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 Cosh

 http://functions.wolfram.com/01.20.21.4496.01

 Input Form

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

 Rule Form

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

 2002-12-18