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 Cosh

 http://functions.wolfram.com/01.20.21.3775.01

 Input Form

 Integrate[E^(p z) Sin[b Sqrt[z]]^m Cosh[c 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[m, m/2] (1 - Mod[m, 2]) Sum[(E^((p - c (-2 s + v)) z)/(p - c (-2 s + v)) + E^((p + c (-2 s + v)) z)/(p + c (-2 s + v))) Binomial[v, s], {s, 0, Floor[(1/2) (-1 + v)]}] + 2^(-1 - m - v) E^((I m Pi)/2) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(-1)^k Binomial[m, k] ((2 E^((-I) b (-2 k + m) Sqrt[z] + p z))/p + (2 E^((-I) m Pi + I b (-2 k + m) Sqrt[z] + p z))/p - (I b E^((b^2 (-2 k + m)^2)/(4 p)) (-2 k + m) Sqrt[Pi] Erfi[(I b (-2 k + m) - 2 p Sqrt[z])/(2 Sqrt[p])])/p^(3/2) - (I b E^((b^2 (-2 k + m)^2)/(4 p) - I m Pi) (-2 k + m) Sqrt[Pi] Erfi[(I b (-2 k + m) + 2 p Sqrt[z])/(2 Sqrt[p])])/p^(3/2)), {k, 0, Floor[(1/2) (-1 + m)]}] + (2^(-1 - m - v) Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] ((2 E^((-I) b (2 k - m) Sqrt[z] + (p - c (-2 s + v)) z))/(p - c (-2 s + v)) + (2 E^(I m Pi + I b (2 k - m) Sqrt[z] + (p + c (-2 s + v)) z))/ (p + c (-2 s + v)) - (I b E^((b^2 (2 k - m)^2)/ (4 (p - c (-2 s + v)))) (2 k - m) Sqrt[Pi] Erfi[(I b (2 k - m) + 2 (-p + c (-2 s + v)) Sqrt[z])/ (2 Sqrt[p - c (-2 s + v)])])/(p - c (-2 s + v))^(3/2) - (I b E^(I m Pi + (b^2 (2 k - m)^2)/(4 (p + c (-2 s + v)))) (2 k - m) Sqrt[Pi] Erfi[(I b (2 k - m) + 2 (p + c (-2 s + v)) Sqrt[z])/(2 Sqrt[p + c (-2 s + v)])])/(p + c (-2 s + v))^ (3/2) + (-1)^m ((2 E^((-I) b (-2 k + m) Sqrt[z] + (p - c (-2 s + v)) z))/(p - c (-2 s + v)) + (2 E^((-I) m Pi + I b (-2 k + m) Sqrt[z] + (p + c (-2 s + v)) z))/ (p + c (-2 s + v)) - (I b E^((b^2 (-2 k + m)^2)/ (4 (p - c (-2 s + v)))) (-2 k + m) Sqrt[Pi] Erfi[(I b (-2 k + m) + 2 (-p + c (-2 s + v)) Sqrt[z])/ (2 Sqrt[p - c (-2 s + v)])])/(p - c (-2 s + v))^(3/2) - (I b E^((-I) m Pi + (b^2 (-2 k + m)^2)/(4 (p + c (-2 s + v)))) (-2 k + m) Sqrt[Pi] Erfi[(I b (-2 k + m) + 2 (p + c (-2 s + v)) Sqrt[z])/(2 Sqrt[p + c (-2 s + v)])])/(p + c (-2 s + v))^ (3/2))), {s, 0, Floor[(1/2) (-1 + v)]}], {k, 0, Floor[(1/2) (-1 + m)]}])/I^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 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 ( 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 p - c ( v - 2 s ) + ( p + c ( v - 2 s ) ) z p + c ( v - 2 s ) ) ( 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 m π 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 - 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]]]]] ( - b b 2 ( m - 2 k ) 2 4 p - m π ( m - 2 k ) π erfi ( b ( m - 2 k ) + 2 p z 2 p ) p 3 / 2 + 2 p z - b ( m - 2 k ) z p + 2 - π m + p z + b ( m - 2 k ) z p - b b 2 ( m - 2 k ) 2 