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 Cosh

 http://functions.wolfram.com/01.20.21.4826.01

 Input Form

 Integrate[z^n E^(b z^2 + d z + e) Sinh[a z^2 + p z + q] Cosh[c z^2 + f z + g]^v, z] == (-I) 2^(-2 - v) Binomial[v, v/2] (1 - Mod[v, 2]) ((-a + b)^(-1 - n) E^(e - (d - p)^2/(4 (-a + b)) + (I Pi)/2 - q) Sum[2^(j - n) (-d + p)^(-j + n) (d - p + 2 (-a + b) z)^ (1 + j) (-((d - p + 2 (-a + b) z)^2/(-a + b)))^((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -((d - p + 2 (-a + b) z)^2/ (4 (-a + b)))], {j, 0, n}] + (a + b)^(-1 - n) E^(e - (d + p)^2/(4 (a + b)) - (I Pi)/2 + q) Sum[2^(j - n) (-d - p)^(-j + n) (d + p + 2 (a + b) z)^(1 + j) (-((d + p + 2 (a + b) z)^2/(a + b)))^((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -((d + p + 2 (a + b) z)^2/(4 (a + b)))], {j, 0, n}]) - I 2^(-2 - v) Sum[Binomial[v, s] (E^(e + (I Pi)/2 - q - (d - p + f (2 s - v))^2/ (4 (-a + b + c (2 s - v))) + g (2 s - v)) (-a + b + c (2 s - v))^ (-1 - n) Sum[2^(j - n) (-d + p - f (2 s - v))^(-j + n) (d - p + f (2 s - v) + 2 (-a + b + c (2 s - v)) z)^(1 + j) (-((d - p + f (2 s - v) + 2 (-a + b + c (2 s - v)) z)^2/ (-a + b + c (2 s - v))))^((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -((d - p + f (2 s - v) + 2 (-a + b + c (2 s - v)) z)^2/(4 (-a + b + c (2 s - v))))], {j, 0, n}] + E^(e - (I Pi)/2 + q - (d + p + f (2 s - v))^2/ (4 (a + b + c (2 s - v))) + g (2 s - v)) (a + b + c (2 s - v))^ (-1 - n) Sum[2^(j - n) (-d - p - f (2 s - v))^(-j + n) (d + p + f (2 s - v) + 2 (a + b + c (2 s - v)) z)^(1 + j) (-((d + p + f (2 s - v) + 2 (a + b + c (2 s - v)) z)^2/ (a + b + c (2 s - v))))^((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -((d + p + f (2 s - v) + 2 (a + b + c (2 s - v)) z)^2/(4 (a + b + c (2 s - v))))], {j, 0, n}] + E^(e + (I Pi)/2 - q + g (-2 s + v) - (d - p + f (-2 s + v))^2/ (4 (-a + b + c (-2 s + v)))) (-a + b + c (-2 s + v))^(-1 - n) Sum[2^(j - n) (-d + p - f (-2 s + v))^(-j + n) (d - p + f (-2 s + v) + 2 (-a + b + c (-2 s + v)) z)^(1 + j) (-((d - p + f (-2 s + v) + 2 (-a + b + c (-2 s + v)) z)^2/ (-a + b + c (-2 s + v))))^((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -((d - p + f (-2 s + v) + 2 (-a + b + c (-2 s + v)) z)^2/(4 (-a + b + c (-2 s + v))))], {j, 0, n}] + E^(e - (I Pi)/2 + q + g (-2 s + v) - (d + p + f (-2 s + v))^2/(4 (a + b + c (-2 s + v)))) (a + b + c (-2 s + v))^(-1 - n) Sum[2^(j - n) (-d - p - f (-2 s + v))^ (-j + n) (d + p + f (-2 s + v) + 2 (a + b + c (-2 s + v)) z)^ (1 + j) (-((d + p + f (-2 s + v) + 2 (a + b + c (-2 s + v)) z)^2/ (a + b + c (-2 s + v))))^((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -((d + p + f (-2 s + v) + 2 (a + b + c (-2 s + v)) z)^2/(4 (a + b + c (-2 s + v))))], {j, 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 b z 2 + d z + e sinh ( a z 2 + p z + q ) cosh v ( c z 2 + f z + g ) z - 2 - v - 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]]]]] ( 1 - v mod 2 \$CellContext`v 2 ) ( - ( d - p ) 2 4 ( b - a ) + e - q + π 2 ( b - a ) - n - 1 j = 0 n 2 j - n ( p - d ) n - j ( d - p + 2 ( b - a ) z ) j + 1 ( - ( d - p + 2 ( b - a ) z ) 2 b - a ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( j + 1 2 , - ( d - p + 2 ( b - a ) z ) 2 4 ( b - a ) ) + ( a + b ) - n - 1 - ( d + p ) 2 4 ( a + b ) + e + q - π 2 j = 0 n 2 j - n ( - d - p ) n - j ( d + p + 2 ( a + b ) z ) j + 1 ( - ( d + p + 2 ( a + b ) z ) 2 a + b ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( j + 1 2 , - ( d + p + 2 ( a + b ) z ) 2 4 ( a + b ) ) ) - 2 - v - 2 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]]]]] ( - ( d - p + f ( 2 s - v ) ) 2 4 ( - a + b + c ( 2 s - v ) ) + e - q + g ( 2 s - v ) + π 2 ( - a + b + c ( 2 s - v ) ) - n - 1 j = 0 n 2 j - n ( - d + p - f ( 2 s - v ) ) n - j ( d - p + f ( 2 s - v ) + 2 ( - a + b + c ( 2 s - v ) ) z ) j + 1 ( - ( d - p + f ( 2 s - v ) + 2 ( - a + b + c ( 2 s - v ) ) z ) 2 - a + b + c ( 2 s - v ) ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( j + 1 2 , - ( d - p + f ( 2 s - v ) + 2 ( - a + b + c ( 2 s - v ) ) z ) 2 4 ( - a + b + c ( 2 s - v ) ) ) + - ( d + p + f ( 2 s - v ) ) 2 4 ( a + b + c ( 2 s - v ) ) + e + q + g ( 2 s - v ) - π 2 ( a + b + c ( 2 s - v ) ) - n - 1 j = 0 n 2 j - n ( - d - p - f ( 2 s - v ) ) n - j ( d + p + f ( 2 s - v ) + 2 ( a + b + c ( 2 s - v ) ) z ) j + 1 ( - ( d + p + f ( 2 s - v ) + 2 ( a + b + c ( 2 s - v ) ) z ) 2 a + b + c ( 2 s - v ) ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( j + 1 2 , - ( d + p + f ( 2 s - v ) + 2 ( a + b + c ( 2 s - v ) ) z ) 2 4 ( a + b + c ( 2 s - v ) ) ) + - ( d - p + f ( v - 2 s ) ) 2 4 ( - a + b + c ( v - 2 s ) ) + e - q + g ( v - 2 s ) + π 2 ( - a + b + c ( v - 2 s ) ) - n - 1 j = 0 n 2 j - n ( - d + p - f ( v - 2 s ) ) n - j ( d - p + f ( v - 2 s ) + 2 ( - a + b + c ( v - 2 s ) ) z ) j + 1 ( - ( d - p + f ( v - 2 s ) + 2 ( - a + b + c ( v - 2 s ) ) z ) 2 - a + b + c ( v - 2 s ) ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( j + 1 2 , - ( d - p + f ( v - 2 s ) + 2 ( - a + b + c ( v - 2 s ) ) z ) 2 4 ( - a + b + c ( v - 2 s ) ) ) + - ( d + p + f ( v - 2 s ) ) 