html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Cosh

 http://functions.wolfram.com/01.20.21.4465.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18