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 Cosh

 http://functions.wolfram.com/01.20.21.1383.01

 Input Form

 Integrate[z^n E^(p Sqrt[z]) Sin[b Sqrt[z]]^m Cosh[c z], z] == (2^(-2 - m - 2 n) ((-E^(p^2/(2 c))) Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[(-1)^(-q + r) 4^r p^(2 n - q - r) (p - 2 c Sqrt[z])^(q + r) ((p - 2 c Sqrt[z])^2/c)^((1/2) (-1 - q - r)) Binomial[n, r] Binomial[r, q] (p (p - 2 c Sqrt[z]) Gamma[(1/2) (1 + q + r), (p - 2 c Sqrt[z])^2/(4 c)] - 2 c Sqrt[(p - 2 c Sqrt[z])^2/c] Gamma[(1/2) (2 + q + r), (p - 2 c Sqrt[z])^2/(4 c)]), {r, 0, n}, {q, 0, r}] - Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[(-1)^(-q + r) 4^r p^(2 n - q - r) (p + 2 c Sqrt[z])^(q + r) (-((p + 2 c Sqrt[z])^2/c))^((1/2) (-1 - q - r)) Binomial[n, r] Binomial[r, q] (p (p + 2 c Sqrt[z]) Gamma[(1/2) (1 + q + r), -((p + 2 c Sqrt[z])^2/(4 c))] + 2 c Sqrt[-((p + 2 c Sqrt[z])^2/c)] Gamma[(1/2) (2 + q + r), -((p + 2 c Sqrt[z])^2/(4 c))]), {r, 0, n}, {q, 0, r}] + Sum[((-1)^k Binomial[m, k] (E^((8 I b k p + 3 p^2 + 4 I c m Pi)/(4 c)) Sum[(-1)^(-q + r) 4^r (2 I b k - I b m + p)^(2 n - q - r) (2 I b k - I b m + p - 2 c Sqrt[z])^(q + r) ((2 I b k - I b m + p - 2 c Sqrt[z])^2/c)^((1/2) (-1 - q - r)) Binomial[n, r] Binomial[r, q] ((2 I b k - I b m + p) (2 I b k - I b m + p - 2 c Sqrt[z]) Gamma[(1/2) (1 + q + r), (2 I b k - I b m + p - 2 c Sqrt[z])^2/(4 c)] - 2 c Sqrt[(2 I b k - I b m + p - 2 c Sqrt[z])^2/c] Gamma[(1/2) (2 + q + r), (2 I b k - I b m + p - 2 c Sqrt[z])^2/ (4 c)]), {r, 0, n}, {q, 0, r}] + E^((p (4 I b m + 3 p))/(4 c)) Sum[(-1)^(-q + r) 4^r (-2 I b k + I b m + p)^(2 n - q - r) (-2 I b k + I b m + p - 2 c Sqrt[z])^(q + r) ((-2 I b k + I b m + p - 2 c Sqrt[z])^2/c)^ ((1/2) (-1 - q - r)) Binomial[n, r] Binomial[r, q] ((-2 I b k + I b m + p) (-2 I b k + I b m + p - 2 c Sqrt[z]) Gamma[(1/2) (1 + q + r), (-2 I b k + I b m + p - 2 c Sqrt[z])^2/ (4 c)] - 2 c Sqrt[(-2 I b k + I b m + p - 2 c Sqrt[z])^2/c] Gamma[(1/2) (2 + q + r), (-2 I b k + I b m + p - 2 c Sqrt[z])^2/ (4 c)]), {r, 0, n}, {q, 0, r}] + E^((2 b^2 (-2 k + m)^2 + p^2)/(4 c)) (E^((I m (b p + c Pi))/c) Sum[(-1)^(-q + r) 4^r (2 I b k - I b m + p)^(2 n - q - r) (2 I b k - I b m + p + 2 c Sqrt[z])^(q + r) (-((2 I b k - I b m + p + 2 c Sqrt[z])^2/c))^((1/2) (-1 - q - r)) Binomial[n, r] Binomial[r, q] ((2 I b k - I b m + p) (2 I b k - I b m + p + 2 c Sqrt[z]) Gamma[(1/2) (1 + q + r), -((2 I b k - I b m + p + 2 c Sqrt[z])^2/(4 c))] + 2 c Sqrt[ -((2 I b k - I b m + p + 2 c Sqrt[z])^2/c)] Gamma[(1/2) (2 + q + r), -((2 I b k - I b m + p + 2 c Sqrt[z])^2/(4 c))]), {r, 0, n}, {q, 0, r}] + E^((2 I b k p)/c) Sum[(-1)^(-q + r) 4^r (-2 I b k + I b m + p)^( 2 n - q - r) (-2 I b k + I b m + p + 2 c Sqrt[z])^(q + r) (-((-2 I b k + I b m + p + 2 c Sqrt[z])^2/c))^((1/2) (-1 - q - r)) Binomial[n, r] Binomial[r, q] ((-2 I b k + I b m + p) (-2 I b k + I b m + p + 2 c Sqrt[z]) Gamma[(1/2) (1 + q + r), -((-2 I b k + I b m + p + 2 c Sqrt[z])^2/(4 c))] + 2 c Sqrt[-((-2 I b k + I b m + p + 2 c Sqrt[z])^2/c)] Gamma[(1/2) (2 + q + r), -((-2 I b k + I b m + p + 2 c Sqrt[z])^2/(4 c))]), {r, 0, n}, {q, 0, r}])))/ E^((b^2 (-2 k + m)^2 + 2 I b (2 k + m) p + p^2 + 2 I c m Pi)/(4 c)), {k, 