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 Cosh

 http://functions.wolfram.com/01.20.21.4468.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18