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 Cosh

 http://functions.wolfram.com/01.20.21.4571.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18