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 Cosh

 http://functions.wolfram.com/01.20.21.4201.01

 Input Form

 Integrate[Sinh[b z^2 + d z]^m Cosh[c z^2 + f z]^v, z] == I^m 2^(-m - v) z Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2]) - (1/b) (2^(-1 - m - v) Sqrt[Pi] Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(1/(-2 k + m)) (((-1)^k Binomial[m, k] ((-1)^m E^((d^2 (-2 k + m))/(2 b)) Sqrt[b (2 k - m)] Erfi[(Sqrt[b (2 k - m)] (d + 2 b z))/(2 b)] + Sqrt[b (-2 k + m)] Erfi[((2 k - m) (d + 2 b z))/(2 Sqrt[b (-2 k + m)])]))/ E^((d^2 (-2 k + m))/(4 b))), {k, 0, Floor[(1/2) (-1 + m)]}]) - (1/c) (I^m 2^(-1 - m - v) Sqrt[Pi] Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(1/(-2 s + v)) (E^((f^2 (2 s - v))/(4 c)) Binomial[v, s] (E^((f^2 (-2 s + v))/(2 c)) Sqrt[c (2 s - v)] Erfi[(Sqrt[c (2 s - v)] (f + 2 c z))/(2 c)] + Sqrt[c (-2 s + v)] Erfi[((2 s - v) (f + 2 c z))/(2 Sqrt[c (-2 s + v)])])), {s, 0, Floor[(1/2) (-1 + v)]}]) + 2^(-1 - m - v) Sqrt[Pi] Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] ((E^((-2 d k + d m - 2 f s + f v)^2/(8 b k - 4 b m + 8 c s - 4 c v)) Erfi[(-2 k (d + 2 b z) + m (d + 2 b z) - (2 s - v) (f + 2 c z))/ (2 Sqrt[b (-2 k + m) + c (-2 s + v)])])/ Sqrt[b (-2 k + m) + c (-2 s + v)] - ((-1)^m Sqrt[2 b k - b m + 2 c s - c v] Erfi[(2 k (d + 2 b z) - m (d + 2 b z) + (2 s - v) (f + 2 c z))/ (2 Sqrt[2 b k - b m + 2 c s - c v])])/ E^((-2 d k + d m - 2 f s + f v)^2/(8 b k - 4 b m + 8 c s - 4 c v))/ (b (-2 k + m) + c (-2 s + v)) + ((-(-1)^m) Sqrt[2 b k - b m - 2 c s + c v] Erfi[(2 k (d + 2 b z) - m (d + 2 b z) - (2 s - v) (f + 2 c z))/(2 Sqrt[2 b k - b m - 2 c s + c v])] + E^((2 d k - d m - 2 f s + f v)^2/(4 b k - 2 b m - 4 c s + 2 c v)) Sqrt[-2 b k + b m + 2 c s - c v] Erfi[(-2 k (d + 2 b z) + m (d + 2 b z) + (2 s - v) (f + 2 c z))/( 2 Sqrt[-2 b k + b m + 2 c s - c v])])/ E^((2 d k - d m + f (-2 s + v))^2/(8 b k - 4 b m - 8 c s + 4 c v))/ (-2 b k + b m + 2 c s - c v)), {s, 0, Floor[(1/2) (-1 + v)]}], {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 sinh m ( b z 2 + d z ) cosh v ( c z 2 + f z ) z m 2 - m - v z ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["m", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["v", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) - 1 b ( 2 - m - v - 1 π ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["v", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 - v mod 2 \$CellContext`v 2 ) k = 0 m - 1 2 1 m - 2 k ( ( - 1 ) k - d 2 ( m - 2 k ) 4 b ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( - 1 ) m d 2 ( m - 2 k ) 2 b b ( 2 k - m ) erfi ( b ( 2 k - m ) ( d + 2 b z ) 2 b ) + b ( m - 2 k ) erfi ( ( 2 k - m ) ( d + 2 b z ) 2 b ( m - 2 k ) ) ) ) ) - 1 c ( m 2 - m - v - 1 π ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["m", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 - m mod 2 \$CellContext`m 2 ) s = 0 v - 1 2 1 v - 2 s ( f 2 ( 2 s - v ) 4 c ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( f 2 ( v - 2 s ) 2 c c ( 2 s - v ) erfi ( c ( 2 s - v ) ( f + 2 c z ) 2 c ) + c ( v - 2 s ) erfi ( ( 2 s - v ) ( f + 2 c z ) 2 c ( v - 2 s ) ) ) ) ) + 2 - m - v - 1 π k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 - 2 b k + b m + 2 c s - c v ( - ( 2 d k - d m + f ( v - 2 s ) ) 2 8 b k - 4 b m - 8 c s + 4 c v ( ( 2 d k - d m - 2 f s + f v ) 2 4 b k - 2 b m - 4 c s + 2 c v - 2 b k + b m + 2 c s - c v erfi ( - 2 k ( d + 2 b z ) + m ( d + 2 b z ) + ( 2 s - v ) ( f + 2 c z ) 2 - 2 b k + b m + 2 c s - c v ) - ( - 1 ) m 2 b k - b m - 2 c s + c v erfi ( 2 k ( d + 2 b z ) - m ( d + 2 b z ) - ( 2 s - v ) ( f + 2 c z ) 2 2 b k - b m - 2 c s + c v ) ) ) + ( - 2 d k + d m - 2 f s + f v ) 2 8 b k - 4 b m + 8 c s - 4 c v erfi ( - 2 k ( d + 2 b z ) + m ( d + 2 b z ) - ( 2 s - v ) ( f + 2 c z ) 2 b ( m - 2 k ) + c ( v - 2 s ) ) b ( m - 2 k ) + c ( v - 2 s ) - ( ( - 1 ) m - ( - 2 d k + d m - 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 erfi ( 2 k ( d + 2 b z ) - m ( d + 2 b z ) + ( 2 s - v ) ( f + 2 c z ) 2 2 b k - b m + 2 c s - c v ) ) / ( b ( m - 2 k ) + c ( v - 2 s ) ) ) /; m + v + Condition z b z 2 d z m c z 2 f z v m 2 -1 m -1 v z Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 -1 1 b -1 2 -1 m -1 v -1 1 2 Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 1 m -1 2 k -1 -1 k -1 d 2 m -1 2 k 4 b -1 Binomial m k -1 m d 2 m -1 2 k 2 b -1 b 2 k -1 m 1 2 Erfi b 2 k -1 m 1 2 d 2 b z 2 b -1 b m -1 2 k 1 2 Erfi 2 k -1 m d 2 b z 2 b m -1 2 k 1 2 -1 -1 1 c -1 m 2 -1 m -1 v -1 1 2 Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 v -1 2 -1 1 v -1 2 s -1 f 2 2 s -1 v 4 c -1 Binomial v s f 2 v -1 2 s 2 c -1 c 2 s -1 v 1 2 Erfi c 2 s -1 v 1 2 f 2 c z 2 c -1 c v -1 2 s 1 2 Erfi 2 s -1 v f 2 c z 2 c v -1 2 s 1 2 -1 2 -1 m -1 v -1 1 2 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s 1 -2 b k b m 2 c s -1 c v -1 -1 2 d k -1 d m f v -1 2 s 2 8 b k -1 4 b m -1 8 c s 4 c v -1 2 d k -1 d m -1 2 f s f v 2 4 b k -1 2 b m -1 4 c s 2 c v -1 -2 b k b m 2 c s -1 c v 1 2 Erfi -2 k d 2 b z m d 2 b z 2 s -1 v f 2 c z 2 -2 b k b m 2 c s -1 c v 1 2 -1 -1 -1 m 2 b k -1 b m -1 2 c s c v 1 2 Erfi 2 k d 2 b z -1 m d 2 b z -1 2 s -1 v f 2 c z 2 2 b k -1 b m -1 2 c s c v 1 2 -1 -2 d k d m -1 2 f s f v 2 8 b k -1 4 b m 8 c s -1 4 c v -1 Erfi -2 k d 2 b z m d 2 b z -1 2 s -1 v f 2 c z 2 b m -1 2 k c v -1 2 s 1 2 -1 b m -1 2 k c v -1 2 s 1 2 -1 -1 -1 m -1 -2 d k d m -1 2 f s f v 2 8 b k -1 4 b m 8 c s -1 4 c v -1 2 b k -1 b m 2 c s -1 c v 1 2 Erfi 2 k d 2 b z -1 m d 2 b z 2 s -1 v f 2 c z 2 2 b k -1 b m 2 c s -1 c v 1 2 -1 b m -1 2 k c v -1 2 s -1 m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18