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 Cosh

 http://functions.wolfram.com/01.20.21.4658.01

 Input Form

 Integrate[E^(p Sqrt[z]) Sinh[b z]^m Cosh[c Sqrt[z]]^v, z] == (1/p^2) ((2^(1 - m - v) E^(p Sqrt[z]) (-1 + p Sqrt[z]) Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2]))/I^m) + 2^(-m - v) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(-1)^k Binomial[m, k] ((-1)^m (-(E^(p Sqrt[z] - b (-2 k + m) z)/(b (-2 k + m))) - (E^(p^2/(4 b (-2 k + m))) p Sqrt[Pi] Erfi[(p - 2 b (-2 k + m) Sqrt[z])/(2 Sqrt[(-b) (-2 k + m)])])/ (2 ((-b) (-2 k + m))^(3/2))) + E^(p Sqrt[z] + b (-2 k + m) z)/ (b (-2 k + m)) - (p Sqrt[Pi] Erfi[(p + 2 b (-2 k + m) Sqrt[z])/ (2 Sqrt[b (-2 k + m)])])/(E^(p^2/(4 b (-2 k + m))) (2 (b (-2 k + m))^(3/2)))), {k, 0, Floor[(1/2) (-1 + m)]}] + (2^(2 - m - v) E^(p Sqrt[z]) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(Binomial[v, s] ((-p^2 - c^2 (-2 s + v)^2 + p^3 Sqrt[z] - c^2 p (-2 s + v)^2 Sqrt[z]) Cosh[c (2 s - v) Sqrt[z]] - c (2 s - v) (-2 p + p^2 Sqrt[z] - c^2 (-2 s + v)^2 Sqrt[z]) Sinh[c (2 s - v) Sqrt[z]]))/(p^2 - c^2 (-2 s + v)^2)^2, {s, 0, Floor[(1/2) (-1 + v)]}])/I^m + 2^(-m - v) Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] ((-1)^m (-(E^((p - c (-2 s + v)) Sqrt[z] - b (-2 k + m) z)/(b (-2 k + m))) - (E^((p - c (-2 s + v))^2/(4 b (-2 k + m))) Sqrt[Pi] (p - c (-2 s + v)) Erfi[(p - c (-2 s + v) - 2 b (-2 k + m) Sqrt[z])/(2 Sqrt[(-b) (-2 k + m)])])/ (2 ((-b) (-2 k + m))^(3/2))) + (-1)^m (-(E^((p + c (-2 s + v)) Sqrt[z] - b (-2 k + m) z)/ (b (-2 k + m))) - (E^((p + c (-2 s + v))^2/(4 b (-2 k + m))) Sqrt[Pi] (p + c (-2 s + v)) Erfi[(p + c (-2 s + v) - 2 b (-2 k + m) Sqrt[z])/(2 Sqrt[(-b) (-2 k + m)])])/ (2 ((-b) (-2 k + m))^(3/2))) + E^((p - c (-2 s + v)) Sqrt[z] + b (-2 k + m) z)/(b (-2 k + m)) - (Sqrt[Pi] (p - c (-2 s + v)) Erfi[(p - c (-2 s + v) + 2 b (-2 k + m) Sqrt[z])/(2 Sqrt[b (-2 k + m)])])/E^((p - c (-2 s + v))^2/ (4 b (-2 k + m)))/(2 (b (-2 k + m))^(3/2)) + E^((p + c (-2 s + v)) Sqrt[z] + b (-2 k + m) z)/(b (-2 k + m)) - (Sqrt[Pi] (p + c (-2 s + v)) Erfi[(p + c (-2 s + v) + 2 b (-2 k + m) Sqrt[z])/(2 Sqrt[b (-2 k + m)])])/E^((p + c (-2 s + v))^2/ (4 b (-2 k + m)))/(2 (b (-2 k + m))^(3/2))), {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

