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 Cosh

 http://functions.wolfram.com/01.20.21.3777.01

 Input Form

 Integrate[E^(p z) Sin[b z]^m Cosh[c Sqrt[z]]^v, z] == (1/p) (2^(-m - v) E^(p z) Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2])) + 2^(-m - v) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(-1)^k E^((I m Pi)/2) (E^(((-I) b (-2 k + m) + p) z)/ ((-I) b (-2 k + m) + p) + E^((-I) m Pi + (I b (-2 k + m) + p) z)/ (I b (-2 k + m) + p)) Binomial[m, k], {k, 0, Floor[(1/2) (-1 + m)]}] + 2^(-1 - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[Binomial[v, s] ((4 E^(p z) Cosh[c (-2 s + v) Sqrt[z]])/p + (1/p^(3/2)) ((c Sqrt[Pi] (2 s - v) (Erfi[((-c) (2 s - v) + 2 p Sqrt[z])/(2 Sqrt[p])] - Erfi[((-c) (-2 s + v) + 2 p Sqrt[z])/(2 Sqrt[p])]))/ E^((c^2 (-2 s + v)^2)/(4 p)))), {s, 0, Floor[(1/2) (-1 + v)]}] + 2^(-1 - m - v) Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] (((2 E^(I m Pi + (2 c s - c v) Sqrt[z] + (2 I b k - I b m + p) z))/ (2 I b k - I b m + p) + (2 E^((-2 c s + c v) Sqrt[z] + (-2 I b k + I b m + p) z))/(-2 I b k + I b m + p) + (c E^(I m Pi - (2 c s - c v)^2/(4 (2 I b k - I b m + p))) Sqrt[Pi] (-2 s + v) Erfi[((-c) (-2 s + v) + 2 (2 I b k - I b m + p) Sqrt[z])/(2 Sqrt[2 I b k - I b m + p])])/ (2 I b k - I b m + p)^(3/2) + (c Sqrt[Pi] (2 s - v) Erfi[((-c) (2 s - v) + 2 (-2 I b k + I b m + p) Sqrt[z])/ (2 Sqrt[-2 I b k + I b m + p])])/E^((-2 c s + c v)^2/(4 (-2 I b k + I b m + p)))/(-2 I b k + I b m + p)^(3/2))/ E^((1/2) I m Pi) + E^((I m Pi)/2) ((2 E^((-2 c s + c v) Sqrt[z] + (2 I b k - I b m + p) z))/ (2 I b k - I b m + p) + (2 E^((-I) m Pi + (2 c s - c v) Sqrt[z] + (-2 I b k + I b m + p) z))/(-2 I b k + I b m + p) + (c Sqrt[Pi] (2 s - v) Erfi[((-c) (2 s - v) + 2 (2 I b k - I b m + p) Sqrt[z])/(2 Sqrt[2 I b k - I b m + p])])/ E^((-2 c s + c v)^2/(4 (2 I b k - I b m + p)))/ (2 I b k - I b m + p)^(3/2) + (c E^((-I) m Pi - (c^2 (-2 s + v)^2)/(4 (-2 I b k + I b m + p))) Sqrt[Pi] (-2 s + v) Erfi[((-c) (-2 s + v) + 2 (-2 I b k + I b m + p) Sqrt[z])/(2 Sqrt[-2 I b k + I b m + p])])/ (-2 I b k + I b m + p)^(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 sin m ( b z ) cosh v ( c z ) z 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]]]]] ( k = 0 m - 1 2 ( - 1 ) k m π 2 ( ( b ( m - 2 k ) + p ) z - m π b ( m - 2 k ) + p + ( p - b ( m - 2 k ) ) z p - b ( m - 2 k ) ) ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ) ( 1 - v mod 2 \$CellContext`v 2 ) + 2 - m - v 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]]]]] ( 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 - 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]]]]] ( 