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 Cosh

 http://functions.wolfram.com/01.20.21.3841.01

 Input Form

 Integrate[E^(p Sqrt[z]) Cos[b z]^m Cosh[c 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[v, v/2] (1 - Mod[v, 2]) Sum[Binomial[m, s] ((2 I E^(p Sqrt[z] - I b (m - 2 s) z))/(b (m - 2 s)) - (2 I E^(p Sqrt[z] + I b (m - 2 s) z))/(b (m - 2 s)) - (p Sqrt[Pi] Erfi[(p - 2 I b (m - 2 s) Sqrt[z])/ (2 Sqrt[(-I) b (m - 2 s)])])/(E^((I p^2)/(4 b (m - 2 s))) ((-I) b (m - 2 s))^(3/2)) - (E^((I p^2)/(4 b (m - 2 s))) p Sqrt[Pi] Erfi[(p + 2 I b (m - 2 s) Sqrt[z])/(2 Sqrt[I b (m - 2 s)])])/ (I b (m - 2 s))^(3/2)), {s, 0, Floor[(1/2) (-1 + m)]}] + 2^(-1 - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[Binomial[v, s] (-((2 E^(p Sqrt[z] - c (-2 s + v) z))/ (c (-2 s + v))) + (2 E^(p Sqrt[z] + c (-2 s + v) z))/ (c (-2 s + v)) - (E^(p^2/(4 c (-2 s + v))) p Sqrt[Pi] Erfi[(p - 2 c (-2 s + v) Sqrt[z])/(2 Sqrt[(-c) (-2 s + v)])])/ ((-c) (-2 s + v))^(3/2) - (p Sqrt[Pi] Erfi[(p + 2 c (-2 s + v) Sqrt[z])/(2 Sqrt[c (-2 s + v)])])/ (E^(p^2/(4 c (-2 s + v))) (c (-2 s + v))^(3/2))), {s, 0, Floor[(1/2) (-1 + v)]}] + 2^(-1 - m - v) Sum[Binomial[m, s] Sum[Binomial[v, k] (2 E^(p Sqrt[z]) ((2 Sinh[((-I) b (m - 2 s) + c (2 k - v)) z])/ ((-I) b (m - 2 s) + c (2 k - v)) + (2 Sinh[(I b (m - 2 s) + c (2 k - v)) z])/(I b (m - 2 s) + c (2 k - v))) - (p Sqrt[Pi] Erfi[(p + 2 (2 c k + I b m - 2 I b s - c v) Sqrt[z])/ (2 Sqrt[2 c k + I b m - 2 I b s - c v])])/ E^(p^2/(4 (2 c k + I b m - 2 I b s - c v)))/ (2 c k + I b m - 2 I b s - c v)^(3/2) - (p Sqrt[Pi] Erfi[(p + 2 (2 c k - I b m + 2 I b s - c v) Sqrt[z])/ (2 Sqrt[2 c k - I b m + 2 I b s - c v])])/ E^(p^2/(4 (2 c k - I b m + 2 I b s - c v)))/ (2 c k - I b m + 2 I b s - c v)^(3/2) - (p Sqrt[Pi] Erfi[(p + 2 (-2 c k + I b m - 2 I b s + c v) Sqrt[z])/ (2 Sqrt[-2 c k + I b m - 2 I b s + c v])])/ E^(p^2/(4 (-2 c k + I b m - 2 I b s + c v)))/ (-2 c k + I b m - 2 I b s + c v)^(3/2) - (p Sqrt[Pi] Erfi[(p + 2 (-2 c k - I b m + 2 I b s + c v) Sqrt[z])/ (2 Sqrt[-2 c k - I b m + 2 I b s + c v])])/ E^(p^2/(4 (-2 c k - I b m + 2 I b s + c v)))/ (-2 c k - I b m + 2 I b s + c v)^(3/2)), {k, 0, Floor[(1/2) (-1 + v)]}], {s, 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 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]]]]] ( s = 0 m - 1 2 ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 p z - b ( m - 2 s ) z b ( m - 2 s ) - 2 z p + b ( m - 2 s ) z b ( m - 2 s ) - - p 2 4 b ( m - 2 s ) p π erfi ( p - 2 b ( m - 2 s ) z 2 - b ( m - 2 s ) ) ( - b ( m - 2 s ) ) 3 / 2 - p 2 4 b ( m - 2 s ) p π erfi ( p + 2 b ( m - 2 s ) z 2 b ( m - 2 s ) ) ( b ( m - 2 s ) ) 3 / 2 ) ) ( 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 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - p 2 4 c ( v - 2 s ) p π erfi ( p - 2 c ( v - 2 s ) z 2 - c ( v - 2 s ) ) ( - c ( v - 2 s ) ) 3 / 2 + 2 z p + c ( v - 2 s ) z c ( v - 2 s ) - 2 p z - c ( v - 2 s ) z c ( v - 2 s ) - - p 2 4 c ( v - 2 s ) p π erfi ( p + 2 c ( v - 2 s ) z 2 c ( v - 2 s ) ) ( c ( v - 2 s ) ) 3 / 2 ) + 2 - m - v - 1 s = 0 m - 1 2 ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] k = 0 v - 1 2 ( v k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 p z ( 2 sinh ( ( c ( 2 k - v ) - b ( m - 2 s ) ) z ) c ( 2 k - v ) - b ( m - 2 s ) + 2 sinh ( ( b ( m - 2 s ) + c ( 2 k - v ) ) z ) b ( m - 2 s ) + c ( 2 k - v ) ) - - p 2 4 ( - 2 c k - b m + 2 b s + c v ) p π erfi ( p + 2 ( - 2 c k - b m + 2 b s + c v ) z 2 - 2 c k - b m + 2 b s + c v ) ( - 2 c k - b m + 2 b s + c v ) 3 / 2 - - p 2 4 ( - 2 c k + b m - 2 b s + c v ) p π erfi ( p + 2 ( - 2 c k + b m - 2 b s + c v ) z 2 - 2 c k + b m - 2 b s + c v ) ( - 2 c k + b m - 2 b s + c v ) 3 / 2 - - p 2 4 ( 2 c k - b m + 2 b s - c v ) p π erfi ( p + 2 ( 2 c k - b m + 2 b s - c v ) z 2 2 c k - b m + 2 b s - c v ) ( 2 c k - b m + 2 b s - c v ) 3 / 2 - - p 2 4 ( 2 c k + b m - 2 b s - c v ) p π erfi ( p + 2 ( 2 c k + b m - 2 b s - c v ) z 2 2 c k + b m - 2 b s - c v ) ( 2 c k + b m - 2 b s - c v ) 3 / 2 ) /; m TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] m > 0 v TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] v > 0 Condition z p z 1 2 b z m c z v 2 -1 m -1 v -1 Binomial v v 2 -1 s 0 m -1 2 -1 Binomial m s 2 p z 1 2 -1 b m -1 2 s z b m -1 2 s -1 -1 2 z 1 2 p b m -1 2 s z b m -1 2 s -1 -1 -1 p 2 4 b m -1 2 s -1 p 1 2 Erfi p -1 2 b m -1 2 s z 1 2 2 -1 b m -1 2 s 1 2 -1 -1 b m -1 2 s 3 2 -1 -1 p 2 4 b m -1 2 s -1 p 1 2 Erfi p 2 b m -1 2 s z 1 2 2 b m -1 2 s 1 2 -1 b m -1 2 s 3 2 -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 Binomial v s -1 p 2 4 c v -1 2 s -1 p 1 2 Erfi p -1 2 c v -1 2 s z 1 2 2 -1 c v -1 2 s 1 2 -1 -1 c v -1 2 s 3 2 -1 2 z 1 2 p c v -1 2 s z c v -1 2 s -1 -1 2 p z 1 2 -1 c v -1 2 s z c v -1 2 s -1 -1 -1 p 2 4 c v -1 2 s -1 p 1 2 Erfi p 2 c v -1 2 s z 1 2 2 c v -1 2 s 1 2 -1 c v -1 2 s 3 2 -1 2 -1 m -1 v -1 s 0 m -1 2 -1 Binomial m s k 0 v -1 2 -1 Binomial v k 2 p z 1 2 2 c 2 k -1 v -1 b m -1 2 s z c 2 k -1 v -1 b m -1 2 s -1 2 b m -1 2 s c 2 k -1 v z b m -1 2 s c 2 k -1 v -1 -1 -1 p 2 4 -2 c k -1 b m 2 b s c v -1 p 1 2 Erfi p 2 -2 c k -1 b m 2 b s c v z 1 2 2 -2 c k -1 b m 2 b s c v 1 2 -1 -2 c k -1 b m 2 b s c v 3 2 -1 -1 -1 p 2 4 -2 c k b m -1 2 b s c v -1 p 1 2 Erfi p 2 -2 c k b m -1 2 b s c v z 1 2 2 -2 c k b m -1 2 b s c v 1 2 -1 -2 c k b m -1 2 b s c v 3 2 -1 -1 -1 p 2 4 2 c k -1 b m 2 b s -1 c v -1 p 1 2 Erfi p 2 2 c k -1 b m 2 b s -1 c v z 1 2 2 2 c k -1 b m 2 b s -1 c v 1 2 -1 2 c k -1 b m 2 b s -1 c v 3 2 -1 -1 -1 p 2 4 2 c k b m -1 2 b s -1 c v -1 p 1 2 Erfi p 2 2 c k b m -1 2 b s -1 c v z 1 2 2 2 c k b m -1 2 b s -1 c v 1 2 -1 2 c k b m -1 2 b s -1 c v 3 2 -1 m m 0 v v 0 [/itex]

 Rule Form

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

 2002-12-18