 p z sinh m ( b z ) cosh v ( c z ) z 2 - m - v + 1 - m p z ( p z - 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 ) p 2 + 2 - m - v ( 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 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - 1 ) m ( - p 2 4 b ( m - 2 k ) p π erfi ( p - 2 b ( m - 2 k ) z 2 - b ( m - 2 k ) ) 2 ( - b ( m - 2 k ) ) 3 / 2 - p z - b ( m - 2 k ) z b ( m - 2 k ) ) + z p + b ( m - 2 k ) z b ( m - 2 k ) - - p 2 4 b ( m - 2 k ) p π erfi ( 2 b z ( m - 2 k ) + p 2 b ( m - 2 k ) ) 2 ( b ( m - 2 k ) ) 3 / 2 ) + 2 - m - v + 2 - m p z ( 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 1 ( p 2 - c 2 ( v - 2 s ) 2 ) 2 ( ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( z p 3 - p 2 - c 2 ( v - 2 s ) 2 z p - c 2 ( v - 2 s ) 2 ) cosh ( c ( 2 s - v ) z ) - c ( 2 s - v ) ( z p 2 - 2 p - c 2 ( v - 2 s ) 2 z ) sinh ( c ( 2 s - v ) z ) ) ) + 2 - m - v 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 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - 1 ) m ( - ( p + c ( v - 2 s ) ) z - b ( m - 2 k ) z b ( m - 2 k ) - ( p + c ( v - 2 s ) ) 2 4 b ( m - 2 k ) π ( p + c ( v - 2 s ) ) erfi ( - 2 b z ( m - 2 k ) + p + c ( v - 2 s ) 2 - b ( m - 2 k ) ) 2 ( - b ( m - 2 k ) ) 3 / 2 ) + ( - 1 ) m ( - ( p - c ( v - 2 s ) ) z - b ( m - 2 k ) z b ( m - 2 k ) - ( p - c ( v - 2 s ) ) 2 4 b ( m - 2 k ) π ( p - c ( v - 2 s ) ) erfi ( - 2 b z ( m - 2 k ) + p - c ( v - 2 s ) 2 - b ( m - 2 k ) ) 2 ( - b ( m - 2 k ) ) 3 / 2 ) + z ( p + c ( v - 2 s ) ) + b ( m - 2 k ) z b ( m - 2 k ) + z ( p - c ( v - 2 s ) ) + b ( m - 2 k ) z b ( m - 2 k ) - - ( p + c ( v - 2 s ) ) 2 4 b ( m - 2 k ) π ( p + c ( v - 2 s ) ) erfi ( 2 b z ( m - 2 k ) + p + c ( v - 2 s ) 2 b ( m - 2 k ) ) 2 ( b ( m - 2 k ) ) 3 / 2 - - ( p - c ( v - 2 s ) ) 2 4 b ( m - 2 k ) π ( p - c ( v - 2 s ) ) erfi ( 2 b z ( m - 2 k ) + p - c ( v - 2 s ) 2 b ( m - 2 k ) ) 2 ( b ( m - 2 k ) ) 3 / 2 ) /; m + v + Condition z p z 1 2 b z m c z 1 2 v 2 -1 m -1 v 1 -1 m p z 1 2 p z 1 2 -1 Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 p 2 -1 2 -1 m -1 v Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 -1 k Binomial m k -1 m -1 p 2 4 b m -1 2 k -1 p 1 2 Erfi p -1 2 b m -1 2 k z 1 2 2 -1 b m -1 2 k 1 2 -1 2 -1 b m -1 2 k 3 2 -1 -1 p z 1 2 -1 b m -1 2 k z b m -1 2 k -1 z 1 2 p b m -1 2 k z b m -1 2 k -1 -1 -1 p 2 4 b m -1 2 k -1 p 1 2 Erfi 2 b z 1 2 m -1 2 k p 2 b m -1 2 k 1 2 -1 2 b m -1 2 k 3 2 -1 2 -1 m -1 v 2 -1 m p z 1 2 Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 v -1 2 -1 1 p 2 -1 c 2 v -1 2 s 2 2 -1 Binomial v s z 1 2 p 3 -1 p 2 -1 c 2 v -1 2 s 2 z 1 2 p -1 c 2 v -1 2 s 2 c 2 s -1 v z 1 2 -1 c 2 s -1 v z 1 2 p 2 -1 2 p -1 c 2 v -1 2 s 2 z 1 2 c 2 s -1 v z 1 2 2 -1 m -1 v k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s -1 m -1 p c v -1 2 s z 1 2 -1 b m -1 2 k z b m -1 2 k -1 -1 p c v -1 2 s 2 4 b m -1 2 k -1 1 2 p c v -1 2 s Erfi -2 b z 1 2 m -1 2 k p c v -1 2 s 2 -1 b m -1 2 k 1 2 -1 2 -1 b m -1 2 k 3 2 -1 -1 m -1 p -1 c v -1 2 s z 1 2 -1 b m -1 2 k z b m -1 2 k -1 -1 p -1 c v -1 2 s 2 4 b m -1 2 k -1 1 2 p -1 c v -1 2 s Erfi -2 b z 1 2 m -1 2 k p -1 c v -1 2 s 2 -1 b m -1 2 k 1 2 -1 2 -1 b m -1 2 k 3 2 -1 z 1 2 p c v -1 2 s b m -1 2 k z b m -1 2 k -1 z 1 2 p -1 c v -1 2 s b m -1 2 k z b m -1 2 k -1 -1 -1 p c v -1 2 s 2 4 b m -1 2 k -1 1 2 p c v -1 2 s Erfi 2 b z 1 2 m -1 2 k p c v -1 2 s 2 b m -1 2 k 1 2 -1 2 b m -1 2 k 3 2 -1 -1 -1 p -1 c v -1 2 s 2 4 b m -1 2 k -1 1 2 p -1 c v -1 2 s Erfi 2 b z 1 2 m -1 2 k p -1 c v -1 2 s 2 b m -1 2 k 1 2 -1 2 b m -1 2 k 3 2 -1 m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18