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 Cosh

 http://functions.wolfram.com/01.20.21.4844.01

 Input Form

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

 Standard Form

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RowBox[List["(", RowBox[List[RowBox[List["8", " ", "k"]], "-", RowBox[List["4", " ", "m"]]]], ")"]]]]]]], "]"]]]]]]]], ")"]]]]]], ")"]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2002-12-18