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 Cosh

 http://functions.wolfram.com/01.20.21.4535.01

 Input Form

 Integrate[z^n Sinh[b z^2 + d z]^m Cosh[c z]^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 + (2 c^(-n - 1) Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[Binomial[v, s] (v - 2 s)^(-1 - n) ((-(-1)^n) Gamma[1 + n, c (2 s - v) z] + Gamma[1 + n, c (-2 s + v) z]), {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)]}] + b^(-n - 1) Sum[(-1)^k Binomial[m, k] Sum[((-2 k + m)^(-n - 1) Binomial[v, s] ((-1)^(n + m) E^((2 d c s)/b) Sum[2^(-n + q) (-2 d k + d m + 2 c s - c v)^(n - q) (2 d k - d m - 2 c s + c v + 4 b k z - 2 b m z)^(1 + q) (-(2 d k - d m - 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, -(2 d k - d m - 2 c s + c v + 4 b k z - 2 b m z)^2/(8 b k - 4 b m)], {q, 0, n}] - E^((d c v)/b) (E^((d^2 (2 k - m))/(2 b) + (c^2 (-2 s + v)^2)/ (4 b k - 2 b m)) (-1)^(m + m) Sum[2^(-n + q) (2 d k - d m - 2 c s + c v)^(n - q) ((2 d k - d m - 2 c s + c v + 4 b k z - 2 b m z)^2/(2 b k - b m))^((1/2) (-1 - q)) (-2 d k + d m + 2 c s - c v - 4 b k z + 2 b m z)^ (1 + q) Binomial[n, q] Gamma[(1 + q)/2, (2 d k - d m - 2 c s + c v + 4 b k z - 2 b m z)^2/ (8 b k - 4 b m)], {q, 0, n}] - (-1)^(n + m) Sum[2^(-n + q) (-2 d k + d m - 2 c s + c v)^(n - q) (2 d k - d m + 2 c s - c v + 4 b k z - 2 b m z)^(1 + q) (-((1/(b (2 k - m))) (d (-2 k + m) + c (-2 s + v) + 2 b (-2 k + m) z)^2))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -(-2 d k + d m - 2 c s + c v - 4 b k z + 2 b m z)^2/(8 b k - 4 b m)], {q, 0, n}] + E^((-2 d k + d m - 2 c s + c v)^2/(4 b k - 2 b m)) Sum[(d (k - m/2) + c s - (c v)/2)^(n - q) ((-2 d k + d m - 2 c s + c v - 4 b k z + 2 b m z)^2/ (2 b k - b m))^((1/2) (-1 - q)) (d (-2 k + m) + c (-2 s + v) + 2 b (-2 k + m) z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, (-2 d k + d m - 2 c s + c v - 4 b k z + 2 b m z)^2/(8 b k - 4 b m)], {q, 0, n}])))/ E^((d^2 (-2 k + m)^2 + c^2 (-2 s + v)^2 + 2 d c (2 k - m) (2 s + v))/ (b (8 k - 4 m))), {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 ) z 2 - m - v - 1 ( ( 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]]]]] s = 0 v - 1 2 ( m - 2 k ) - n - 1 ( 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 + c 2 ( v - 2 s ) 2 + 2 c d ( 2 k - m ) ( 2 s + v ) b ( 8 k - 4 m ) ( ( - 1 ) m + n 2 d c s b q = 0 n 2 q - n ( - 2 d k + d m + 2 c s - c v ) n - q ( 2 d k + 4 b z k - d m - 2 c s + c v - 2 b m z ) q + 1 ( - ( 2 d k + 4 b z k - d m - 2 c s + c v - 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 , - ( 2 d k + 4 b z k - d m - 2 c s + c v - 2 b m z ) 2 8 b k - 4 b m ) - d c v b ( - ( - 1 ) m + n q = 0 n 2 q - n ( - 2 d k + d m - 2 c s + c v ) n - q ( 2 d k + 4 b z k - d m + 2 c s - c v - 2 b m z ) q + 1 ( - ( d ( m - 2 k ) + 2 b z ( m - 2 k ) + c ( v - 2 s ) ) 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 , - ( - 2 d k - 4 b z k + d m - 2 c s + c v + 2 b m z ) 2 8 b k - 4 b m ) + ( - 2 d k + d m - 2 c s + c v ) 2 4 b k - 2 b m q = 0 n ( ( k - m 2 ) d + c s - c v 2 ) n - q ( ( - 2 d k - 4 b z k + d m - 2 c s + c v + 2 b m z ) 2 2 b k - b m ) 1 2 ( - q - 1 ) ( d ( m - 2 k ) + 2 b z ( m - 2 k ) + c ( v - 2 s ) ) 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 d k - 4 b z k + d m - 2 c s + c v + 2 b m z ) 2 8 b k - 4 b m ) + ( - 1 ) m + m ( 2 k - m ) d 2 2 b + c 2 ( v - 2 s ) 2 4 b k - 2 b m q = 0 n 2 q - n ( 2 d k - d m - 2 c s + c v ) n - q ( ( 2 d k + 4 b z k - d m - 2 c s + c v - 2 b m z ) 2 2 b k - b m ) 1 2 ( - q - 1 ) ( - 2 d k - 4 b z k + d m + 2 c s - c v + 2 b m 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 d k + 4 b z k - d m - 2 c s + c v - 2 b m z ) 2 8 b k - 4 b m ) ) ) ) b - n - 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 + 2 c - 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]]]]] ( m mod 2 \$CellContext`m 2 - 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]]]]] ( v - 2 s ) - n - 1 ( Γ ( n + 1 , c ( v - 2 s ) z ) - ( - 1 ) n Γ ( n + 1 , c ( 2 s - v ) z ) ) + ( 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 ) ) ) /; n m + v + Condition z z n b z 2 d z m c z v 2 -1 m -1 v -1 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 m -1 2 k -1 n -1 Binomial v s -1 d 2 m -1 2 k 2 c 2 v -1 2 s 2 2 c d 2 k -1 m 2 s v b 8 k -1 4 m -1 -1 m n 2 d c s b -1 q 0 n 2 q -1 n -2 d k d m 2 c s -1 c v n -1 q 2 d k 4 b z k -1 d m -1 2 c s c v -1 2 b m z q 1 -1 2 d k 4 b z k -1 d m -1 2 c s c v -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 2 d k 4 b z k -1 d m -1 2 c s c v -1 2 b m z 2 8 b k -1 4 b m -1 -1 d c v b -1 -1 -1 m n q 0 n 2 q -1 n -2 d k d m -1 2 c s c v n -1 q 2 d k 4 b z k -1 d m 2 c s -1 c v -1 2 b m z q 1 -1 d m -1 2 k 2 b z m -1 2 k c v -1 2 s 2 b 2 k -1 m -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 -2 d k -1 4 b z k d m -1 2 c s c v 2 b m z 2 8 b k -1 4 b m -1 -2 d k d m -1 2 c s c v 2 4 b k -1 2 b m -1 q 0 n k -1 m 2 -1 d c s -1 c v 2 -1 n -1 q -2 d k -1 4 b z k d m -1 2 c s c v 2 b m z 2 2 b k -1 b m -1 1 2 -1 q -1 d m -1 2 k 2 b z m -1 2 k c v -1 2 s q 1 Binomial n q Gamma q 1 2 -1 -2 d k -1 4 b z k d m -1 2 c s c v 2 b m z 2 8 b k -1 4 b m -1 -1 m m 2 k -1 m d 2 2 b -1 c 2 v -1 2 s 2 4 b k -1 2 b m -1 q 0 n 2 q -1 n 2 d k -1 d m -1 2 c s c v n -1 q 2 d k 4 b z k -1 d m -1 2 c s c v -1 2 b m z 2 2 b k -1 b m -1 1 2 -1 q -1 -2 d k -1 4 b z k d m 2 c s -1 c v 2 b m z q 1 Binomial n q Gamma q 1 2 -1 2 d k 4 b z k -1 d m -1 2 c s c v -1 2 b m z 2 8 b k -1 4 b m -1 b -1 n -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 2 c -1 n -1 -1 m Binomial m m 2 -1 \$CellContext`m 2 -1 s 0 v -1 2 -1 Binomial v s v -1 2 s -1 n -1 Gamma n 1 c v -1 2 s z -1 -1 n Gamma n 1 c 2 s -1 v z 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 n m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18