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 Cosh

 http://functions.wolfram.com/01.20.21.1391.01

 Input Form

 Integrate[z^n E^(p Sqrt[z]) Sin[b z]^m Cosh[c Sqrt[z]], z] == (-2^(-m)) Binomial[m, m/2] (Gamma[2 (1 + n), (-c - p) Sqrt[z]]/ (c + p)^(2 (1 + n)) + Gamma[2 (1 + n), (c - p) Sqrt[z]]/ (c - p)^(2 (1 + n))) (1 - Mod[m, 2]) + 2^(-2 - m - 2 n) Sum[((-1)^k Binomial[m, k] (E^(-((I (-c + p)^2)/(4 b (-2 k + m))) + (I m Pi)/2) Sum[(-1)^(-h + i) 4^i (-c + p)^(-h - i + 2 n) (-c + p - 2 I b (-2 k + m) Sqrt[z])^(h + i) (-((I (-c + p - 2 I b (-2 k + m) Sqrt[z])^2)/(b (-2 k + m))))^ ((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] ((-c + p) (-c + p - 2 I b (-2 k + m) Sqrt[z]) Gamma[(1/2) (1 + h + i), -((I (-c + p - 2 I b (-2 k + m) Sqrt[z])^2)/(4 b (-2 k + m)))] - 2 I b (-2 k + m) Sqrt[-((I (-c + p - 2 I b (-2 k + m) Sqrt[z])^2)/ (b (-2 k + m)))] Gamma[(1/2) (2 + h + i), -((I (-c + p - 2 I b (-2 k + m) Sqrt[z])^2)/(4 b (-2 k + m)))]), {i, 0, n}, {h, 0, i}] + E^(-((I (c + p)^2)/(4 b (-2 k + m))) + (I m Pi)/2) Sum[(-1)^(-h + i) 4^i (c + p)^(-h - i + 2 n) (c + p - 2 I b (-2 k + m) Sqrt[z])^(h + i) (-((I (c + p - 2 I b (-2 k + m) Sqrt[z])^2)/(b (-2 k + m))))^ ((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] ((c + p) (c + p - 2 I b (-2 k + m) Sqrt[z]) Gamma[(1/2) (1 + h + i), -((I (c + p - 2 I b (-2 k + m) Sqrt[z])^ 2)/(4 b (-2 k + m)))] - 2 I b (-2 k + m) Sqrt[-((I (c + p - 2 I b (-2 k + m) Sqrt[z])^2)/(b (-2 k + m)))] Gamma[(1/2) (2 + h + i), -((I (c + p - 2 I b (-2 k + m) Sqrt[z])^ 2)/(4 b (-2 k + m)))]), {i, 0, n}, {h, 0, i}] + E^((I (-c + p)^2)/(4 b (-2 k + m)) - (I m Pi)/2) Sum[(-1)^(-h + i) 4^i (-c + p)^(-h - i + 2 n) (-c + p + 2 I b (-2 k + m) Sqrt[z])^(h + i) ((I (-c + p + 2 I b (-2 k + m) Sqrt[z])^2)/(b (-2 k + m)))^ ((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] ((-c + p) (-c + p + 2 I b (-2 k + m) Sqrt[z]) Gamma[(1/2) (1 + h + i), (I (-c + p + 2 I b (-2 k + m) Sqrt[z])^ 2)/(4 b (-2 k + m))] + 2 I b (-2 k + m) Sqrt[(I (-c + p + 2 I b (-2 k + m) Sqrt[z])^2)/(b (-2 k + m))] Gamma[(1/2) (2 + h + i), (I (-c + p + 2 I b (-2 k + m) Sqrt[z])^ 2)/(4 b (-2 k + m))]), {i, 0, n}, {h, 0, i}] + E^((I (c + p)^2)/(4 b (-2 k + m)) - (I m Pi)/2) Sum[(-1)^(-h + i) 4^i (c + p)^(-h - i + 2 n) (c + p + 2 I b (-2 k + m) Sqrt[z])^(h + i) ((I (c + p + 2 I b (-2 k + m) Sqrt[z])^2)/(b (-2 k + m)))^ ((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] ((c + p) (c + p + 2 I b (-2 k + m) Sqrt[z]) Gamma[(1/2) (1 + h + i), (I (c + p + 2 I b (-2 k + m) Sqrt[z])^ 2)/(4 b (-2 k + m))] + 2 I b (-2 k + m) Sqrt[(I (c + p + 2 I b (-2 k + m) Sqrt[z])^2)/(b (-2 k + m))] Gamma[(1/2) (2 + h + i), (I (c + p + 2 I b (-2 k + m) Sqrt[z])^ 2)/(4 b (-2 k + m))]), {i, 0, n}, {h, 0, i}]))/ (I b (-2 k + m))^(2 (1 + n)), {k, 0, Floor[(1/2) (-1 + m)]}] /; 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 k = 0 m - 1 2 ( - 1 ) k ( b ( m - 2 k ) ) - 2 ( n + 1 ) ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( p - c ) 2 4 b ( m - 2 k ) - m π 2 i = 0 n h = 0 i ( - 1 ) i - h 4 i ( p - c ) - h - i + 2 n ( - c + p + 2 b ( m - 2 k ) z ) h + i ( ( - c + p + 2 b ( m - 2 k ) z ) 2 b ( m - 2 k ) ) 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]]]]] ( ( p - c ) ( - c + p + 2 b ( m - 2 k ) z ) Γ ( 1 2 ( h + i + 1 ) , ( - c + p + 2 b ( m - 2 k ) z ) 2 4 b ( m - 2 k ) ) + 2 b ( m - 2 k ) ( - c + p + 2 b ( m - 2 k ) z ) 2 b ( m - 2 k ) Γ ( 1 2 ( h + i + 2 ) , ( - c + p + 2 b ( m - 2 k ) z ) 2 4 b ( m - 2 k ) ) ) + ( c + p ) 2 4 b ( m - 2 k ) - m π 2 i = 0 n h = 0 i ( - 1 ) i - h 4 i ( c + p ) - h - i + 2 n ( c + p + 2 b ( m - 2 k ) z ) h + i ( ( c + p + 2 b ( m - 2 k ) z ) 2 b ( m - 2 k ) ) 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 + p ) ( c + p + 2 b ( m - 2 k ) z ) Γ ( 1 2 ( h + i + 1 ) , ( c + p + 2 b ( m - 2 k ) z ) 2 4 b ( m - 2 k ) ) + 2 b ( m - 2 k ) ( c + p + 2 b ( m - 2 k ) z ) 2 b ( m - 2 k ) Γ ( 1 2 ( h + i + 2 ) , ( c + p + 2 b ( m - 2 k ) z ) 2 4 b ( m - 2 k ) ) ) + m π 2 - ( p - c ) 2 4 b ( m - 2 k ) i = 0 n h = 0 i ( - 1 ) i - h 4 i ( p - c ) - h - i + 2 n ( - c + p - 2 b ( m - 2 k ) z ) h + i ( - ( - c + p - 2 b ( m - 2 k ) z ) 2 b ( m - 2 k ) ) 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]]]]] ( ( p - c ) ( - c + p - 2 b ( m - 2 k ) z ) Γ ( 1 2 ( h + i + 1 ) , - ( - c + p - 2 b ( m - 2 k ) z ) 2 4 b ( m - 2 k ) ) - 2 b ( m - 2 k ) - ( - c + p - 2 b ( m - 2 k ) z ) 2 b ( m - 2 k ) Γ ( 1 2 ( h + i + 2 ) , - ( - c + p - 2 b ( m - 2 k ) z ) 2 4 b ( m - 2 k ) ) ) + m π 2 - ( c + p ) 2 4 b ( m - 2 k ) i = 0 n h = 0 i ( - 1 ) i - h 4 i ( c + p ) - h - i + 2 n ( c + p - 2 b ( m - 2 k ) z ) h + i ( - ( c + p - 2 b ( m - 2 k ) z ) 2 b ( m - 2 k ) ) 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 + p ) ( c + p - 2 b ( m - 2 k ) z ) Γ ( 1 2 ( h + i + 1 ) , - ( c + p - 2 b ( m - 2 k ) z ) 2 4 b ( m - 2 k ) ) - 2 b ( m - 2 k ) - ( c + p - 2 b ( m - 2 k ) z ) 2 b ( m - 2 k ) Γ ( 1 2 ( h + i + 2 ) , - ( c + p - 2 b ( m - 2 k ) z ) 2 4 b ( m - 2 k ) ) ) ) - 2 - m ( 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]]]]] ( Γ ( 2 ( n + 1 ) , ( c - p ) z ) ( c - p ) - 2 ( n + 1 ) + ( c + p ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( - c - p ) z ) ) ( 1 - m mod 2 \$CellContext`m 2 ) /; m + n Condition z z n p z 1 2 b z m c z 1 2 2 -1 m -1 2 n -2 k 0 m -1 2 -1 -1 k b m -1 2 k -2 n 1 Binomial m k p -1 c 2 4 b m -1 2 k -1 -1 m 2 -1 h 0 i i 0 n -1 i -1 h 4 i p -1 c -1 h -1 i 2 n -1 c p 2 b m -1 2 k z 1 2 h i -1 c p 2 b m -1 2 k z 1 2 2 b m -1 2 k -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i p -1 c -1 c p 2 b m -1 2 k z 1 2 Gamma 1 2 h i 1 -1 c p 2 b m -1 2 k z 1 2 2 4 b m -1 2 k -1 2 b m -1 2 k -1 c p 2 b m -1 2 k z 1 2 2 b m -1 2 k -1 1 2 Gamma 1 2 h i 2 -1 c p 2 b m -1 2 k z 1 2 2 4 b m -1 2 k -1 c p 2 4 b m -1 2 k -1 -1 m 2 -1 h 0 i i 0 n -1 i -1 h 4 i c p -1 h -1 i 2 n c p 2 b m -1 2 k z 1 2 h i c p 2 b m -1 2 k z 1 2 2 b m -1 2 k -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i c p c p 2 b m -1 2 k z 1 2 Gamma 1 2 h i 1 c p 2 b m -1 2 k z 1 2 2 4 b m -1 2 k -1 2 b m -1 2 k c p 2 b m -1 2 k z 1 2 2 b m -1 2 k -1 1 2 Gamma 1 2 h i 2 c p 2 b m -1 2 k z 1 2 2 4 b m -1 2 k -1 m 2 -1 -1 p -1 c 2 4 b m -1 2 k -1 h 0 i i 0 n -1 i -1 h 4 i p -1 c -1 h -1 i 2 n -1 c p -1 2 b m -1 2 k z 1 2 h i -1 -1 c p -1 2 b m -1 2 k z 1 2 2 b m -1 2 k -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i p -1 c -1 c p -1 2 b m -1 2 k z 1 2 Gamma 1 2 h i 1 -1 -1 c p -1 2 b m -1 2 k z 1 2 2 4 b m -1 2 k -1 -1 2 b m -1 2 k -1 -1 c p -1 2 b m -1 2 k z 1 2 2 b m -1 2 k -1 1 2 Gamma 1 2 h i 2 -1 -1 c p -1 2 b m -1 2 k z 1 2 2 4 b m -1 2 k -1 m 2 -1 -1 c p 2 4 b m -1 2 k -1 h 0 i i 0 n -1 i -1 h 4 i c p -1 h -1 i 2 n c p -1 2 b m -1 2 k z 1 2 h i -1 c p -1 2 b m -1 2 k z 1 2 2 b m -1 2 k -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i c p c p -1 2 b m -1 2 k z 1 2 Gamma 1 2 h i 1 -1 c p -1 2 b m -1 2 k z 1 2 2 4 b m -1 2 k -1 -1 2 b m -1 2 k -1 c p -1 2 b m -1 2 k z 1 2 2 b m -1 2 k -1 1 2 Gamma 1 2 h i 2 -1 c p -1 2 b m -1 2 k z 1 2 2 4 b m -1 2 k -1 -1 2 -1 m Binomial m m 2 -1 Gamma 2 n 1 c -1 p z 1 2 c -1 p -2 n 1 c p -2 n 1 Gamma 2 n 1 -1 c -1 p z 1 2 1 -1 \$CellContext`m 2 m SuperPlus n [/itex]

 Rule Form

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

 2002-12-18