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 Cosh

 http://functions.wolfram.com/01.20.21.4815.01

 Input Form

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

 Standard Form

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 MathML Form

 z n p z sinh ( b z ) cosh v ( c z ) z 2 - v - 1 ( 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]]]]] ( ( b - p ) - n - 1 Γ ( n + 1 , ( b - p ) z ) - ( - b - p ) - n - 1 Γ ( n + 1 , ( - b - p ) z ) ) ( 1 - v mod 2 \$CellContext`v 2 ) + 2 - 2 n - v - 2 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 ( p - b ) ( i = 0 n h = 0 i ( - 1 ) i - h 4 i ( - c ( v - 2 s ) ) - h - i + 2 n ( 2 ( p - b ) z - c ( v - 2 s ) ) h + i ( - ( 2 ( p - b ) z - c ( v - 2 s ) ) 2 p - b ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 ( p - b ) - ( 2 ( p - b ) z - c ( v - 2 s ) ) 2 p - b Γ ( 1 2 ( h + i + 2 ) , - ( 2 ( p - b ) z - c ( v - 2 s ) ) 2 4 ( p - b ) ) - c ( v - 2 s ) ( 2 ( p - b ) z - c ( v - 2 s ) ) Γ ( 1 2 ( h + i + 1 ) , - ( 2 ( p - b ) z - c ( v - 2 s ) ) 2 4 ( p - b ) ) ) ) ( p - b ) - 2 ( n + 1 ) - - c 2 ( v - 2 s ) 2 4 ( p - b ) ( i = 0 n h = 0 i ( - 1 ) i - h 4 i ( c ( v - 2 s ) ) - h - i + 2 n ( 2 z ( p - b ) + c ( v - 2 s ) ) h + i ( - ( 2 z ( p - b ) + c ( v - 2 s ) ) 2 p - b ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( c ( v - 2 s ) ( 2 z ( p - b ) + c ( v - 2 s ) ) Γ ( 1 2 ( h + i + 1 ) , - ( 2 z ( p - b ) + c ( v - 2 s ) ) 2 4 ( p - b ) ) + 2 - ( 2 z ( p - b ) + c ( v - 2 s ) ) 2 p - b ( p - b ) Γ ( 1 2 ( h + i + 2 ) , - ( 2 z ( p - b ) + c ( v - 2 s ) ) 2 4 ( p - b ) ) ) ) ( p - b ) - 2 ( n + 1 ) + - c 2 ( v - 2 s ) 2 4 ( b + p ) ( b + p ) - 2 ( n + 1 ) i = 0 n h = 0 i ( - 1 ) i - h 4 i ( - c ( v - 2 s ) ) - h - i + 2 n ( 2 ( b + p ) z - c ( v - 2 s ) ) h + i ( - ( 2 ( b + p ) z - c ( v - 2 s ) ) 2 b + p ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 ( b + p ) - ( 2 ( b + p ) z - c ( v - 2 s ) ) 2 b + p Γ ( 1 2 ( h + i + 2 ) , - ( 2 ( b + p ) z - c ( v - 2 s ) ) 2 4 ( b + p ) ) - c ( v - 2 s ) ( 2 ( b + p ) z - c ( v - 2 s ) ) Γ ( 1 2 ( h + i + 1 ) , - ( 2 ( b + p ) z - c ( v - 2 s ) ) 2 4 ( b + p ) ) ) + - c 2 ( v - 2 s ) 2 4 ( b + p ) ( b + p ) - 2 ( n + 1 ) i = 0 n h = 0 i ( - 1 ) i - h 4 i ( c ( v - 2 s ) ) - h - i + 2 n ( 2 z ( b + p ) + c ( v - 2 s ) ) h + i ( - ( 2 z ( b + p ) + c ( v - 2 s ) ) 2 b + p ) 