2 4 ( a + b + c ( v - 2 s ) ) + e + q + g ( v - 2 s ) - π 2 ( a + b + c ( v - 2 s ) ) - n - 1 j = 0 n 2 j - n ( - d - p - f ( v - 2 s ) ) n - j ( d + p + f ( v - 2 s ) + 2 ( a + b + c ( v - 2 s ) ) z ) j + 1 ( - ( d + p + f ( v - 2 s ) + 2 ( a + b + c ( v - 2 s ) ) z ) 2 a + b + c ( v - 2 s ) ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( j + 1 2 , - ( d + p + f ( v - 2 s ) + 2 ( a + b + c ( v - 2 s ) ) z ) 2 4 ( a + b + c ( v - 2 s ) ) ) ) /; n v + Condition z z n b z 2 d z e a z 2 p z q c z 2 f z g v -1 2 -1 v -2 Binomial v v 2 -1 1 -1 \$CellContext`v 2 -1 d -1 p 2 4 b -1 a -1 e -1 q 2 -1 b -1 a -1 n -1 j 0 n 2 j -1 n p -1 d n -1 j d -1 p 2 b -1 a z j 1 -1 d -1 p 2 b -1 a z 2 b -1 a -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 d -1 p 2 b -1 a z 2 4 b -1 a -1 a b -1 n -1 -1 d p 2 4 a b -1 e q -1 2 -1 j 0 n 2 j -1 n -1 d -1 p n -1 j d p 2 a b z j 1 -1 d p 2 a b z 2 a b -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 d p 2 a b z 2 4 a b -1 -1 2 -1 v -2 s 0 v -1 2 -1 Binomial v s -1 d -1 p f 2 s -1 v 2 4 -1 a b c 2 s -1 v -1 e -1 q g 2 s -1 v 2 -1 -1 a b c 2 s -1 v -1 n -1 j 0 n 2 j -1 n -1 d p -1 f 2 s -1 v n -1 j d -1 p f 2 s -1 v 2 -1 a b c 2 s -1 v z j 1 -1 d -1 p f 2 s -1 v 2 -1 a b c 2 s -1 v z 2 -1 a b c 2 s -1 v -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 d -1 p f 2 s -1 v 2 -1 a b c 2 s -1 v z 2 4 -1 a b c 2 s -1 v -1 -1 d p f 2 s -1 v 2 4 a b c 2 s -1 v -1 e q g 2 s -1 v -1 2 -1 a b c 2 s -1 v -1 n -1 j 0 n 2 j -1 n -1 d -1 p -1 f 2 s -1 v n -1 j d p f 2 s -1 v 2 a b c 2 s -1 v z j 1 -1 d p f 2 s -1 v 2 a b c 2 s -1 v z 2 a b c 2 s -1 v -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 d p f 2 s -1 v 2 a b c 2 s -1 v z 2 4 a b c 2 s -1 v -1 -1 d -1 p f v -1 2 s 2 4 -1 a b c v -1 2 s -1 e -1 q g v -1 2 s 2 -1 -1 a b c v -1 2 s -1 n -1 j 0 n 2 j -1 n -1 d p -1 f v -1 2 s n -1 j d -1 p f v -1 2 s 2 -1 a b c v -1 2 s z j 1 -1 d -1 p f v -1 2 s 2 -1 a b c v -1 2 s z 2 -1 a b c v -1 2 s -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 d -1 p f v -1 2 s 2 -1 a b c v -1 2 s z 2 4 -1 a b c v -1 2 s -1 -1 d p f v -1 2 s 2 4 a b c v -1 2 s -1 e q g v -1 2 s -1 2 -1 a b c v -1 2 s -1 n -1 j 0 n 2 j -1 n -1 d -1 p -1 f v -1 2 s n -1 j d p f v -1 2 s 2 a b c v -1 2 s z j 1 -1 d p f v -1 2 s 2 a b c v -1 2 s z 2 a b c v -1 2 s -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 d p f v -1 2 s 2 a b c v -1 2 s z 2 4 a b c v -1 2 s -1 n v SuperPlus [/itex]

 Rule Form

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

 2002-12-18