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/01.20.21.4540.01

 Input Form

 Integrate[z^n Sinh[b Sqrt[z]]^m Cosh[f z + g]^v, z] == 2^(-1 - m - v) ((1/(1 + n)) ((2 z^(1 + n) Binomial[m, m/2] Binomial[v, v/2] (-1 + Mod[m, 2]) (-1 + Mod[v, 2]))/I^m) + (2 Binomial[m, m/2] (-1 + Mod[m, 2]) f^(-n - 1) Sum[Binomial[v, s] (-2 s + v)^(-1 - n) ((-E^(g (-2 s + v))) (-1)^n Gamma[1 + n, f (2 s - v) z] + E^(2 g s - g v) Gamma[1 + n, f (-2 s + v) z]), {s, 0, Floor[(1/2) (-1 + v)]}])/ I^m + 4 b^(-2 n - 2) Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[((-1)^k Binomial[m, k] ((-1)^m Gamma[2 (1 + n), (-b) (2 k - m) Sqrt[z]] + Gamma[2 (1 + n), b (2 k - m) Sqrt[z]]))/ (-2 k + m)^(2 (1 + n)), {k, 0, Floor[(1/2) (-1 + m)]}] + Sum[(-1)^u Binomial[m, u] Sum[(f^(-2 - 2 n) (-2 s + v)^(-2 - 2 n) Binomial[v, s] (E^(4 g s) Sum[(-1)^(-h + k) 4^k (b (m - 2 u))^(-h - k + 2 n) (b (m - 2 u) + 2 f (2 s - v) Sqrt[z])^(h + k) (-((b (m - 2 u) + 2 f (2 s - v) Sqrt[z])^2/(f (2 s - v))))^( (1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (b (m - 2 u) (b (m - 2 u) + 2 f (2 s - v) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((b (m - 2 u) + 2 f (2 s - v) Sqrt[z])^2/(f (8 s - 4 v)))] + 2 f (2 s - v) Sqrt[-((b (m - 2 u) + 2 f (2 s - v) Sqrt[z])^2/ (f (2 s - v)))] Gamma[(1/2) (2 + h + k), -((b (m - 2 u) + 2 f (2 s - v) Sqrt[z])^2/(f (8 s - 4 v)))]), {k, 0, n}, {h, 0, k}] + (-1)^m E^(2 g v + (b^2 (m - 2 u)^2)/(4 f s - 2 f v)) Sum[(-1)^(-h + k) 4^k ((-b) (m - 2 u))^(-h - k + 2 n) ((b (m - 2 u) + 2 f (2 s - v) Sqrt[z])^2/(f (2 s - v)))^((1/2) (-1 - h - k)) ((-b) (m - 2 u) + 2 f (-2 s + v) Sqrt[z])^(h + k) Binomial[k, h] Binomial[n, k] (b (m - 2 u) (b (m - 2 u) + 2 f (2 s - v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (b (m - 2 u) + 2 f (2 s - v) Sqrt[z])^2/(f (8 s - 4 v))] - 2 f (2 s - v) Sqrt[(b (m - 2 u) + 2 f (2 s - v) Sqrt[z])^2/ (f (2 s - v))] Gamma[(1/2) (2 + h + k), (b (m - 2 u) + 2 f (2 s - v) Sqrt[z])^2/(f (8 s - 4 v))]), {k, 0, n}, {h, 0, k}] + (-1)^m E^(4 g s) Sum[(-1)^(-h + k) 4^k ((-b) (m - 2 u))^(-h - k + 2 n) ((-b) (m - 2 u) + 2 f (2 s - v) Sqrt[z])^(h + k) (-((b (m - 2 u) + 2 f (-2 s + v) Sqrt[z])^2/(f (2 s - v))))^( (1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (b (m - 2 u) (b (m - 2 u) + 2 f (-2 s + v) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((b (m - 2 u) + 2 f (-2 s + v) Sqrt[z])^2/(f (8 s - 4 v)))] - 2 f (-2 s + v) Sqrt[-((b (m - 2 u) + 2 f (-2 s + v) Sqrt[z])^2/ (f (2 s - v)))] Gamma[(1/2) (2 + h + k), -((b (m - 2 u) + 2 f (-2 s + v) Sqrt[z])^2/(f (8 s - 4 v)))]), {k, 0, n}, {h, 0, k}] + E^(2 g v + (b^2 (m - 2 u)^2)/(4 f s - 2 f v)) Sum[(-1)^(-h + k) 4^k (b (m - 2 u))^(-h - k + 2 n) (b (m - 2 u) + 2 f (-2 s + v) Sqrt[z])^(h + k) ((b (m - 2 u) + 2 f (-2 s + v) Sqrt[z])^2/(f (2 s - v)))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (b (m - 2 u) (b (m - 2 u) + 2 f (-2 s + v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (b (m - 2 u) + 2 f (-2 s + v) Sqrt[z])^2/ (f (8 s - 4 v))] + 2 f (-2 s + v) Sqrt[ (b (m - 2 u) + 2 f (-2 s + v) Sqrt[z])^2/(f (2 s - v))] Gamma[(1/2) (2 + h + k), (b (m - 2 u) + 2 f (-2 s + v) Sqrt[z])^2/(f (8 s - 4 v))]), {k, 0, n}, {h, 0, k}]))/ E^((b^2 (m - 2 u)^2 + 4 f (2 s - v) g (2 s + v))/(f (8 s - 4 v))), {s, 0, Floor[(1/2) (-1 + v)]}], {u, 0, Floor[(1/2) (-1 + m)]}]/ 4^n) /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 z n sinh m ( b z ) cosh v ( g + f z ) z 2 - m - v - 1 ( 4 ( 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]]]]] ( v mod 2 \$CellContext`v 2 - 1 ) ( k = 0 m - 1 2 ( - 1 ) k ( m - 2 k ) - 2 ( n + 1 ) ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - 1 ) m Γ ( 2 ( n + 1 ) , - b ( 2 k - m ) z ) + Γ ( 2 ( n + 1 ) , b ( 2 k - m ) z ) ) ) b - 2 n - 2 + 2 - m z n + 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]]]]] ( 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]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) ( v mod 2 \$CellContext`v 2 - 1 ) n + 1 + 2 f - n - 1 - m ( 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]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) 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]]]]] ( v - 2 s ) - n - 1 ( 2 g s - g v Γ ( n + 1 , f ( v - 2 s ) z ) - g ( v - 2 s ) ( - 1 ) n Γ ( n + 1 , f ( 2 s - v ) z ) ) + 4 - n u = 0 m - 1 2 ( - 1 ) u ( m u ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["u", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] s = 0 v - 1 2 - b 2 ( m - 2 u ) 2 + 4 f g ( 2 s - v ) ( 2 s + v ) f ( 8 s - 4 v ) f - 2 n - 2 ( v - 2 s ) - 2 n - 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 b 2 ( m - 2 u ) 2 4 f s - 2 f v + 2 g v k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b ( m - 2 u ) ) - h - k + 2 n ( ( b ( m - 2 u ) + 2 f ( 2 s - v ) z ) 2 f ( 2 s - v ) ) 1 2 ( - h - k - 1 ) ( 2 f ( v - 2 s ) z - b ( m - 2 u ) ) h + k ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( b ( m - 2 u ) ( b ( m - 2 u ) + 2 f ( 2 s - v ) z ) Γ ( 1 2 ( h + k + 1 ) , ( b ( m - 2 u ) + 2 f ( 2 s - v ) z ) 2 f ( 8 s - 4 v ) ) - 2 f ( 2 s - v ) ( b ( m - 2 u ) + 2 f ( 2 s - v ) z ) 2 f ( 2 s - v ) Γ ( 1 2 ( h + k + 2 ) , ( b ( m - 2 u ) + 2 f ( 2 s - v ) z ) 2 f ( 8 s - 4 v ) ) ) + b 2 ( m - 2 u ) 2 4 f s - 2 f v + 2 g v k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b ( m - 2 u ) ) - h - k + 2 n ( b ( m - 2 u ) + 2 f ( v - 2 s ) z ) h + k ( ( b ( m - 2 u ) + 2 f ( v - 2 s ) z ) 2 f ( 2 s - v ) ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( b ( m - 2 u ) ( b ( m - 2 u ) + 2 f ( v - 2 s ) z ) Γ ( 1 2 ( h + k + 1 ) , ( b ( m - 2 u ) + 2 f ( v - 2 s ) z ) 2 f ( 8 