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 Cosh

 http://functions.wolfram.com/01.20.21.3847.01

 Input Form

 Integrate[E^(p Sqrt[z]) Cos[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])) + 2^(1 - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[E^((p + c (-2 s + v)) Sqrt[z]) (-(1/(-p - c (-2 s + v))^2) + (-1 + p Sqrt[z] - c (-2 s + v) Sqrt[z])/(E^(2 c (-2 s + v) Sqrt[z]) (p - c (-2 s + v))^2) + Sqrt[z]/(p + c (-2 s + v))) Binomial[v, s], {s, 0, Floor[(1/2) (-1 + v)]}] - 2^(-1 - m - v) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[Binomial[m, k] ((4 E^(p Sqrt[z]) Sin[b (-2 k + m) z])/ (b (2 k - m)) + (p Sqrt[Pi] Erfi[(p - 2 I b (-2 k + m) Sqrt[z])/ (2 Sqrt[(-I) b (-2 k + m)])])/(E^((I p^2)/(4 b (-2 k + m))) ((-I) b (-2 k + m))^(3/2)) + (E^((I p^2)/(4 b (-2 k + m))) p Sqrt[Pi] Erfi[(p + 2 I b (-2 k + m) Sqrt[z])/ (2 Sqrt[I b (-2 k + m)])])/(I b (-2 k + m))^(3/2)), {k, 0, Floor[(1/2) (-1 + m)]}] + 2^(-1 - m - v) Sum[Binomial[v, s] Sum[Binomial[m, k] ((2 I E^((p + c (-2 s + v)) Sqrt[z] - I b (2 k - m) z))/ (b (2 k - m)) - (2 I E^((p - c (-2 s + v)) Sqrt[z] + I b (2 k - m) z))/(b (2 k - m)) + (2 I E^((p + c (-2 s + v)) Sqrt[z] - I b (-2 k + m) z))/ (b (-2 k + m)) - (2 I E^((p - c (-2 s + v)) Sqrt[z] + I b (-2 k + m) z))/(b (-2 k + m)) - (p Sqrt[Pi] Erfi[(p + c (-2 s + v) - 2 I b (2 k - m) Sqrt[z])/ (2 Sqrt[(-I) b (2 k - m)])])/(E^((I (p + c (-2 s + v))^2)/ (4 b (2 k - m))) ((-I) b (2 k - m))^(3/2)) + (c Sqrt[Pi] (-2 s + v) Erfi[(-p - c (-2 s + v) + 2 I b (2 k - m) Sqrt[z])/(2 Sqrt[(-I) b (2 k - m)])])/ E^((I (p + c (-2 s + v))^2)/(4 b (2 k - m)))/((-I) b (2 k - m))^ (3/2) - (E^((I (p - c (-2 s + v))^2)/(4 b (2 k - m))) p Sqrt[Pi] Erfi[(p - c (-2 s + v) + 2 I b (2 k - m) Sqrt[z])/ (2 Sqrt[I b (2 k - m)])])/(I b (2 k - m))^(3/2) + (c E^((I (p - c (-2 s + v))^2)/(4 b (2 k - m))) Sqrt[Pi] (-2 s + v) Erfi[(p - c (-2 s + v) + 2 I b (2 k - m) Sqrt[z])/ (2 Sqrt[I b (2 k - m)])])/(I b (2 k - m))^(3/2) - (p Sqrt[Pi] Erfi[(p + c (-2 s + v) - 2 I b (-2 k + m) Sqrt[z])/ (2 Sqrt[(-I) b (-2 k + m)])])/(E^((I (p + c (-2 s + v))^2)/ (4 b (-2 k + m))) ((-I) b (-2 k + m))^(3/2)) + (c Sqrt[Pi] (-2 s + v) Erfi[(-p - c (-2 s + v) + 2 I b (-2 k + m) Sqrt[z])/(2 Sqrt[(-I) b (-2 k + m)])])/ E^((I (p + c (-2 s + v))^2)/(4 b (-2 k + m)))/((-I) b (-2 k + m))^ (3/2) - (E^((I (p - c (-2 s + v))^2)/(4 b (-2 k + m))) p Sqrt[Pi] Erfi[(p - c (-2 s + v) + 2 I b (-2 k + m) Sqrt[z])/ (2 Sqrt[I b (-2 k + m)])])/(I b (-2 k + m))^(3/2) + (c E^((I (p - c (-2 s + v))^2)/(4 b (-2 k + m))) Sqrt[Pi] (-2 s + v) Erfi[(p - c (-2 s + v) + 2 I b (-2 k + m) Sqrt[z])/ (2 Sqrt[I b (-2 k + m)])])/(I b (-2 k + m))^(3/2)), {k, 0, Floor[(1/2) (-1 + m)]}], {s, 0, Floor[(1/2) (-1 + v)]}] /; Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

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