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 Cosh

 http://functions.wolfram.com/01.20.21.4212.01

 Input Form

 Integrate[Sinh[b Sqrt[z] + e]^m Cosh[c Sqrt[z] + f z + g]^v, z] == I^m 2^(-m - v) z Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2]) + (2^(-1 - m - v)/b^2) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[((-1)^k Binomial[m, k] (1/(-2 k + m)^2) (E^(-4 e k - 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]))))/E^(e (-2 k + m)), {k, 0, Floor[(1/2) (-1 + m)]}] - I^m 2^(-1 - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(Binomial[v, s] ((1/(f (2 s - v))) (2 E^((2 s - v) (c + f Sqrt[z]) Sqrt[z]) (-1 + E^(2 (-2 s + v) (g + c Sqrt[z] + f z)))) + (c E^((c^2 (-2 s + v))/(4 f)) Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])/ (2 Sqrt[(-f) (-2 s + v)])])/((-f) (-2 s + v))^(3/2) + (c E^(-((c^2 (-2 s + v))/(4 f)) + 2 g (-2 s + v)) Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])/ (2 Sqrt[f (-2 s + v)])])/(f (-2 s + v))^(3/2)))/E^(g (-2 s + v)), {s, 0, Floor[(1/2) (-1 + v)]}] + 2^(-1 - m - v) Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] (E^((-e) (2 k - m) - g (-2 s + v)) (-((2 E^(((-b) (2 k - m) - c (-2 s + v)) Sqrt[z] - f (-2 s + v) z))/ (f (-2 s + v))) + (1/(f (-2 s + v))) (2 (-1)^m E^(2 (e (2 k - m) + g (-2 s + v)) + (b (2 k - m) + c (-2 s + v)) Sqrt[z] + f (-2 s + v) z)) - (E^(((-b) (2 k - m) - c (-2 s + v))^2/(4 f (-2 s + v))) Sqrt[Pi] (b (2 k - m) + c (-2 s + v)) Erfi[(b (2 k - m) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])/(2 Sqrt[(-f) (-2 s + v)])])/ ((-f) (-2 s + v))^(3/2) - ((-1)^m E^(-((b (2 k - m) + c (-2 s + v))^2/(4 f (-2 s + v))) + 2 (e (2 k - m) + g (-2 s + v))) Sqrt[Pi] (b (2 k - m) + c (-2 s + v)) Erfi[(b (2 k - m) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])/(2 Sqrt[f (-2 s + v)])])/(f (-2 s + v))^(3/2)) + (-1)^m E^((-e) (-2 k + m) - g (-2 s + v)) (-((2 E^(((-b) (-2 k + m) - c (-2 s + v)) Sqrt[z] - f (-2 s + v) z))/(f (-2 s + v))) + (1/(f (-2 s + v))) (2 (-1)^m E^(2 (e (-2 k + m) + g (-2 s + v)) + (b (-2 k + m) + c (-2 s + v)) Sqrt[z] + f (-2 s + v) z)) - (E^(((-b) (-2 k + m) - c (-2 s + v))^2/(4 f (-2 s + v))) Sqrt[Pi] (b (-2 k + m) + c (-2 s + v)) Erfi[(b (-2 k + m) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])/(2 Sqrt[ (-f) (-2 s + v)])])/((-f) (-2 s + v))^(3/2) - ((-1)^m E^(-((b (-2 k + m) + c (-2 s + v))^2/(4 f (-2 s + v))) + 2 (e (-2 k + m) + g (-2 s + v))) Sqrt[Pi] (b (-2 k + m) + c (-2 s + v)) Erfi[(b (-2 k + m) + c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])/(2 Sqrt[f (-2 s + v)])])/ (f (-2 s + v))^(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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SuperscriptBox[RowBox[List["(", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], ")"]]]]]], ")"]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