1 2 ( - h - i - 1 ) ( i h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( c ( v - 2 s ) ( 2 z ( b + p ) + c ( v - 2 s ) ) Γ ( 1 2 ( h + i + 1 ) , - ( 2 z ( b + p ) + c ( v - 2 s ) ) 2 4 ( b + p ) ) + 2 - ( 2 z ( b + p ) + c ( v - 2 s ) ) 2 b + p ( b + p ) Γ ( 1 2 ( h + i + 2 ) , - ( 2 z ( b + p ) + c ( v - 2 s ) ) 2 4 ( b + p ) ) ) ) /; v + n Condition z z n p z b z c z 1 2 v 2 -1 v -1 Binomial v v 2 -1 b -1 p -1 n -1 Gamma n 1 b -1 p z -1 -1 b -1 p -1 n -1 Gamma n 1 -1 b -1 p z 1 -1 \$CellContext`v 2 2 -2 n -1 v -2 s 0 v -1 2 -1 Binomial v s -1 -1 c 2 v -1 2 s 2 4 p -1 b -1 h 0 i i 0 n -1 i -1 h 4 i -1 c v -1 2 s -1 h -1 i 2 n 2 p -1 b z 1 2 -1 c v -1 2 s h i -1 2 p -1 b z 1 2 -1 c v -1 2 s 2 p -1 b -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i 2 p -1 b -1 2 p -1 b z 1 2 -1 c v -1 2 s 2 p -1 b -1 1 2 Gamma 1 2 h i 2 -1 2 p -1 b z 1 2 -1 c v -1 2 s 2 4 p -1 b -1 -1 c v -1 2 s 2 p -1 b z 1 2 -1 c v -1 2 s Gamma 1 2 h i 1 -1 2 p -1 b z 1 2 -1 c v -1 2 s 2 4 p -1 b -1 p -1 b -2 n 1 -1 -1 c 2 v -1 2 s 2 4 p -1 b -1 h 0 i i 0 n -1 i -1 h 4 i c v -1 2 s -1 h -1 i 2 n 2 z 1 2 p -1 b c v -1 2 s h i -1 2 z 1 2 p -1 b c v -1 2 s 2 p -1 b -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i c v -1 2 s 2 z 1 2 p -1 b c v -1 2 s Gamma 1 2 h i 1 -1 2 z 1 2 p -1 b c v -1 2 s 2 4 p -1 b -1 2 -1 2 z 1 2 p -1 b c v -1 2 s 2 p -1 b -1 1 2 p -1 b Gamma 1 2 h i 2 -1 2 z 1 2 p -1 b c v -1 2 s 2 4 p -1 b -1 p -1 b -2 n 1 -1 c 2 v -1 2 s 2 4 b p -1 b p -2 n 1 h 0 i i 0 n -1 i -1 h 4 i -1 c v -1 2 s -1 h -1 i 2 n 2 b p z 1 2 -1 c v -1 2 s h i -1 2 b p z 1 2 -1 c v -1 2 s 2 b p -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i 2 b p -1 2 b p z 1 2 -1 c v -1 2 s 2 b p -1 1 2 Gamma 1 2 h i 2 -1 2 b p z 1 2 -1 c v -1 2 s 2 4 b p -1 -1 c v -1 2 s 2 b p z 1 2 -1 c v -1 2 s Gamma 1 2 h i 1 -1 2 b p z 1 2 -1 c v -1 2 s 2 4 b p -1 -1 c 2 v -1 2 s 2 4 b p -1 b p -2 n 1 h 0 i i 0 n -1 i -1 h 4 i c v -1 2 s -1 h -1 i 2 n 2 z 1 2 b p c v -1 2 s h i -1 2 z 1 2 b p c v -1 2 s 2 b p -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i c v -1 2 s 2 z 1 2 b p c v -1 2 s Gamma 1 2 h i 1 -1 2 z 1 2 b p c v -1 2 s 2 4 b p -1 2 -1 2 z 1 2 b p c v -1 2 s 2 b p -1 1 2 b p Gamma 1 2 h i 2 -1 2 z 1 2 b p c v -1 2 s 2 4 b p -1 v SuperPlus n [/itex]

 Rule Form

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

 2002-12-18