4 p ( m - 2 k ) π erfi ( b ( m - 2 k ) - 2 p z 2 p ) p 3 / 2 ) + 2 - m - v - 1 - m 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 ( 2 - π m + ( p + c ( v - 2 s ) ) z + b ( m - 2 k ) z p + c ( v - 2 s ) - b b 2 ( m - 2 k ) 2 4 ( p + c ( v - 2 s ) ) - m π ( m - 2 k ) π erfi ( b ( m - 2 k ) + 2 ( p + c ( v - 2 s ) ) z 2 p + c ( v - 2 s ) ) ( p + c ( v - 2 s ) ) 3 / 2 + 2 ( p - c ( v - 2 s ) ) z - b ( m - 2 k ) z p - c ( v - 2 s ) - b b 2 ( m - 2 k ) 2 4 ( p - c ( v - 2 s ) ) ( m - 2 k ) π erfi ( b ( m - 2 k ) + 2 ( c ( v - 2 s ) - p ) z 2 p - c ( v - 2 s ) ) ( p - c ( v - 2 s ) ) 3 / 2 ) + 2 b z ( 2 k - m ) + ( p + c ( v - 2 s ) ) z + m π p + c ( v - 2 s ) - b b 2 ( 2 k - m ) 2 4 ( p + c ( v - 2 s ) ) + m π ( 2 k - m ) π erfi ( b ( 2 k - m ) + 2 ( p + c ( v - 2 s ) ) z 2 p + c ( v - 2 s ) ) ( p + c ( v - 2 s ) ) 3 / 2 + 2 ( p - c ( v - 2 s ) ) z - b ( 2 k - m ) z p - c ( v - 2 s ) - b b 2 ( 2 k - m ) 2 4 ( p - c ( v - 2 s ) ) ( 2 k - m ) π erfi ( b ( 2 k - m ) + 2 ( c ( v - 2 s ) - p ) z 2 p - c ( v - 2 s ) ) ( p - c ( v - 2 s ) ) 3 / 2 ) /; m TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] m > 0 v TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] v > 0 Condition z p z b z 1 2 m c z v 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 Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 v -1 2 -1 p -1 c v -1 2 s z p -1 c v -1 2 s -1 p c v -1 2 s z p c v -1 2 s -1 Binomial v s 2 -1 m -1 v -1 m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 -1 k Binomial m k -1 b b 2 m -1 2 k 2 4 p -1 -1 m m -1 2 k 1 2 Erfi b m -1 2 k 2 p z 1 2 2 p 1 2 -1 p 3 2 -1 2 p z -1 b m -1 2 k z 1 2 p -1 2 -1 m p z b m -1 2 k z 1 2 p -1 -1 b b 2 m -1 2 k 2 4 p -1 m -1 2 k 1 2 Erfi b m -1 2 k -1 2 p z 1 2 2 p 1 2 -1 p 3 2 -1 2 -1 m -1 v -1 -1 m k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s -1 m 2 -1 m p c v -1 2 s z b m -1 2 k z 1 2 p c v -1 2 s -1 -1 b b 2 m -1 2 k 2 4 p c v -1 2 s -1 -1 m m -1 2 k 1 2 Erfi b m -1 2 k 2 p c v -1 2 s z 1 2 2 p c v -1 2 s 1 2 -1 p c v -1 2 s 3 2 -1 2 p -1 c v -1 2 s z -1 b m -1 2 k z 1 2 p -1 c v -1 2 s -1 -1 b b 2 m -1 2 k 2 4 p -1 c v -1 2 s -1 m -1 2 k 1 2 Erfi b m -1 2 k 2 c v -1 2 s -1 p z 1 2 2 p -1 c v -1 2 s 1 2 -1 p -1 c v -1 2 s 3 2 -1 2 b z 1 2 2 k -1 m p c v -1 2 s z m p c v -1 2 s -1 -1 b b 2 2 k -1 m 2 4 p c v -1 2 s -1 m 2 k -1 m 1 2 Erfi b 2 k -1 m 2 p c v -1 2 s z 1 2 2 p c v -1 2 s 1 2 -1 p c v -1 2 s 3 2 -1 2 p -1 c v -1 2 s z -1 b 2 k -1 m z 1 2 p -1 c v -1 2 s -1 -1 b b 2 2 k -1 m 2 4 p -1 c v -1 2 s -1 2 k -1 m 1 2 Erfi b 2 k -1 m 2 c v -1 2 s -1 p z 1 2 2 p -1 c v -1 2 s 1 2 -1 p -1 c v -1 2 s 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