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 Cosh

 http://functions.wolfram.com/01.20.21.1363.01

 Input Form

 Integrate[z^n E^(p Sqrt[z]) Sin[b z] Cosh[c Sqrt[z]], z] == I (-1)^(-1 + n) 2^(-3 - 2 n) b^(-2 - 2 n) (Sum[(-1)^(-h + j) 4^j (-c + p)^(-h - j + 2 n) (-c + p - 2 I b Sqrt[z])^ (h + j) (-((I (-c + p - 2 I b Sqrt[z])^2)/b))^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((-c + p) (-c + p - 2 I b Sqrt[z]) Gamma[(1/2) (1 + h + j), -((I (-c + p - 2 I b Sqrt[z])^2)/(4 b))] - 2 I b Sqrt[-((I (-c + p - 2 I b Sqrt[z])^2)/b)] Gamma[(1/2) (2 + h + j), -((I (-c + p - 2 I b Sqrt[z])^2)/(4 b))]), {j, 0, n}, {h, 0, j}]/E^((I (-c + p)^2)/(4 b)) + Sum[(-1)^(-h + j) 4^j (c + p)^(-h - j + 2 n) (c + p - 2 I b Sqrt[z])^ (h + j) (-((I (c + p - 2 I b Sqrt[z])^2)/b))^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((c + p) (c + p - 2 I b Sqrt[z]) Gamma[(1/2) (1 + h + j), -((I (c + p - 2 I b Sqrt[z])^2)/(4 b))] - 2 I b Sqrt[-((I (c + p - 2 I b Sqrt[z])^2)/b)] Gamma[(1/2) (2 + h + j), -((I (c + p - 2 I b Sqrt[z])^2)/(4 b))]), {j, 0, n}, {h, 0, j}]/E^((I (c + p)^2)/(4 b)) - E^((I (-c + p)^2)/(4 b)) Sum[(-1)^(-h + j) 4^j (-c + p)^(-h - j + 2 n) (-c + p + 2 I b Sqrt[z])^(h + j) ((I (-c + p + 2 I b Sqrt[z])^2)/b)^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((-c + p) (-c + p + 2 I b Sqrt[z]) Gamma[(1/2) (1 + h + j), (I (-c + p + 2 I b Sqrt[z])^2)/(4 b)] + 2 I b Sqrt[(I (-c + p + 2 I b Sqrt[z])^2)/b] Gamma[(1/2) (2 + h + j), (I (-c + p + 2 I b Sqrt[z])^2)/(4 b)]), {j, 0, n}, {h, 0, j}] - E^((I (c + p)^2)/(4 b)) Sum[(-1)^(-h + j) 4^j (c + p)^(-h - j + 2 n) (c + p + 2 I b Sqrt[z])^(h + j) ((I (c + p + 2 I b Sqrt[z])^2)/b)^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((c + p) (c + p + 2 I b Sqrt[z]) Gamma[(1/2) (1 + h + j), (I (c + p + 2 I b Sqrt[z])^2)/(4 b)] + 2 I b Sqrt[(I (c + p + 2 I b Sqrt[z])^2)/b] Gamma[(1/2) (2 + h + j), (I (c + p + 2 I b Sqrt[z])^2)/(4 b)]), {j, 0, n}, {h, 0, j}]) /; Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n p z sin ( b z ) cosh ( c z ) z ( - 1 ) n - 1 2 - 2 n - 3 b - 2 n - 2 ( - ( p - c ) 2 4 b j = 0 n h = 0 j ( - 1 ) j - h 4 j ( p - c ) - h - j + 2 n ( - c + p + 2 b z ) h + j ( ( - c + p + 2 b z ) 2 b ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( p - c ) ( - c + p + 2 b z ) Γ ( 1 2 ( h + j + 1 ) , ( - c + p + 2 b z ) 2 4 b ) + 2 ( - c + p + 2 b z ) 2 b b Γ ( 1 2 ( h + j + 2 ) , ( - c + p + 2 b z ) 2 4 b ) ) - ( c + p ) 2 4 b j = 0 n h = 0 j ( - 1 ) j - h 4 j ( c + p ) - h - j + 2 n ( c + p + 2 b z ) h + j ( ( c + p + 2 b z ) 2 b ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( c + p ) ( c + p + 2 b z ) Γ ( 1 2 ( h + j + 1 ) , ( c + p + 2 b z ) 2 4 b ) + 2 ( c + p + 2 b z ) 2 b b Γ ( 1 2 ( h + j + 2 ) , ( c + p + 2 b z ) 2 4 b ) ) + - ( p - c ) 2 4 b j = 0 n h = 0 j ( - 1 ) j - h 4 j ( p - c ) - h - j + 2 n ( - c + p - 2 b z ) h + j ( - ( - c + p - 2 b z ) 2 b ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( p - c ) ( - c + p - 2 b z ) Γ ( 1 2 ( h + j + 1 ) , - ( - c + p - 2 b z ) 2 4 b ) - 2 b - ( - c + p - 2 b z ) 2 b Γ ( 1 2 ( h + j + 2 ) , - ( - c + p - 2 b z ) 2 4 b ) ) + - ( c + p ) 2 4 b j = 0 n h = 0 j ( - 1 ) j - h 4 j ( c + p ) - h - j + 2 n ( c + p - 2 b z ) h + j ( - ( c + p - 2 b z ) 2 b ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( c + p ) ( c + p - 2 b z ) Γ ( 1 2 ( h + j + 1 ) , - ( c + p - 2 b z ) 2 4 b ) - 2 b - ( c + p - 2 b z ) 2 b Γ ( 1 2 ( h + j + 2 ) , - ( c + p - 2 b z ) 2 4 b ) ) ) /; n Condition z z n p z 1 2 b z c z 1 2 -1 n -1 2 -2 n -3 b -2 n -2 -1 p -1 c 2 4 b -1 h 0 j j 0 n -1 j -1 h 4 j p -1 c -1 h -1 j 2 n -1 c p 2 b z 1 2 h j -1 c p 2 b z 1 2 2 b -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j p -1 c -1 c p 2 b z 1 2 Gamma 1 2 h j 1 -1 c p 2 b z 1 2 2 4 b -1 2 -1 c p 2 b z 1 2 2 b -1 1 2 b Gamma 1 2 h j 2 -1 c p 2 b z 1 2 2 4 b -1 -1 c p 2 4 b -1 h 0 j j 0 n -1 j -1 h 4 j c p -1 h -1 j 2 n c p 2 b z 1 2 h j c p 2 b z 1 2 2 b -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j c p c p 2 b z 1 2 Gamma 1 2 h j 1 c p 2 b z 1 2 2 4 b -1 2 c p 2 b z 1 2 2 b -1 1 2 b Gamma 1 2 h j 2 c p 2 b z 1 2 2 4 b -1 -1 p -1 c 2 4 b -1 h 0 j j 0 n -1 j -1 h 4 j p -1 c -1 h -1 j 2 n -1 c p -1 2 b z 1 2 h j -1 -1 c p -1 2 b z 1 2 2 b -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j p -1 c -1 c p -1 2 b z 1 2 Gamma 1 2 h j 1 -1 -1 c p -1 2 b z 1 2 2 4 b -1 -1 2 b -1 -1 c p -1 2 b z 1 2 2 b -1 1 2 Gamma 1 2 h j 2 -1 -1 c p -1 2 b z 1 2 2 4 b -1 -1 c p 2 4 b -1 h 0 j j 0 n -1 j -1 h 4 j c p -1 h -1 j 2 n c p -1 2 b z 1 2 h j -1 c p -1 2 b z 1 2 2 b -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j c p c p -1 2 b z 1 2 Gamma 1 2 h j 1 -1 c p -1 2 b z 1 2 2 4 b -1 -1 2 b -1 c p -1 2 b z 1 2 2 b -1 1 2 Gamma 1 2 h j 2 -1 c p -1 2 b z 1 2 2 4 b -1 n [/itex]

 Rule Form

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

 2002-12-18