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 Cosh

 http://functions.wolfram.com/01.20.21.4848.01

 Input Form

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

 z n p z 2 sinh m ( b z ) cosh v ( c z 2 ) z - m 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]]]]] ( 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 ) z n + 1 ( - p z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , - p z 2 ) + m 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 ) z n + 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]]]]] ( - ( - ( p + 2 c s - c v ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , - ( p + 2 c s - c v ) z 2 ) - ( - ( p + c ( v - 2 s ) ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , - ( p + c ( v - 2 s ) ) z 2 ) ) - 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]]]]] p - n - 1 - b 2 ( m - 2 k ) 2 4 p ( ( - 1 ) m q = 0 n 2 q - n ( b ( m - 2 k ) ) n - q ( 2 b k - b m + 2 p z ) q + 1 ( - ( 2 b k - b m + 2 p z ) 2 p ) 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 + 2 p z ) 2 4 p ) + q = 0 n ( b ( k - m 2 ) ) n - q ( b ( m - 2 k ) + 2 p z ) q + 1 ( - ( b ( m - 2 k ) + 2 p z ) 2 p ) 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 ) + 2 p z ) 2 4 p ) ) + 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 4 ( - b 2 ( m - 2 k ) 2 p + 2 c s - c v - b 2 ( m - 2 k ) 2 p + c ( v - 2 s ) ) ( - b 2 ( m - 2 k ) 2 4 ( p + 2 c s - c v ) p + 2 c s - c v ( m π q = 0 n 2 q - n ( b ( m - 2 k ) ) n - q ( p - 2 c s + c v ) - n - 1 2 ( 2 b k - b m + 2 ( p - 2 c s + c v ) z ) q + 1 ( - ( 2 b k - b m + 2 ( p - 2 c s + c v ) z ) 2 p - 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 + 2 ( p - 2 c s + c v ) z ) 2 4 ( p - 2 c s + c v ) ) + q = 0 n ( b ( k - m 2 ) ) n - q ( p - 2 c s + c v ) - n - 1 2 ( b ( m - 2 k ) + 2 ( p - 2 c s + c v ) z ) q + 1 ( - ( b ( m - 2 k ) + 2 ( p - 2 c s + c v ) z ) 2 p - 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 , - ( b ( m - 2 k ) + 2 ( p - 2 c s + c v ) z ) 2 4 ( p - 2 c s + c v ) ) ) - b 2 ( m - 2 k ) 2 4 ( p + c ( v - 2 s ) ) + m π p + c ( v - 2 s ) q = 0 n 2 q - n ( b ( m - 2 k ) ) n - q ( p + 2 c s - c v ) - n - 1 2 ( 2 b k - b m + 2 ( p + 2 c s - c v ) z ) q + 1 ( - ( 2 b k - b m + 2 ( p + 2 c s - c v ) z ) 2 p + 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 + 2 ( p + 2 c s - c v ) z ) 2 4 ( p + 2 c s - c v ) ) - b 2 ( m - 2 k ) 2 4 ( p - 2 c s + c v ) p - 2 c s + c v q = 0 n ( b ( k - m 2 ) ) n - q ( p + 2 c s - c v ) - n - 1 2 ( b ( m - 2 k ) + 2 ( p + 2 c s - c v ) z ) q + 1 ( - ( b ( m - 2 k ) + 2 ( p + 2 c s - c v ) z ) 2 p + 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 , - ( b ( m - 2 k ) + 2 ( p + 2 c s - c v ) z ) 2 4 ( p + 2 c s - c v ) ) ) ) / ( p + 2 c s - c v p + c ( v - 2 s ) ) /; m + v + n Condition z z n p z 2 b z m c z 2 v -1 m 2 -1 m -1 v -1 Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 z n 1 -1 p z 2 1 2 -1 n -1 Gamma n 1 2 -1 -1 p z 2 m 2 -1 m -1 v -1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 z n 1 s 0 v -1 2 -1 Binomial v s -1 -1 p 2 c s -1 c v z 2 1 2 -1 n -1 Gamma n 1 2 -1 -1 p 2 c s -1 c v z 2 -1 -1 p c v -1 2 s z 2 1 2 -1 n -1 Gamma n 1 2 -1 -1 p c v -1 2 s z 2 -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 p -1 n -1 -1 b 2 m -1 2 k 2 4 p -1 -1 m q 0 n 2 q -1 n b m -1 2 k n -1 q 2 b k -1 b m 2 p z q 1 -1 2 b k -1 b m 2 p z 2 p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 2 b k -1 b m 2 p z 2 4 p -1 q 0 n b k -1 m 2 -1 n -1 q b m -1 2 k 2 p z q 1 -1 b m -1 2 k 2 p z 2 p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 b m -1 2 k 2 p z 2 4 p -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 4 -1 b 2 m -1 2 k 2 p 2 c s -1 c v -1 -1 b 2 m -1 2 k 2 p c v -1 2 s -1 -1 b 2 m -1 2 k 2 4 p 2 c s -1 c v -1 p 2 c s -1 c v 1 2 m q 0 n 2 q -1 n b m -1 2 k n -1 q p -1 2 c s c v -1 n -1 1 2 2 b k -1 b m 2 p -1 2 c s c v z q 1 -1 2 b k -1 b m 2 p -1 2 c s c v z 2 p -1 2 c s c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 2 b k -1 b m 2 p -1 2 c s c v z 2 4 p -1 2 c s c v -1 q 0 n b k -1 m 2 -1 n -1 q p -1 2 c s c v -1 n -1 1 2 b m -1 2 k 2 p -1 2 c s c v z q 1 -1 b m -1 2 k 2 p -1 2 c s c v z 2 p -1 2 c s c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 b m -1 2 k 2 p -1 2 c s c v z 2 4 p -1 2 c s c v -1 -1 b 2 m -1 2 k 2 4 p c v -1 2 s -1 m p c v -1 2 s 1 2 q 0 n 2 q -1 n b m -1 2 k n -1 q p 2 c s -1 c v -1 n -1 1 2 2 b k -1 b m 2 p 2 c s -1 c v z q 1 -1 2 b k -1 b m 2 p 2 c s -1 c v z 2 p 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 2 p 2 c s -1 c v z 2 4 p 2 c s -1 c v -1 -1 b 2 m -1 2 k 2 4 p -1 2 c s c v -1 p -1 2 c s c v 1 2 q 0 n b k -1 m 2 -1 n -1 q p 2 c s -1 c v -1 n -1 1 2 b m -1 2 k 2 p 2 c s -1 c v z q 1 -1 b m -1 2 k 2 p 2 c s -1 c v z 2 p 2 c s -1 c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 b m -1 2 k 2 p 2 c s -1 c v z 2 4 p 2 c s -1 c v -1 p 2 c s -1 c v 1 2 p c v -1 2 s 1 2 -1 m SuperPlus v SuperPlus n [/itex]

 Rule Form

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

 2002-12-18