0, Floor[(1/2) (-1 + m)]}]))/(c^(2 (1 + n)) E^(p^2/(4 c))) /; Element[m, Integers] && m > 0 && Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n p z sin m ( b z ) cosh ( c z ) z 2 - m - 2 n - 2 c - 2 ( n + 1 ) - p 2 4 c ( - p 2 2 c ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) r = 0 n q = 0 r ( - 1 ) r - q 4 r p 2 n - q - r ( p - 2 c z ) q + r ( ( p - 2 c z ) 2 c ) 1 2 ( - q - r - 1 ) ( n r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["r", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( r q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["r", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( p - 2 c z ) Γ ( 1 2 ( q + r + 1 ) , ( p - 2 c z ) 2 4 c ) - 2 c ( p - 2 c z ) 2 c Γ ( 1 2 ( q + r + 2 ) , ( p - 2 c z ) 2 4 c ) ) - ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) r = 0 n q = 0 r ( - 1 ) r - q 4 r p 2 n - q - r ( 2 z c + p ) q + r ( - ( 2 z c + p ) 2 c ) 1 2 ( - q - r - 1 ) ( n r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["r", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( r q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["r", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( 2 z c + p ) Γ ( 1 2 ( q + r + 1 ) , - ( 2 z c + p ) 2 4 c ) + 2 - ( 2 z c + p ) 2 c c Γ ( 1 2 ( q + r + 2 ) , - ( 2 z c + p ) 2 4 c ) ) + k = 0 m - 1 2 ( - 1 ) k - b 2 ( m - 2 k ) 2 + p 2 + 2 b ( 2 k + m ) p + 2 π c m 4 c ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( 4 b m + 3 p ) 4 c r = 0 n q = 0 r ( - 1 ) r - q 4 r ( - 2 b k + b m + p ) 2 n - q - r ( - 2 z c - 2 b k + b m + p ) q + r ( ( - 2 z c - 2 b k + b m + p ) 2 c ) 1 2 ( - q - r - 1 ) ( n r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["r", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( r q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["r", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - 2 b k + b m + p ) ( - 2 z c - 2 b k + b m + p ) Γ ( 1 2 ( q + r + 1 ) , ( - 2 z c - 2 b k + b m + p ) 2 4 c ) - 2 c ( - 2 z c - 2 b k + b m + p ) 2 c Γ ( 1 2 ( q + r + 2 ) , ( - 2 z c - 2 b k + b m + p ) 2 4 c ) ) + 3 p 2 + 8 b k p + 4 π c m 4 c r = 0 n q = 0 r ( - 1 ) r - q 4 r ( 2 b k - b m + p ) 2 n - q - r ( - 2 z c + 2 b k - b m + p ) q + r ( ( - 2 z c + 2 b k - b m + p ) 2 c ) 1 2 ( - q - r - 1 ) ( n r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["r", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( r q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["r", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( 2 b k - b m + p ) ( - 2 z c + 2 b k - b m + p ) Γ ( 1 2 ( q + r + 1 ) , ( - 2 z c + 2 b k - b m + p ) 2 4 c ) - 2 c ( - 2 z c + 2 b k - b m + p ) 2 c Γ ( 1 2 ( q + r + 2 ) , ( - 2 z c + 2 b k - b m + p ) 2 4 c ) ) + 2 b 2 ( m - 2 k ) 2 + p 2 4 c ( 2 b k p c r = 0 n q = 0 r ( - 1 ) r - q 4 r ( - 2 b k + b m + p ) 2 n - q - r ( 2 z c - 2 b k + b m + p ) q + r ( - ( 2 z c - 2 b k + b m + p ) 2 c ) 1 2 ( - q - r - 1 ) ( n r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["r", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( r q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["r", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - 2 b k + b m + p ) ( 2 z