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 Cosh

 http://functions.wolfram.com/01.20.21.4539.01

 Input Form

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

 Rule Form

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

 2002-12-18