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 Cosh

 http://functions.wolfram.com/01.20.21.3950.01

 Input Form

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

 Standard Form

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"Integers"]], "\[And]", RowBox[List["v", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2002-12-18