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 Cosh

 http://functions.wolfram.com/01.20.21.4456.01

 Input Form

 Integrate[z^n Sinh[b Sqrt[z] + d z] Cosh[c z]^v, z] == 2^(-2 - v) ((d^(-2 - 2 n) Binomial[v, v/2] (-1 + Mod[v, 2]) (-Sum[(-1)^(-h + k) 4^k b^(-h - k + 2 n) (b + 2 d Sqrt[z])^(h + k) (-((b + 2 d Sqrt[z])^2/d))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (b (b + 2 d Sqrt[z]) Gamma[(1/2) (1 + h + k), -((b + 2 d Sqrt[z])^2/(4 d))] + 2 d Sqrt[-((b + 2 d Sqrt[z])^2/d)] Gamma[(1/2) (2 + h + k), -((b + 2 d Sqrt[z])^2/(4 d))]), {k, 0, n}, {h, 0, k}] + E^(b^2/(2 d)) Sum[(-1)^(-h + k) 4^k (-b)^(-h - k + 2 n) (-b - 2 d Sqrt[z])^(h + k) ((b + 2 d Sqrt[z])^2/d)^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (b (b + 2 d Sqrt[z]) Gamma[(1/2) (1 + h + k), (b + 2 d Sqrt[z])^2/ (4 d)] - 2 d Sqrt[(b + 2 d Sqrt[z])^2/d] Gamma[(1/2) (2 + h + k), (b + 2 d Sqrt[z])^2/(4 d)]), {k, 0, n}, {h, 0, k}]))/(4^n E^(b^2/(4 d))) - Sum[Binomial[v, s] (((-E^(-(b^2/(4 d + 8 c s - 4 c v)))) Sum[(-1)^(-h + k) 4^k b^(-h - k + 2 n) (b + 2 (d + 2 c s - c v) Sqrt[z])^(h + k) (-((b + 2 (d + 2 c s - c v) Sqrt[z])^2/(d + 2 c s - c v)))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (b (b + 2 (d + 2 c s - c v) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((b + 2 (d + 2 c s - c v) Sqrt[z])^2/(4 d + 8 c s - 4 c v))] + 2 (d + 2 c s - c v) Sqrt[-((b + 2 (d + 2 c s - c v) Sqrt[z])^2/(d + 2 c s - c v))] Gamma[(1/2) (2 + h + k), -((b + 2 (d + 2 c s - c v) Sqrt[z])^2/ (4 d + 8 c s - 4 c v))]), {k, 0, n}, {h, 0, k}])/ (d + 2 c s - c v)^(2 (1 + n)) + (E^(b^2/(4 d + 8 c s - 4 c v)) Sum[(-1)^(-h + k) 4^k (-b)^(-h - k + 2 n) ((b + 2 (d + 2 c s - c v) Sqrt[z])^2/(d + 2 c s - c v))^ ((1/2) (-1 - h - k)) (-b + 2 (-d - 2 c s + c v) Sqrt[z])^(h + k) Binomial[k, h] Binomial[n, k] (b (b + 2 (d + 2 c s - c v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (b + 2 (d + 2 c s - c v) Sqrt[z])^2/(4 d + 8 c s - 4 c v)] - 2 (d + 2 c s - c v) Sqrt[(b + 2 (d + 2 c s - c v) Sqrt[z])^2/(d + 2 c s - c v)] Gamma[(1/2) (2 + h + k), (b + 2 (d + 2 c s - c v) Sqrt[z])^2/ (4 d + 8 c s - 4 c v)]), {k, 0, n}, {h, 0, k}])/ (-d - 2 c s + c v)^(2 (1 + n)) + ((-(d - 2 c s + c v)^2)^(-1 - 2 n) ((-d + 2 c s - c v)^(2 n) Sum[(-1)^(-h + k) 4^k b^(-h - k + 2 n) (b + 2 (d - 2 c s + c v) Sqrt[z])^(h + k) (-((b + 2 (d - 2 c s + c v) Sqrt[z])^2/(d - 2 c s + c v)))^( (1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (b (b + 2 (d - 2 c s + c v) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((b + 2 (d - 2 c s + c v) Sqrt[z])^2/(4 (d - 2 c s + c v)))] + 2 (d - 2 c s + c v) Sqrt[ -((b + 2 (d - 2 c s + c v) Sqrt[z])^2/(d - 2 c s + c v))] Gamma[(1/2) (2 + h + k), -((b + 2 (d - 2 c s + c v) Sqrt[z])^ 2/(4 (d - 2 c s + c v)))]), {k, 0, n}, {h, 0, k}] - E^(b^2/(2 (d - 2 c s + c v))) (d - 2 c s + c v)^(2 n) Sum[(-1)^(-h + k) 4^k (-b)^(-h - k + 2 n) (-b - 2 (d - 2 c s + c v) Sqrt[z])^(h + k) ((b + 2 (d - 2 c s + c v) Sqrt[z])^2/(d - 2 c s + c v))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (b (b + 2 (d - 2 c s + c