html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Cosh

 http://functions.wolfram.com/01.20.21.4172.01

 Input Form

 Integrate[Sinh[b z^r + e]^m Cosh[c z^r]^v, z] == 2^(-m - v) z ((Binomial[m, m/2] Binomial[v, v/2] (-1 + Mod[m, 2]) (-1 + Mod[v, 2]))/I^m + (1/r) (Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[(-1)^k E^(2 e k - e m) Binomial[m, k] ((E^(-4 e k + 2 e m) Gamma[1/r, b (2 k - m) z^r])/ (b (2 k - m) z^r)^r^(-1) + ((-1)^m Gamma[1/r, b (-2 k + m) z^r])/ ((-b) (2 k - m) z^r)^r^(-1)), {k, 0, Floor[(1/2) (-1 + m)]}]) + (1/r) ((Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[Binomial[v, s] (Gamma[1/r, c (2 s - v) z^r]/(c (2 s - v) z^r)^ r^(-1) + Gamma[1/r, c (-2 s + v) z^r]/(c (-2 s + v) z^r)^r^(-1)), {s, 0, Floor[(1/2) (-1 + v)]}])/I^m) - (1/r) Sum[(-1)^k Binomial[m, k] Sum[E^(-2 e k - e m) Binomial[v, s] ((E^(2 e m) Gamma[1/r, (2 b k - b m + 2 c s - c v) z^r])/ ((2 b k - b m + 2 c s - c v) z^r)^r^(-1) + ((-1)^m E^(4 e k) Gamma[1/r, (-2 b k + b m + 2 c s - c v) z^r])/ ((-2 b k + b m + 2 c s - c v) z^r)^r^(-1) + (E^(2 e m) Gamma[1/r, (2 b k - b m - 2 c s + c v) z^r])/ ((2 b k - b m - 2 c s + c v) z^r)^r^(-1) + ((-1)^m E^(4 e k) Gamma[1/r, (-2 b k + b m - 2 c s + c v) z^r])/ ((-2 b k + b m - 2 c s + c v) z^r)^r^(-1)), {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 ( b z r + e ) cosh v ( c z r ) z 2 - m - v z ( - m ( 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]] ( m mod 2 \$CellContext`m 2 - 1 ) ( v mod 2 \$CellContext`v 2 - 1 ) + 1 r ( ( 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]] ( v mod 2 \$CellContext`v 2 - 1 ) k = 0 m - 1 2 ( - 1 ) k 2 e k - e m ( 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]] ( ( - 1 ) m Γ ( 1 r , b ( m - 2 k ) z r ) ( - b ( 2 k - m ) z r ) - 1 / r + 2 e m - 4 e k ( b ( 2 k - m ) z r ) - 1 / r Γ ( 1 r , b ( 2 k - m ) z r ) ) ) + 1 r ( - m ( 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]] ( m mod 2 \$CellContext`m 2 - 1 ) 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]] ( Γ ( 1 r , c ( 2 s - v ) z r ) ( c ( 2 s - v ) z r ) - 1 / r + ( c ( v - 2 s ) z r ) - 1 / r Γ ( 1 r , c ( v - 2 s ) z r ) ) ) - 1 r ( 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 - 2 e k - e m ( 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]] ( ( - 1 ) m 4 e k Γ ( 1 r , ( - 2 b k + b m - 2 c s + c v ) z r ) ( ( - 2 b k + b m - 2 c s + c v ) z r ) - 1 / r + 2 e m ( ( 2 b k - b m - 2 c s + c v ) z r ) - 1 / r Γ ( 1 r , ( 2 b k - b m - 2 c s + c v ) z r ) + ( - 1 ) m 4 e k ( ( - 2 b k + b m + 2 c s - c v ) z r ) - 1 / r Γ ( 1 r , ( - 2 b k + b m + 2 c s - c v ) z r ) + 2 e m ( ( 2 b k - b m + 2 c s - c v ) z r ) - 1 / r Γ ( 1 r , ( 2 b k - b m + 2 c s - c v ) z r ) ) ) ) /; m + v + Condition z b z r e m c z r v 2 -1 m -1 v z -1 m Binomial m m 2 -1 Binomial v v 2 -1 \$CellContext`m 2 -1 \$CellContext`v 2 -1 1 r -1 Binomial v v 2 -1 \$CellContext`v 2 -1 k 0 m -1 2 -1 -1 k 2 e k -1 e m Binomial m k -1 m Gamma 1 r -1 b m -1 2 k z r -1 b 2 k -1 m z r -1 r -1 2 e m -1 4 e k b 2 k -1 m z r -1 r -1 Gamma 1 r -1 b 2 k -1 m z r 1 r -1 -1 m Binomial m m 2 -1 \$CellContext`m 2 -1 s 0 v -1 2 -1 Binomial v s Gamma 1 r -1 c 2 s -1 v z r c 2 s -1 v z r -1 r -1 c v -1 2 s z r -1 r -1 Gamma 1 r -1 c v -1 2 s z r -1 1 r -1 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 -2 e k -1 e m Binomial v s -1 m 4 e k Gamma 1 r -1 -2 b k b m -1 2 c s c v z r -2 b k b m -1 2 c s c v z r -1 r -1 2 e m 2 b k -1 b m -1 2 c s c v z r -1 r -1 Gamma 1 r -1 2 b k -1 b m -1 2 c s c v z r -1 m 4 e k -2 b k b m 2 c s -1 c v z r -1 r -1 Gamma 1 r -1 -2 b k b m 2 c s -1 c v z r 2 e m 2 b k -1 b m 2 c s -1 c v z r -1 r -1 Gamma 1 r -1 2 b k -1 b m 2 c s -1 c v z r m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18