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 Cosh

 http://functions.wolfram.com/01.20.21.4581.01

 Input Form

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

 Rule Form

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

 2002-12-18