v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (b + 2 (d - 2 c s + c v) Sqrt[z])^2/(4 (d - 2 c s + c v))] - 2 (d - 2 c s + c v) Sqrt[(b + 2 (d - 2 c s + c v) Sqrt[z])^2/ (d - 2 c s + c v)] Gamma[(1/2) (2 + h + k), (b + 2 (d - 2 c s + c v) Sqrt[z])^2/(4 (d - 2 c s + c v))]), {k, 0, n}, {h, 0, k}]))/E^(b^2/(4 (d - 2 c s + c v)))), {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 ( z b + d z ) cosh v ( c z ) z 2 - v - 2 ( 4 - n d - 2 n - 2 - b 2 4 d ( 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]]]]] ( v mod 2 \$CellContext`v 2 - 1 ) ( b 2 2 d k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b ) - h - k + 2 n ( - b - 2 d z ) h + k ( ( b + 2 d z ) 2 d ) 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 ( b + 2 d z ) Γ ( 1 2 ( h + k + 1 ) , ( b + 2 d z ) 2 4 d ) - 2 d ( b + 2 d z ) 2 d Γ ( 1 2 ( h + k + 2 ) , ( b + 2 d z ) 2 4 d ) ) - k = 0 n h = 0 k ( - 1 ) k - h 4 k b - h - k + 2 n ( b + 2 d z ) h + k ( - ( b + 2 d z ) 2 d ) 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 ( b + 2 d z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + 2 d z ) 2 4 d ) + 2 - ( b + 2 d z ) 2 d d Γ ( 1 2 ( h + k + 2 ) , - ( b + 2 d z ) 2 4 d ) ) ) - 4 - n 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]]]]] ( b 2 4 d + 8 c s - 4 c v ( k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b ) - h - k + 2 n ( ( b + 2 ( d + 2 c s - c v ) z ) 2 d + 2 c s - c v ) 1 2 ( - h - k - 1 ) ( 2 ( - d - 2 c s + c v ) z - b ) h + k ( 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 ( b + 2 ( d + 2 c s - c v ) z ) Γ ( 1 2 ( h + k + 1 ) , ( b + 2 ( d + 2 c s - c v ) z ) 2 4 d + 8 c s - 4 c v ) - 2 ( d + 2 c s - c v ) ( ( b + 2 ( d + 2 c s - c v ) z ) 2 d + 2 c s - c v ) Γ ( 1 2 ( h + k + 2 ) , ( b + 2 ( d + 2 c s - c v ) z ) 2 4 d + 8 c s - 4 c v ) ) ) ( - d - 2 c s + c v ) - 2 ( n + 1 ) + - b 2 4 ( d - 2 c s + c v ) ( - ( d - 2 c s + c v ) 2 ) - 2 n - 1 ( ( - d + 2 c s - c v ) 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k b - h - k + 2 n ( b + 2 ( d - 2 c s + c v ) z ) h + k ( - ( b + 2 ( d - 2 c s + c v ) z ) 2 d - 2 c s + c 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 ( b + 2 ( d - 2 c s + c v ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + 2 ( d - 2 c s + c v ) z ) 2 4 ( d - 2 c s + c v ) ) + 2 ( - ( b + 2 ( d - 2 c s + c v ) z ) 2 d - 2 c s + c v ) ( d - 2 c s + c v ) Γ ( 1 2 ( h + k + 2 ) , - ( b + 2 ( d - 2 c s + c v ) z ) 2 4 ( d - 2 c s + c v ) ) ) - b 2 2 ( d - 2 c s + c v ) ( d - 2 c s + c v ) 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b ) - h - k + 2 n ( - b - 2 ( d - 2 c s + c v ) z ) h + k ( ( b + 2 ( d - 2 c s + c v ) z ) 2 d - 2 c s + c 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 ( b + 2 ( d - 2 c s + c v ) z ) Γ ( 1 2 ( h + k + 1 ) , ( b + 2 ( d - 2 c s + c v ) z ) 2 4 ( d - 2 c s + c v ) ) - 2 ( d - 2 c s + c v ) ( ( b + 2 ( d - 2 c s + c v ) z ) 2 d - 2 c s + c v ) Γ ( 1 2 ( h + k + 2 ) , ( b + 2 ( d - 2 c s + c v ) z ) 2 4 ( d - 2 c s + c v ) ) ) ) - - b 2 4 d + 8 c s - 4 c v ( d + 2 c s - c v ) - 2 ( n + 1 ) k = 0 n h = 0 k ( - 1 ) k - h 