c - 2 b k + b m + p ) Γ ( 1 2 ( q + r + 1 ) , - ( 2 z c - 2 b k + b m + p ) 2 4 c ) + 2 - ( 2 z c - 2 b k + b m + p ) 2 c c Γ ( 1 2 ( q + r + 2 ) , - ( 2 z c - 2 b k + b m + p ) 2 4 c ) ) + m ( π c + b p ) c r = 0 n q = 0 r ( - 1 ) r - q 4 r ( 2 b k - b m + p ) 2 n - q - r ( 2 z c + 2 b k - b m + p ) q + r ( - ( 2 z c + 2 b k - b m + p ) 2 c ) 1 2 ( - q - r - 1 ) ( n r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["r", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( r q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["r", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( 2 b k - b m + p ) ( 2 z c + 2 b k - b m + p ) Γ ( 1 2 ( q + r + 1 ) , - ( 2 z c + 2 b k - b m + p ) 2 4 c ) + 2 - ( 2 z c + 2 b k - b m + p ) 2 c c Γ ( 1 2 ( q + r + 2 ) , - ( 2 z c + 2 b k - b m + p ) 2 4 c ) ) ) ) ) /; m + n Condition z z n p z 1 2 b z 1 2 m c z 2 -1 m -1 2 n -2 c -2 n 1 -1 p 2 4 c -1 -1 p 2 2 c -1 Binomial m m 2 -1 \$CellContext`m 2 -1 q 0 r r 0 n -1 r -1 q 4 r p 2 n -1 q -1 r p -1 2 c z 1 2 q r p -1 2 c z 1 2 2 c -1 1 2 -1 q -1 r -1 Binomial n r Binomial r q p p -1 2 c z 1 2 Gamma 1 2 q r 1 p -1 2 c z 1 2 2 4 c -1 -1 2 c p -1 2 c z 1 2 2 c -1 1 2 Gamma 1 2 q r 2 p -1 2 c z 1 2 2 4 c -1 -1 Binomial m m 2 -1 \$CellContext`m 2 -1 q 0 r r 0 n -1 r -1 q 4 r p 2 n -1 q -1 r 2 z 1 2 c p q r -1 2 z 1 2 c p 2 c -1 1 2 -1 q -1 r -1 Binomial n r Binomial r q p 2 z 1 2 c p Gamma 1 2 q r 1 -1 2 z 1 2 c p 2 4 c -1 2 -1 2 z 1 2 c p 2 c -1 1 2 c Gamma 1 2 q r 2 -1 2 z 1 2 c p 2 4 c -1 k 0 m -1 2 -1 -1 k -1 b 2 m -1 2 k 2 p 2 2 b 2 k m p 2 c m 4 c -1 Binomial m k p 4 b m 3 p 4 c -1 q 0 r r 0 n -1 r -1 q 4 r -2 b k b m p 2 n -1 q -1 r -2 z 1 2 c -1 2 b k b m p q r -2 z 1 2 c -1 2 b k b m p 2 c -1 1 2 -1 q -1 r -1 Binomial n r Binomial r q -2 b k b m p -2 z 1 2 c -1 2 b k b m p Gamma 1 2 q r 1 -2 z 1 2 c -1 2 b k b m p 2 4 c -1 -1 2 c -2 z 1 2 c -1 2 b k b m p 2 c -1 1 2 Gamma 1 2 q r 2 -2 z 1 2 c -1 2 b k b m p 2 4 c -1 3 p 2 8 b k p 4 c m 4 c -1 q 0 r r 0 n -1 r -1 q 4 r 2 b k -1 b m p 2 n -1 q -1 r -2 z 1 2 c 2 b k -1 b m p q r -2 z 1 2 c 2 b k -1 b m p 2 c -1 1 2 -1 q -1 r -1 Binomial n r Binomial r q 2 b k -1 b m p -2 z 1 2 c 2 b k -1 b m p Gamma 1 2 q r 1 -2 z 1 2 c 2 b k -1 b m p 2 4 c -1 -1 2 c -2 z 1 2 c 2 b k -1 b m p 2 c -1 1 2 Gamma 1 2 q r 2 -2 z 1 2 c 2 b k -1 b m p 2 4 c -1 2 b 2 m -1 2 k 2 p 2 4 c -1 2 b k p c -1 q 0 r r 0 n -1 r -1 q 4 r -2 b k b m p 2 n -1 q -1 r 2 z 1 2 c -1 2 b k b m p q r -1 2 z 1 2 c -1 2 b k b m p 2 c -1 1 2 -1 q -1 r -1 Binomial n r Binomial r q -2 b k b m p 2 z 1 2 c -1 2 b k b m p Gamma 1 2 q r 1 -1 2 z 1 2 c -1 2 b k b m p 2 4 c -1 2 -1 2 z 1 2 c -1 2 b k b m p 2 c -1 1 2 c Gamma 1 2 q r 2 -1 2 z 1 2 c -1 2 b k b m p 2 4 c -1 m c b p c -1 q 0 r r 0 n -1 r -1 q 4 r 2 b k -1 b m p 2 n -1 q -1 r 2 z 1 2 c 2 b k -1 b m p q r -1 2 z 1 2 c 2 b k -1 b m p 2 c -1 1 2 -1 q -1 r -1 Binomial n r Binomial r q 2 b k -1 b m p 2 z 1 2 c 2 b k -1 b m p Gamma 1 2 q r 1 -1 2 z 1 2 c 2 b k -1 b m p 2 4 c -1 2 -1 2 z 1 2 c 2 b k -1 b m p 2 c -1 1 2 c Gamma 1 2 q r 2 -1 2 z 1 2 c 2 b k -1 b m p 2 4 c -1 m SuperPlus n [/itex]

 Rule Form

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

 2002-12-18