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 Cosh

 http://functions.wolfram.com/01.20.21.1113.01

 Input Form

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

 Standard Form

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 MathML Form

 z n cos m ( b z ) cosh ( c z ) z 2 - m - 2 n - 2 ( s = 0 m - 1 2 ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - b 2 ( m - 2 s ) 2 4 c j = 0 n h = 0 j ( - 1 ) j - h 4 j ( - b ( m - 2 s ) ) - h - j + 2 n ( - 2 z c - b ( m - 2 s ) ) h + j ( ( - 2 z c - b ( m - 2 s ) ) 2 c ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox["h", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - b ( m - 2 s ) ( - 2 z c - b ( m - 2 s ) ) Γ ( 1 2 ( h + j + 1 ) , ( - 2 z c - b ( m - 2 s ) ) 2 4 c ) - 2 c ( - 2 z c - b ( m - 2 s ) ) 2 c Γ ( 1 2 ( h + j + 2 ) , ( - 2 z c - b ( m - 2 s ) ) 2 4 c ) ) + - b 2 ( m - 2 s ) 2 4 c j = 0 n h = 0 j ( - 1 ) j - h 4 j ( b ( m - 2 s ) ) - h - j + 2 n ( b ( m - 2 s ) - 2 c z ) h + j ( ( b ( m - 2 s ) - 2 c z ) 2 c ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox["h", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( b ( m - 2 s ) ( b ( m - 2 s ) - 2 c z ) Γ ( 1 2 ( h + j + 1 ) , ( b ( m - 2 s ) - 2 c z ) 2 4 c ) - 2 c ( b ( m - 2 s ) - 2 c z ) 2 c Γ ( 1 2 ( h + j + 2 ) , ( b ( m - 2 s ) - 2 c z ) 2 4 c ) ) + b 2 ( m - 2 s ) 2 4 c j = 0 n h = 0 j ( - 1 ) j - h 4 j ( - b ( m - 2 s ) ) - h - j + 2 n ( 2 c z - b ( m - 2 s ) ) h + j ( - ( 2 c z - b ( m - 2 s ) ) 2 c ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox["h", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 2 c - ( 2 c z - b ( m - 2 s ) ) 2 c Γ ( 1 2 ( h + j + 2 ) , - ( 2 c z - b ( m - 2 s ) ) 2 4 c ) - b ( m - 2 s ) ( 2 c z - b ( m - 2 s ) ) Γ ( 1 2 ( h + j + 1 ) , - ( 2 c z - b ( m - 2 s ) ) 2 4 c ) ) + b 2 ( m - 2 s ) 2 4 c j = 0 n h = 0 j ( - 1 ) j - h 4 j ( b ( m - 2 s ) ) - h - j + 2 n ( 2 z c + b ( m - 2 s ) ) h + j ( - ( 2 z c + b ( m - 2 s ) ) 2 c ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox["h", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( b ( m - 2 s ) ( 2 z c + b ( m - 2 s ) ) Γ ( 1 2 ( h + j + 1 ) , - ( 2 z c + b ( m - 2 s ) ) 2 4 c ) + 2 - ( 2 z c + b ( m - 2 s ) ) 2 c c Γ ( 1 2 ( h + j + 2 ) , - ( 2 z c + b ( m - 2 s ) ) 2 4 c ) ) ) ) c - 2 ( n + 1 ) + ( - 1 ) n 2 - m - 1 ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["m", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( Γ ( n + 1 , c z ) ( - c ) - n - 1 + c - n - 1 Γ ( n + 1 , - c z ) ) ( 1 - m mod 2 \$CellContext`m 2 ) /; n m + Condition z z n b z 1 2 m c z 2 -1 m -1 2 n -2 s 0 m -1 2 -1 Binomial m s -1 b 2 m -1 2 s 2 4 c -1 h 0 j j 0 n -1 j -1 h 4 j -1 b m -1 2 s -1 h -1 j 2 n -2 z 1 2 c -1 b m -1 2 s h j -2 z 1 2 c -1 b m -1 2 s 2 c -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j -1 b m -1 2 s -2 z 1 2 c -1 b m -1 2 s Gamma 1 2 h j 1 -2 z 1 2 c -1 b m -1 2 s 2 4 c -1 -1 2 c -2 z 1 2 c -1 b m -1 2 s 2 c -1 1 2 Gamma 1 2 h j 2 -2 z 1 2 c -1 b m -1 2 s 2 4 c -1 -1 b 2 m -1 2 s 2 4 c -1 h 0 j j 0 n -1 j -1 h 4 j b m -1 2 s -1 h -1 j 2 n b m -1 2 s -1 2 c z 1 2 h j b m -1 2 s -1 2 c z 1 2 2 c -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j b m -1 2 s b m -1 2 s -1 2 c z 1 2 Gamma 1 2 h j 1 b m -1 2 s -1 2 c z 1 2 2 4 c -1 -1 2 c b m -1 2 s -1 2 c z 1 2 2 c -1 1 2 Gamma 1 2 h j 2 b m -1 2 s -1 2 c z 1 2 2 4 c -1 b 2 m -1 2 s 2 4 c -1 h 0 j j 0 n -1 j -1 h 4 j -1 b m -1 2 s -1 h -1 j 2 n 2 c z 1 2 -1 b m -1 2 s h j -1 2 c z 1 2 -1 b m -1 2 s 2 c -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j 2 c -1 2 c z 1 2 -1 b m -1 2 s 2 c -1 1 2 Gamma 1 2 h j 2 -1 2 c z 1 2 -1 b m -1 2 s 2 4 c -1 -1 b m -1 2 s 2 c z 1 2 -1 b m -1 2 s Gamma 1 2 h j 1 -1 2 c z 1 2 -1 b m -1 2 s 2 4 c -1 b 2 m -1 2 s 2 4 c -1 h 0 j j 0 n -1 j -1 h 4 j b m -1 2 s -1 h -1 j 2 n 2 z 1 2 c b m -1 2 s h j -1 2 z 1 2 c b m -1 2 s 2 c -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j b m -1 2 s 2 z 1 2 c b m -1 2 s Gamma 1 2 h j 1 -1 2 z 1 2 c b m -1 2 s 2 4 c -1 2 -1 2 z 1 2 c b m -1 2 s 2 c -1 1 2 c Gamma 1 2 h j 2 -1 2 z 1 2 c b m -1 2 s 2 4 c -1 c -2 n 1 -1 n 2 -1 m -1 Binomial m m 2 -1 Gamma n 1 c z -1 c -1 n -1 c -1 n -1 Gamma n 1 -1 c z 1 -1 \$CellContext`m 2 n m SuperPlus [/itex]

 Rule Form

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

 2002-12-18