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 Cosh

 http://functions.wolfram.com/01.20.21.4561.01

 Input Form

 Integrate[z^n Sinh[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] + E^(2 g (-2 s + v)) ((-c) (2 s - v) z^2)^ ((1 + n)/2) Gamma[(1 + n)/2, c (2 s - v) z^2]), {s, 0, Floor[(1/2) (-1 + v)]}])/I^m + 2 d^(-n - 1) Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[(-1)^k (m - 2 k)^(-1 - n) Binomial[m, k] ((-(-1)^n) Gamma[1 + n, d (2 k - m) z] + (-1)^m Gamma[1 + n, d (-2 k + m) z]), {k, 0, Floor[(1/2) (-1 + m)]}] - c^(-n - 1) Sum[(-1)^(k + n) Binomial[m, k] Sum[E^(((-d^2) (-2 k + m)^2 + 4 c (2 s - v) (2 g s - g v))/ (c (8 s - 4 v))) (-2 s + v)^(-n - 1) Binomial[v, s] ((-1)^(m + n) E^((d^2 (-2 k + m)^2)/(c (4 s - 2 v)) + 2 g (-2 s + v)) Sum[2^(-n + q) (d (-2 k + m))^(n - q) ((-2 d k + d m + 4 c s z - 2 c v z)^2/(2 c s - c v))^ ((1/2) (-1 - q)) (d (2 k - m) + 2 c (-2 s + v) z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, (-2 d k + d m + 4 c s z - 2 c v z)^2/(c (8 s - 4 v))], {q, 0, n}] - Sum[(d (k - m/2))^(n - q) (-2 d k + d m + 4 c s z - 2 c v z)^ (1 + q) (-((-2 d k + d m + 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 d k + d m + 4 c s z - 2 c v z)^2/(8 c s - 4 c v))], {q, 0, n}] - (-1)^m Sum[2^(-n + q) (d (-2 k + m))^(n - q) (2 d k - d m + 4 c s z - 2 c v z)^(1 + q) (-((d (-2 k + m) + 2 c (-2 s + v) z)^2/(c (2 s - v))))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((-2 d k + d m - 4 c s z + 2 c v z)^2/(8 c s - 4 c v))], {q, 0, n}] - (-1)^n E^(2 g (-2 s + v) + (d^2 (-2 k + m)^2)/ (4 c s - 2 c v)) Sum[(d (k - m/2))^(n - q) (2 d k - d m + 4 c s z - 2 c v z) (d (-2 k + m) + 2 c (-2 s + v) z)^q ((d (-2 k + m) + 2 c (-2 s + v) z)^2/(c (2 s - v)))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (-2 d k + d m - 4 c s z + 2 c v z)^2/(8 c s - 4 c v)], {q, 0, n}]), {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 ( d z ) cosh v ( c z 2 + g ) z 2 - m - v - 1 ( - k = 0 m - 1 2 ( - 1 ) k + n ( 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 4 c ( 2 s - v ) ( 2 g s - g v ) - d 2 ( m - 2 k ) 2 c ( 8 s - 4 v ) ( v - 2 s ) - n - 1 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - 1 ) m + n d 2 ( m - 2 k ) 2 c ( 4 s - 2 v ) + 2 g ( v - 2 s ) q = 0 n 2 q - n ( d ( m - 2 k ) ) n - q ( ( - 2 d k + d m + 4 c s z - 2 c v z ) 2 2 c s - c v ) 1 2 ( - q - 1 ) ( d ( 2 k - m ) + 2 c ( v - 2 s ) 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 + d m + 4 c s z - 2 c v z ) 2 c ( 8 s - 4 v ) ) - ( - 1 ) n d 2 ( m - 2 k ) 2 4 c s - 2 c v + 2 g ( v - 2 s ) q = 0 n ( d ( k - m 2 ) ) n - q ( 2 d k - d m + 4 c s z - 2 c v z ) ( d ( m - 2 k ) + 2 c ( v - 2 s ) z ) q ( ( d ( m - 2 k ) + 2 c ( v - 2 s ) 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 , ( - 2 d k + d m - 4 c s z + 2 c v z ) 2 8 c s - 4 c v ) - ( - 1 ) m q = 0 n 2 q - n ( d ( m - 2 k ) ) n - q ( 2 d k - d m + 4 c s z - 2 c v z ) q + 1 ( - ( d ( m - 2 k ) + 2 c ( v - 2 s ) 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 , - ( - 2 d k + d m - 4 c s z + 2 c v z ) 2 8 c s - 4 c v ) - q = 0 n ( d ( k - m 2 ) ) n - q ( - 2 d k + d m + 4 c s z - 2 c v z ) q + 1 ( - ( - 2 d k + d m + 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 d k + d m + 4 c s z - 2 c v z ) 2 8 c s - 4 c v ) ) c - 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 + - 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]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) 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 ( v - 2 s ) 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 ) ) + 2 d - n - 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]]]]] ( v mod 2 \$CellContext`v 2 - 1 ) k = 0 m - 1 2 ( - 1 ) k ( 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]]]]] ( ( - 1 ) m Γ ( n + 1 , d ( m - 2 k ) z ) - ( - 1 ) n Γ ( n + 1 , d ( 2 k - m ) z ) ) ) /; n m + v + Condition z z n d z m c z 2 g v 2 -1 m -1 v -1 -1 k 0 m -1 2 -1 -1 k n Binomial m k s 0 v -1 2 -1 4 c 2 s -1 v 2 g s -1 g v -1 d 2 m -1 2 k 2 c 8 s -1 4 v -1 v -1 2 s -1 n -1 Binomial v s -1 m n d 2 m -1 2 k 2 c 4 s -1 2 v -1 2 g v -1 2 s q 0 n 2 q -1 n d m -1 2 k n -1 q -2 d k d m 4 c s z -1 2 c v z 2 2 c s -1 c v -1 1 2 -1 q -1 d 2 k -1 m 2 c v -1 2 s z q 1 Binomial n q Gamma q 1 2 -1 -2 d k d m 4 c s z -1 2 c v z 2 c 8 s -1 4 v -1 -1 -1 n d 2 m -1 2 k 2 4 c s -1 2 c v -1 2 g v -1 2 s q 0 n d k -1 m 2 -1 n -1 q 2 d k -1 d m 4 c s z -1 2 c v z d m -1 2 k 2 c v -1 2 s z q d m -1 2 k 2 c v -1 2 s z 2 c 2 s -1 v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -2 d k d m -1 4 c s z 2 c v z 2 8 c s -1 4 c v -1 -1 -1 m q 0 n 2 q -1 n d m -1 2 k n -1 q 2 d k -1 d m 4 c s z -1 2 c v z q 1 -1 d m -1 2 k 2 c v -1 2 s z 2 c 2 s -1 v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 -2 d k d m -1 4 c s z 2 c v z 2 8 c s -1 4 c v -1 -1 q 0 n d k -1 m 2 -1 n -1 q -2 d k d m 4 c s z -1 2 c v z q 1 -1 -2 d k d m 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 d k d m 4 c s z -1 2 c v z 2 8 c s -1 4 c v -1 c -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 -1 m z n 1 Binomial m m 2 -1 \$CellContext`m 2 -1 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 v -1 2 s 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 2 d -1 n -1 Binomial v v 2 -1 \$CellContext`v 2 -1 k 0 m -1 2 -1 -1 k m -1 2 k -1 n -1 Binomial m k -1 m Gamma n 1 d m -1 2 k z -1 -1 n Gamma n 1 d 2 k -1 m z n m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18