4 k b - h - k + 2 n ( b + 2 ( d + 2 c s - c v ) z ) h + k ( - ( b + 2 ( d + 2 c s - c v ) z ) 2 d + 2 c s - c 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 ( b + 2 ( d + 2 c s - c v ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + 2 ( d + 2 c s - c v ) z ) 2 4 d + 8 c s - 4 c v ) + 2 ( - ( b + 2 ( d + 2 c s - c v ) z ) 2 d + 2 c s - c v ) ( d + 2 c s - c v ) Γ ( 1 2 ( h + k + 2 ) , - ( b + 2 ( d + 2 c s - c v ) z ) 2 4 d + 8 c s - 4 c v ) ) ) ) /; n v + Condition z z n z 1 2 b d z c z v 2 -1 v -2 4 -1 n d -2 n -2 -1 b 2 4 d -1 Binomial v v 2 -1 \$CellContext`v 2 -1 b 2 2 d -1 h 0 k k 0 n -1 k -1 h 4 k -1 b -1 h -1 k 2 n -1 b -1 2 d z 1 2 h k b 2 d z 1 2 2 d -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b b 2 d z 1 2 Gamma 1 2 h k 1 b 2 d z 1 2 2 4 d -1 -1 2 d b 2 d z 1 2 2 d -1 1 2 Gamma 1 2 h k 2 b 2 d z 1 2 2 4 d -1 -1 h 0 k k 0 n -1 k -1 h 4 k b -1 h -1 k 2 n b 2 d z 1 2 h k -1 b 2 d z 1 2 2 d -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b b 2 d z 1 2 Gamma 1 2 h k 1 -1 b 2 d z 1 2 2 4 d -1 2 -1 b 2 d z 1 2 2 d -1 1 2 d Gamma 1 2 h k 2 -1 b 2 d z 1 2 2 4 d -1 -1 4 -1 n s 0 v -1 2 -1 Binomial v s b 2 4 d 8 c s -1 4 c v -1 h 0 k k 0 n -1 k -1 h 4 k -1 b -1 h -1 k 2 n b 2 d 2 c s -1 c v z 1 2 2 d 2 c s -1 c v -1 1 2 -1 h -1 k -1 2 -1 d -1 2 c s c v z 1 2 -1 b h k Binomial k h Binomial n k b b 2 d 2 c s -1 c v z 1 2 Gamma 1 2 h k 1 b 2 d 2 c s -1 c v z 1 2 2 4 d 8 c s -1 4 c v -1 -1 2 d 2 c s -1 c v b 2 d 2 c s -1 c v z 1 2 2 d 2 c s -1 c v -1 Gamma 1 2 h k 2 b 2 d 2 c s -1 c v z 1 2 2 4 d 8 c s -1 4 c v -1 -1 d -1 2 c s c v -2 n 1 -1 b 2 4 d -1 2 c s c v -1 -1 d -1 2 c s c v 2 -2 n -1 -1 d 2 c s -1 c v 2 n h 0 k k 0 n -1 k -1 h 4 k b -1 h -1 k 2 n b 2 d -1 2 c s c v z 1 2 h k -1 b 2 d -1 2 c s c v z 1 2 2 d -1 2 c s c v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b b 2 d -1 2 c s c v z 1 2 Gamma 1 2 h k 1 -1 b 2 d -1 2 c s c v z 1 2 2 4 d -1 2 c s c v -1 2 -1 b 2 d -1 2 c s c v z 1 2 2 d -1 2 c s c v -1 d -1 2 c s c v Gamma 1 2 h k 2 -1 b 2 d -1 2 c s c v z 1 2 2 4 d -1 2 c s c v -1 -1 b 2 2 d -1 2 c s c v -1 d -1 2 c s c v 2 n h 0 k k 0 n -1 k -1 h 4 k -1 b -1 h -1 k 2 n -1 b -1 2 d -1 2 c s c v z 1 2 h k b 2 d -1 2 c s c v z 1 2 2 d -1 2 c s c v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b b 2 d -1 2 c s c v z 1 2 Gamma 1 2 h k 1 b 2 d -1 2 c s c v z 1 2 2 4 d -1 2 c s c v -1 -1 2 d -1 2 c s c v b 2 d -1 2 c s c v z 1 2 2 d -1 2 c s c v -1 Gamma 1 2 h k 2 b 2 d -1 2 c s c v z 1 2 2 4 d -1 2 c s c v -1 -1 -1 b 2 4 d 8 c s -1 4 c v -1 d 2 c s -1 c v -2 n 1 h 0 k k 0 n -1 k -1 h 4 k b -1 h -1 k 2 n b 2 d 2 c s -1 c v z 1 2 h k -1 b 2 d 2 c s -1 c v z 1 2 2 d 2 c s -1 c v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b b 2 d 2 c s -1 c v z 1 2 Gamma 1 2 h k 1 -1 b 2 d 2 c s -1 c v z 1 2 2 4 d 8 c s -1 4 c v -1 2 -1 b 2 d 2 c s -1 c v z 1 2 2 d 2 c s -1 c v -1 d 2 c s -1 c v Gamma 1 2 h k 2 -1 b 2 d 2 c s -1 c v z 1 2 2 4 d 8 c s -1 4 c v -1 n v SuperPlus [/itex]

 Rule Form

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

 2002-12-18