b -1 h 0 i i 0 n -1 i -1 h 4 i p c v -1 2 s -1 h -1 i 2 n 2 z 1 2 b p c v -1 2 s h i -1 2 z 1 2 b p c v -1 2 s 2 b -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i p c v -1 2 s 2 z 1 2 b p c v -1 2 s Gamma 1 2 h i 1 -1 2 z 1 2 b p c v -1 2 s 2 4 b -1 2 -1 2 z 1 2 b p c v -1 2 s 2 b -1 1 2 b Gamma 1 2 h i 2 -1 2 z 1 2 b p c v -1 2 s 2 4 b -1 -1 p -1 c v -1 2 s 2 4 b -1 h 0 i i 0 n -1 i -1 h 4 i p -1 c v -1 2 s -1 h -1 i 2 n 2 z 1 2 b p -1 c v -1 2 s h i -1 2 z 1 2 b p -1 c v -1 2 s 2 b -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i p -1 c v -1 2 s 2 z 1 2 b p -1 c v -1 2 s Gamma 1 2 h i 1 -1 2 z 1 2 b p -1 c v -1 2 s 2 4 b -1 2 -1 2 z 1 2 b p -1 c v -1 2 s 2 b -1 1 2 b Gamma 1 2 h i 2 -1 2 z 1 2 b p -1 c v -1 2 s 2 4 b -1 b -2 n 1 v SuperPlus n [/itex]

 Rule Form

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

 2002-12-18