 sinh m ( z b + e ) cosh v ( z c + g + f 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 ( m - 2 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]] ( - 4 e k - b ( 2 k + m ) z ( 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 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 - g ( v - 2 s ) ( 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]] ( 2 ( 2 s - v ) ( c + f z ) z ( - 1 + 2 ( v - 2 s ) ( z c + g + f z ) ) f ( 2 s - v ) + c c 2 ( v - 2 s ) 4 f π ( v - 2 s ) erfi ( c ( v - 2 s ) + 2 f z ( v - 2 s ) 2 - f ( v - 2 s ) ) ( - f ( v - 2 s ) ) 3 / 2 + c 2 g ( v - 2 s ) - c 2 ( v - 2 s ) 4 f π ( v - 2 s ) erfi ( c ( v - 2 s ) + 2 f z ( v - 2 s ) 2 f ( v - 2 s ) ) ( f ( v - 2 s ) ) 3 / 2 ) + 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 ) - g ( v - 2 s ) ( 2 ( - 1 ) m z ( b ( 2 k - m ) + c ( v - 2 s ) ) + 2 ( e ( 2 k - m ) + g ( v - 2 s ) ) + f ( v - 2 s ) z f ( v - 2 s ) - 2 ( - b ( 2 k - m ) - c ( v - 2 s ) ) z - f ( v - 2 s ) z f ( v - 2 s ) - 1 ( - f ( v - 2 s ) ) 3 / 2 ( ( - b ( 2 k - m ) - c ( v - 2 s ) ) 2 4 f ( v - 2 s ) π ( b ( 2 k - m ) + c ( v - 2 s ) ) erfi ( b ( 2 k - m ) + c ( v - 2 s ) + 2 f ( v - 2 s ) z 2 - f ( v - 2 s ) ) ) - 1 ( f ( v - 2 s ) ) 3 / 2 ( ( - 1 ) m 2 ( e ( 2 k - m ) + g ( v - 2 s ) ) - ( b ( 2 k - m ) + c ( v - 2 s ) ) 2 4 f ( v - 2 s ) π ( b ( 2 k - m ) + c ( v - 2 s ) ) erfi ( b ( 2 k - m ) + c ( v - 2 s ) + 2 f ( v - 2 s ) z 2 f ( v - 2 s ) ) ) ) + ( - 1 ) m - e ( m - 2 k ) - g ( v - 2 s ) ( 2 ( - 1 ) m z ( b ( m - 2 k ) + c ( v - 2 s ) ) + 2 ( e ( m - 2 k ) + g ( v - 2 s ) ) + f ( v - 2 s ) z f ( v - 2 s ) - 2 ( - b ( m - 2 k ) - c ( v - 2 s ) ) z - f ( v - 2 s ) z f ( v - 2 s ) - 1 ( - f ( v - 2 s ) ) 3 / 2 ( ( - b ( m - 2 k ) - c ( v - 2 s ) ) 2 4 f ( v - 2 s ) π ( b ( m - 2 k ) + c ( v - 2 s ) ) erfi ( b ( m - 2 k ) + c ( v - 2 s ) + 2 f ( v - 2 s ) z 2 - f ( v - 2 s ) ) ) - 1 ( f ( v - 2 s ) ) 3 / 2 ( ( - 1 ) m 2 ( e ( m - 2 k ) + g ( v - 2 s ) ) - ( b ( m - 2 k ) + c ( v - 2 s ) ) 2 4 f ( v - 2 s ) π ( b ( m - 2 k ) + c ( v - 2 s ) ) erfi ( b ( m - 2 k ) + c ( v - 2 s ) + 2 f ( v - 2 s ) z 2 f ( v - 2 s ) ) ) ) ) /; m + v + Condition z z 1 2 b e m z 1 2 c g f 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 m -1 2 k Binomial m k -4 e k -1 b 2 k m z 1 2 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 -1 m 2 -1 m -1 v -1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 v -1 2 -1 -1 g v -1 2 s Binomial v s 2 2 s -1 v c f z 1 2 z 1 2 -1 2 v -1 2 s z 1 2 c g f z f 2 s -1 v -1 c c 2 v -1 2 s 4 f -1 1 2 v -1 2 s Erfi c v -1 2 s 2 f z 1 2 v -1 2 s 2 -1 f v -1 2 s 1 2 -1 -1 f v -1 2 s 3 2 -1 c 2 g v -1 2 s -1 c 2 v -1 2 s 4 f -1 1 2 v -1 2 s Erfi c v -1 2 s 2 f z 1 2 v -1 2 s 2 f v -1 2 s 1 2 -1 f v -1 2 s 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 e 2 k -1 m -1 g v -1 2 s 2 -1 m z 1 2 b 2 k -1 m c v -1 2 s 2 e 2 k -1 m g v -1 2 s f v -1 2 s z f v -1 2 s -1 -1 2 -1 b 2 k -1 m -1 c v -1 2 s z 1 2 -1 f v -1 2 s z f v -1 2 s -1 -1 1 -1 f v -1 2 s 3 2 -1 -1 b 2 k -1 m -1 c v -1 2 s 2 4 f v -1 2 s -1 1 2 b 2 k -1 m c v -1 2 s Erfi b 2 k -1 m c v -1 2 s 2 f v -1 2 s z 1 2 2 -1 f v -1 2 s 1 2 -1 -1 1 f v -1 2 s 3 2 -1 -1 m 2 e 2 k -1 m g v -1 2 s -1 b 2 k -1 m c v -1 2 s 2 4 f v -1 2 s -1 1 2 b 2 k -1 m c v -1 2 s Erfi b 2 k -1 m c v -1 2 s 2 f v -1 2 s z 1 2 2 f v -1 2 s 1 2 -1 -1 m -1 e m -1 2 k -1 g v -1 2 s 2 -1 m z 1 2 b m -1 2 k c v -1 2 s 2 e m -1 2 k g v -1 2 s f v -1 2 s z f v -1 2 s -1 -1 2 -1 b m -1 2 k -1 c v -1 2 s z 1 2 -1 f v -1 2 s z f v -1 2 s -1 -1 1 -1 f v -1 2 s 3 2 -1 -1 b m -1 2 k -1 c v -1 2 s 2 4 f v -1 2 s -1 1 2 b m -1 2 k c v -1 2 s Erfi b m -1 2 k c v -1 2 s 2 f v -1 2 s z 1 2 2 -1 f v -1 2 s 1 2 -1 -1 1 f v -1 2 s 3 2 -1 -1 m 2 e m -1 2 k g v -1 2 s -1 b m -1 2 k c v -1 2 s 2 4 f v -1 2 s -1 1 2 b m -1 2 k c v -1 2 s Erfi b m -1 2 k c v -1 2 s 2 f v -1 2 s z 1 2 2 f v -1 2 s 1 2 -1 m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18