4 p z cosh ( c ( v - 2 s ) z ) p + c - c 2 ( v - 2 s ) 2 4 p π ( 2 s - v ) ( erfi ( 2 p z - c ( 2 s - v ) 2 p ) - erfi ( 2 p z - c ( v - 2 s ) 2 p ) ) p 3 / 2 ) + 2 - m - v - 1 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 2 m π ( c - ( c v - 2 c s ) 2 4 ( - 2 b k + b m + p ) π ( 2 s - v ) erfi ( 2 ( - 2 b k + b m + p ) z - c ( 2 s - v ) 2 - 2 b k + b m + p ) ( - 2 b k + b m + p ) 3 / 2 + ( c m π - ( 2 c s - c v ) 2 4 ( 2 b k - b m + p ) π ( v - 2 s ) erfi ( 2 ( 2 b k - b m + p ) z - c ( v - 2 s ) 2 2 b k - b m + p ) ) / ( 2 b k - b m + p ) 3 / 2 + 2 z ( c v - 2 c s ) + ( - 2 b k + b m + p ) z - 2 b k + b m + p + 2 π m + ( 2 b k - b m + p ) z + ( 2 c s - c v ) z 2 b k - b m + p ) + m π 2 ( ( c - c 2 ( v - 2 s ) 2 4 ( - 2 b k + b m + p ) - m π π ( v - 2 s ) erfi ( 2 ( - 2 b k + b m + p ) z - c ( v - 2 s ) 2 - 2 b k + b m + p ) ) / ( - 2 b k + b m + p ) 3 / 2 + c - ( c v - 2 c s ) 2 4 ( 2 b k - b m + p ) π ( 2 s - v ) erfi ( 2 ( 2 b k - b m + p ) z - c ( 2 s - v ) 2 2 b k - b m + p ) ( 2 b k - b m + p ) 3 / 2 + 2 - π m + ( - 2 b k + b m + p ) z + ( 2 c s - c v ) z - 2 b k + b m + p + 2 z ( c v - 2 c s ) + ( 2 b k - b m + p ) z 2 b k - b m + p ) ) /; m + v + Condition z p z b z m c z 1 2 v 2 -1 m -1 v Binomial v v 2 -1 k 0 m -1 2 -1 -1 k m 2 -1 b m -1 2 k p z -1 m b m -1 2 k p -1 p -1 b m -1 2 k z p -1 b m -1 2 k -1 Binomial m k 1 -1 \$CellContext`v 2 2 -1 m -1 v p z Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 p -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 4 p z c v -1 2 s z 1 2 p -1 c -1 c 2 v -1 2 s 2 4 p -1 1 2 2 s -1 v Erfi 2 p z 1 2 -1 c 2 s -1 v 2 p 1 2 -1 -1 Erfi 2 p z 1 2 -1 c v -1 2 s 2 p 1 2 -1 p 3 2 -1 2 -1 m -1 v -1 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s -1 1 2 m c -1 c v -1 2 c s 2 4 -2 b k b m p -1 1 2 2 s -1 v Erfi 2 -2 b k b m p z 1 2 -1 c 2 s -1 v 2 -2 b k b m p 1 2 -1 -2 b k b m p 3 2 -1 c m -1 2 c s -1 c v 2 4 2 b k -1 b m p -1 1 2 v -1 2 s Erfi 2 2 b k -1 b m p z 1 2 -1 c v -1 2 s 2 2 b k -1 b m p 1 2 -1 2 b k -1 b m p 3 2 -1 2 z 1 2 c v -1 2 c s -2 b k b m p z -2 b k b m p -1 2 m 2 b k -1 b m p z 2 c s -1 c v z 1 2 2 b k -1 b m p -1 m 2 -1 c -1 c 2 v -1 2 s 2 4 -2 b k b m p -1 -1 m 1 2 v -1 2 s Erfi 2 -2 b k b m p z 1 2 -1 c v -1 2 s 2 -2 b k b m p 1 2 -1 -2 b k b m p 3 2 -1 c -1 c v -1 2 c s 2 4 2 b k -1 b m p -1 1 2 2 s -1 v Erfi 2 2 b k -1 b m p z 1 2 -1 c 2 s -1 v 2 2 b k -1 b m p 1 2 -1 2 b k -1 b m p 3 2 -1 2 -1 m -2 b k b m p z 2 c s -1 c v z 1 2 -2 b k b m p -1 2 z 1 2 c v -1 2 c s 2 b k -1 b m p z 2 b k -1 b m p -1 m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18