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 Cosh

 http://functions.wolfram.com/01.20.21.4577.01

 Input Form

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