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 Cosh

 http://functions.wolfram.com/01.20.21.4152.01

 Input Form

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

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

 Rule Form

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

 2002-12-18