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 Cosh

 http://functions.wolfram.com/01.20.21.4494.01

 Input Form

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

 Standard Form

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 MathML Form

 z n sinh ( z b + e + d z ) cosh v ( z c + g ) z 2 - 2 n - v - 2 d - 2 n - 2 ( - b 2 + 4 d e 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 ) ) + 2 e k = 0 n h = 0 k ( - 1 ) - h + k + 1 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 ) ) ) + s = 0 v - 1 2 - e - g ( 2 s + v ) - b 2 + 2 c ( v - 2 s ) b + c 2 ( v - 2 s ) 2 4 d ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - 2 g v + b 2 + c 2 ( v - 2 s ) 2 2 d k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b - 2 c s + c v ) - h - k + 2 n ( - b + c ( v - 2 s ) - 2 d z ) h + k ( ( b + c ( 2 s - v ) + 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 + 2 c s - c v ) ( b + c ( 2 s - v ) + 2 d z ) Γ ( 1 2 ( h + k + 1 ) , ( b + c ( 2 s - v ) + 2 d z ) 2 4 d ) - 2 d ( b + c ( 2 s - v ) + 2 d z ) 2 d Γ ( 1 2 ( h + k + 2 ) , ( b + c ( 2 s - v ) + 2 d z ) 2 4 d ) ) - 4 g s + b 2 + 2 c ( v - 2 s ) b + c 2 ( v - 2 s ) 2 2 d k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b + 2 c s - c v ) - h - k + 2 n ( - b + c ( 2 s - v ) - 2 d z ) h + k ( ( b + c ( v - 2 s ) + 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 - 2 c s + c v ) ( b + c ( v - 2 s ) + 2 d z ) Γ ( 1 2 ( h + k + 1 ) , ( b + c ( v - 2 s ) + 2 d z ) 2 4 d ) - 2 d ( b + c ( v - 2 s ) + 2 d z ) 2 d Γ ( 1 2 ( h + k + 2 ) , ( b + c ( v - 2 s ) + 2 d z ) 2 4 d ) ) + 2 e + 4 g s + b c ( v - 2 s ) d k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b + 2 c s - c v ) - h - k + 2 n ( b + c ( 2 s - v ) + 2 d z ) h + k ( - ( b + c ( 2 s - v ) + 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 + 2 c s - c v ) ( b + c ( 2 s - v ) + 2 d z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + c ( 2 s - v ) + 2 d z ) 2 4 d ) + 2 - ( b + c ( 2 s - v ) + 2 d z ) 2 d d Γ ( 1 2 ( h + k + 2 ) , - ( b + c ( 2 s - v ) + 2 d z ) 2 4 d ) ) + 2 e + 2 g v k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b - 2 c s + c v ) - h - k + 2 n ( b + c ( v - 2 s ) + 2 d z ) h + k ( - ( b + c ( v - 2 s ) + 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 - 2 c s + c v ) ( b + c ( v - 2 s ) + 2 d z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + c ( v - 2 s ) + 2 d z ) 2 4 d ) + 2 - ( b + c ( v - 2 s ) + 2 d z ) 2 d d Γ ( 1 2 ( h + k + 2 ) , - ( b + c ( v - 2 s ) + 2 d z ) 2 4 d ) ) ) ) /; n v + Condition z z n z 1 2 b e d z z 1 2 c g v 2 -2 n -1 v -2 d -2 n -2 -1 b 2 4 d e 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 2 e h 0 k k 0 n -1 -1 h k 1 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 s 0 v -1 2 -1 -1 e -1 g 2 s v -1 b 2 2 c v -1 2 s b c 2 v -1 2 s 2 4 d -1 Binomial v s -1 2 g v b 2 c 2 v -1 2 s 2 2 d -1 h 0 k k 0 n -1 k -1 h 4 k -1 b -1 2 c s c v -1 h -1 k 2 n -1 b c v -1 2 s -1 2 d z 1 2 h k b c 2 s -1 v 2 d z 1 2 2 d -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b 2 c s -1 c v b c 2 s -1 v 2 d z 1 2 Gamma 1 2 h k 1 b c 2 s -1 v 2 d z 1 2 2 4 d -1 -1 2 d b c 2 s -1 v 2 d z 1 2 2 d -1 1 2 Gamma 1 2 h k 2 b c 2 s -1 v 2 d z 1 2 2 4 d -1 -1 4 g s b 2 2 c v -1 2 s b c 2 v -1 2 s 2 2 d -1 h 0 k k 0 n -1 k -1 h 4 k -1 b 2 c s -1 c v -1 h -1 k 2 n -1 b c 2 s -1 v -1 2 d z 1 2 h k b c v -1 2 s 2 d z 1 2 2 d -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b -1 2 c s c v b c v -1 2 s 2 d z 1 2 Gamma 1 2 h k 1 b c v -1 2 s 2 d z 1 2 2 4 d -1 -1 2 d b c v -1 2 s 2 d z 1 2 2 d -1 1 2 Gamma 1 2 h k 2 b c v -1 2 s 2 d z 1 2 2 4 d -1 2 e 4 g s b c v -1 2 s d -1 h 0 k k 0 n -1 k -1 h 4 k b 2 c s -1 c v -1 h -1 k 2 n b c 2 s -1 v 2 d z 1 2 h k -1 b c 2 s -1 v 2 d z 1 2 2 d -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b 2 c s -1 c v b c 2 s -1 v 2 d z 1 2 Gamma 1 2 h k 1 -1 b c 2 s -1 v 2 d z 1 2 2 4 d -1 2 -1 b c 2 s -1 v 2 d z 1 2 2 d -1 1 2 d Gamma 1 2 h k 2 -1 b c 2 s -1 v 2 d z 1 2 2 4 d -1 2 e 2 g v h 0 k k 0 n -1 k -1 h 4 k b -1 2 c s c v -1 h -1 k 2 n b c v -1 2 s 2 d z 1 2 h k -1 b c v -1 2 s 2 d z 1 2 2 d -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b -1 2 c s c v b c v -1 2 s 2 d z 1 2 Gamma 1 2 h k 1 -1 b c v -1 2 s 2 d z 1 2 2 4 d -1 2 -1 b c v -1 2 s 2 d z 1 2 2 d -1 1 2 d Gamma 1 2 h k 2 -1 b c v -1 2 s 2 d z 1 2 2 4 d -1 n v SuperPlus [/itex]

 Rule Form

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

 2002-12-18