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 Cosh

 http://functions.wolfram.com/01.20.21.4809.01

 Input Form

 Integrate[z^n E^(p Sqrt[z]) Sinh[b Sqrt[z]] Cosh[c z]^v, z] == (-2^(-v)) z^(1 + n) Binomial[v, v/2] ((-((b - p) Sqrt[z])^(-2 (1 + n))) Gamma[2 (1 + n), (b - p) Sqrt[z]] + Gamma[2 (1 + n), (-(b + p)) Sqrt[z]]/((-(b + p)) Sqrt[z])^(2 (1 + n))) (1 - Mod[v, 2]) + (1/c^2) 2^(-2 - 2 n - v) Sum[(1/(-2 s + v)^2) Binomial[v, s] (Sum[(-1)^(-h + k) 4^k (b + p)^(-h - k + 2 n) (b + p + 2 c (2 s - v) Sqrt[z])^(h + k) (-((b + p + 2 c (2 s - v) Sqrt[z])^2/(c (2 s - v))))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((b + p) (b + p + 2 c (2 s - v) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((b + p + 2 c (2 s - v) Sqrt[z])^2/(c (8 s - 4 v)))] + 2 c (2 s - v) Sqrt[-((b + p + 2 c (2 s - v) Sqrt[z])^2/(c (2 s - v)))] Gamma[(1/2) (2 + h + k), -((b + p + 2 c (2 s - v) Sqrt[z])^2/(c (8 s - 4 v)))]), {k, 0, n}, {h, 0, k}]/(E^((b + p)^2/(8 c s - 4 c v)) (c (2 s - v))^(2 n)) - Sum[(-1)^(-h + k) 4^k (-b + p)^(-h - k + 2 n) (-b + p + 2 c (2 s - v) Sqrt[z])^(h + k) (-((b - p + 2 c (-2 s + v) Sqrt[z])^2/(c (2 s - v))))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((-b + p) (-b + p + 2 c (2 s - v) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((b - p + 2 c (-2 s + v) Sqrt[z])^2/(c (8 s - 4 v)))] + 2 c (2 s - v) Sqrt[-((b - p + 2 c (-2 s + v) Sqrt[z])^2/(c (2 s - v)))] Gamma[(1/2) (2 + h + k), -((b - p + 2 c (-2 s + v) Sqrt[z])^2/(c (8 s - 4 v)))]), {k, 0, n}, {h, 0, k}]/(E^((b - p)^2/(8 c s - 4 c v)) (c (2 s - v))^(2 n)) + (E^((b - p)^2/(8 c s - 4 c v)) (-Sum[(-1)^(-h + k) 4^k (-b + p)^(-h - k + 2 n) (-b + p + 2 c (-2 s + v) Sqrt[z])^(h + k) (-((-b + p + 2 c (-2 s + v) Sqrt[z])^2/(c (-2 s + v))))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((-b + p) (-b + p + 2 c (-2 s + v) Sqrt[z]) Gamma[ (1/2) (1 + h + k), (-b + p + 2 c (-2 s + v) Sqrt[z])^2/ (c (8 s - 4 v))] + 2 c (-2 s + v) Sqrt[ -((-b + p + 2 c (-2 s + v) Sqrt[z])^2/(c (-2 s + v)))] Gamma[ (1/2) (2 + h + k), (-b + p + 2 c (-2 s + v) Sqrt[z])^2/ (c (8 s - 4 v))]), {k, 0, n}, {h, 0, k}] + E^((b p)/(2 c s - c v)) Sum[(-1)^(-h + k) 4^k (b + p)^(-h - k + 2 n) (b + p + 2 c (-2 s + v) Sqrt[z])^(h + k) (-((b + p + 2 c (-2 s + v) Sqrt[z])^2/(c (-2 s + v))))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((b + p) (b + p + 2 c (-2 s + v) Sqrt[z]) Gamma[ (1/2) (1 + h + k), (b + p + 2 c (-2 s + v) Sqrt[z])^2/ (c (8 s - 4 v))] + 2 c (-2 s + v) Sqrt[ -((b + p + 2 c (-2 s + v) Sqrt[z])^2/(c (-2 s + v)))] Gamma[ (1/2) (2 + h + k), (b + p + 2 c (-2 s + v) Sqrt[z])^2/ (c (8 s - 4 v))]), {k, 0, n}, {h, 0, k}]))/ (c (-2 s + v))^(2 n)), {s, 0, Floor[(1/2) (-1 + v)]}] /; Element[v, Integers] && v > 0 && Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n p z sinh ( b z ) cosh v ( c z ) z 2 - 2 n - v - 2 c 2 s = 0 v - 1 2 1 ( v - 2 s ) 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - ( b + p ) 2 8 c s - 4 c v ( c ( 2 s - v ) ) - 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b + p ) - h - k + 2 n ( b + p + 2 c ( 2 s - v ) z ) h + k ( - ( b + p + 2 c ( 2 s - v ) z ) 2 c ( 2 s - v ) ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b + p ) ( b + p + 2 c ( 2 s - v ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + p + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( 2 s - v ) - ( b + p + 2 c ( 2 s - v ) z ) 2 c ( 2 s - v ) Γ ( 1 2 ( h + k + 2 ) , - ( b + p + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) ) - - ( b - p ) 2 8 c s - 4 c v ( c ( 2 s - v ) ) - 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( p - b ) - h - k + 2 n ( - b + p + 2 c ( 2 s - v ) z ) h + k ( - ( b - p + 2 c ( v - 2 s ) z ) 2 c ( 2 s - v ) ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( p - b ) ( - b + p + 2 c ( 2 s - v ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b - p + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( 2 s - v ) - ( b - p + 2 c ( v - 2 s ) z ) 2 c ( 2 s - v ) Γ ( 1 2 ( h + k + 2 ) , - ( b - p + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) ) + ( b - p ) 2 8 c s - 4 c v ( c ( v - 2 s ) ) - 2 n ( b p 2 c s - c v k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b + p ) - h - k + 2 n ( b + p + 2 c ( v - 2 s ) z ) h + k ( - ( b + p + 2 c ( v - 2 s ) z ) 2 c ( v - 2 s ) ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b + p ) ( b + p + 2 c ( v - 2 s ) z ) Γ ( 1 2 ( h + k + 1 ) , ( b + p + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( v - 2 s ) - ( b + p + 2 c ( v - 2 s ) z ) 2 c ( v - 2 s ) Γ ( 1 2 ( h + k + 2 ) , ( b + p + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) ) - k = 0 n h = 0 k ( - 1 ) k - h 4 k ( p - b ) - h - k + 2 n ( - b + p + 2 c ( v - 2 s ) z ) h + k ( - ( - b + p + 2 c ( v - 2 s ) z ) 2 c ( v - 2 s ) ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( p - b ) ( - b + p + 2 c ( v - 2 s ) z ) Γ ( 1 2 ( h + k + 1 ) , ( - b + p + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( v - 2 s ) - ( - b + p + 2 c ( v - 2 s ) z ) 2 c ( v - 2 s ) Γ ( 1 2 ( h + k + 2 ) , ( - b + p + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) ) ) ) - 2 - v z n + 1 ( 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]]]]] ( ( - ( b + p ) z ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , - ( b + p ) z ) - ( ( b - p ) z ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( b - p ) z ) ) ( 1 - v mod 2 \$CellContext`v 2 ) /; v + n Condition z z n p z 1 2 b z 1 2 c z v 2 -2 n -1 v -2 c 2 -1 s 0 v -1 2 -1 1 v -1 2 s 2 -1 Binomial v s -1 b p 2 8 c s -1 4 c v -1 c 2 s -1 v -2 n h 0 k k 0 n -1 k -1 h 4 k b p -1 h -1 k 2 n b p 2 c 2 s -1 v z 1 2 h k -1 b p 2 c 2 s -1 v z 1 2 2 c 2 s -1 v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b p b p 2 c 2 s -1 v z 1 2 Gamma 1 2 h k 1 -1 b p 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 2 c 2 s -1 v -1 b p 2 c 2 s -1 v z 1 2 2 c 2 s -1 v -1 1 2 Gamma 1 2 h k 2 -1 b p 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 -1 -1 b -1 p 2 8 c s -1 4 c v -1 c 2 s -1 v -2 n h 0 k k 0 n -1 k -1 h 4 k p -1 b -1 h -1 k 2 n -1 b p 2 c 2 s -1 v z 1 2 h k -1 b -1 p 2 c v -1 2 s z 1 2 2 c 2 s -1 v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p -1 b -1 b p 2 c 2 s -1 v z 1 2 Gamma 1 2 h k 1 -1 b -1 p 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 2 c 2 s -1 v -1 b -1 p 2 c v -1 2 s z 1 2 2 c 2 s -1 v -1 1 2 Gamma 1 2 h k 2 -1 b -1 p 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 b -1 p 2 8 c s -1 4 c v -1 c v -1 2 s -2 n b p 2 c s -1 c v -1 h 0 k k 0 n -1 k -1 h 4 k b p -1 h -1 k 2 n b p 2 c v -1 2 s z 1 2 h k -1 b p 2 c v -1 2 s z 1 2 2 c v -1 2 s -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b p b p 2 c v -1 2 s z 1 2 Gamma 1 2 h k 1 b p 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 2 c v -1 2 s -1 b p 2 c v -1 2 s z 1 2 2 c v -1 2 s -1 1 2 Gamma 1 2 h k 2 b p 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 -1 h 0 k k 0 n -1 k -1 h 4 k p -1 b -1 h -1 k 2 n -1 b p 2 c v -1 2 s z 1 2 h k -1 -1 b p 2 c v -1 2 s z 1 2 2 c v -1 2 s -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p -1 b -1 b p 2 c v -1 2 s z 1 2 Gamma 1 2 h k 1 -1 b p 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 2 c v -1 2 s -1 -1 b p 2 c v -1 2 s z 1 2 2 c v -1 2 s -1 1 2 Gamma 1 2 h k 2 -1 b p 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 -1 2 -1 v z n 1 Binomial v v 2 -1 -1 b p z 1 2 -2 n 1 Gamma 2 n 1 -1 b p z 1 2 -1 b -1 p z 1 2 -2 n 1 Gamma 2 n 1 b -1 p z 1 2 1 -1 \$CellContext`v 2 v SuperPlus n [/itex]

 Rule Form

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

 2002-12-18