s - 4 v ) ) + 2 f ( b ( m - 2 u ) + 2 f ( v - 2 s ) z ) 2 f ( 2 s - v ) ( v - 2 s ) Γ ( 1 2 ( h + k + 2 ) , ( b ( m - 2 u ) + 2 f ( v - 2 s ) z ) 2 f ( 8 s - 4 v ) ) ) + 4 g s k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b ( m - 2 u ) ) - h - k + 2 n ( b ( m - 2 u ) + 2 f ( 2 s - v ) z ) h + k ( - ( b ( m - 2 u ) + 2 f ( 2 s - v ) z ) 2 f ( 2 s - v ) ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( b ( m - 2 u ) ( b ( m - 2 u ) + 2 f ( 2 s - v ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b ( m - 2 u ) + 2 f ( 2 s - v ) z ) 2 f ( 8 s - 4 v ) ) + 2 f ( 2 s - v ) - ( b ( m - 2 u ) + 2 f ( 2 s - v ) z ) 2 f ( 2 s - v ) Γ ( 1 2 ( h + k + 2 ) , - ( b ( m - 2 u ) + 2 f ( 2 s - v ) z ) 2 f ( 8 s - 4 v ) ) ) + ( - 1 ) m 4 g s k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b ( m - 2 u ) ) - h - k + 2 n ( 2 f ( 2 s - v ) z - b ( m - 2 u ) ) h + k ( - ( b ( m - 2 u ) + 2 f ( v - 2 s ) z ) 2 f ( 2 s - v ) ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( b ( m - 2 u ) ( b ( m - 2 u ) + 2 f ( v - 2 s ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b ( m - 2 u ) + 2 f ( v - 2 s ) z ) 2 f ( 8 s - 4 v ) ) - 2 f ( v - 2 s ) - ( b ( m - 2 u ) + 2 f ( v - 2 s ) z ) 2 f ( 2 s - v ) Γ ( 1 2 ( h + k + 2 ) , - ( b ( m - 2 u ) + 2 f ( v - 2 s ) z ) 2 f ( 8 s - 4 v ) ) ) ) ) /; n m + v + Condition z z n b z 1 2 m g f z v 2 -1 m -1 v -1 4 Binomial v v 2 -1 \$CellContext`v 2 -1 k 0 m -1 2 -1 -1 k m -1 2 k -2 n 1 Binomial m k -1 m Gamma 2 n 1 -1 b 2 k -1 m z 1 2 Gamma 2 n 1 b 2 k -1 m z 1 2 b -2 n -2 2 -1 m z n 1 Binomial m m 2 -1 Binomial v v 2 -1 \$CellContext`m 2 -1 \$CellContext`v 2 -1 n 1 -1 2 f -1 n -1 -1 m Binomial m m 2 -1 \$CellContext`m 2 -1 s 0 v -1 2 -1 Binomial v s v -1 2 s -1 n -1 2 g s -1 g v Gamma n 1 f v -1 2 s z -1 g v -1 2 s -1 n Gamma n 1 f 2 s -1 v z 4 -1 n u 0 m -1 2 -1 -1 u Binomial m u s 0 v -1 2 -1 -1 b 2 m -1 2 u 2 4 f g 2 s -1 v 2 s v f 8 s -1 4 v -1 f -2 n -2 v -1 2 s -2 n -2 Binomial v s -1 m b 2 m -1 2 u 2 4 f s -1 2 f v -1 2 g v h 0 k k 0 n -1 k -1 h 4 k -1 b m -1 2 u -1 h -1 k 2 n b m -1 2 u 2 f 2 s -1 v z 1 2 2 f 2 s -1 v -1 1 2 -1 h -1 k -1 2 f v -1 2 s z 1 2 -1 b m -1 2 u h k Binomial k h Binomial n k b m -1 2 u b m -1 2 u 2 f 2 s -1 v z 1 2 Gamma 1 2 h k 1 b m -1 2 u 2 f 2 s -1 v z 1 2 2 f 8 s -1 4 v -1 -1 2 f 2 s -1 v b m -1 2 u 2 f 2 s -1 v z 1 2 2 f 2 s -1 v -1 1 2 Gamma 1 2 h k 2 b m -1 2 u 2 f 2 s -1 v z 1 2 2 f 8 s -1 4 v -1 b 2 m -1 2 u 2 4 f s -1 2 f v -1 2 g v h 0 k k 0 n -1 k -1 h 4 k b m -1 2 u -1 h -1 k 2 n b m -1 2 u 2 f v -1 2 s z 1 2 h k b m -1 2 u 2 f v -1 2 s z 1 2 2 f 2 s -1 v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b m -1 2 u b m -1 2 u 2 f v -1 2 s z 1 2 Gamma 1 2 h k 1 b m -1 2 u 2 f v -1 2 s z 1 2 2 f 8 s -1 4 v -1 2 f b m -1 2 u 2 f v -1 2 s z 1 2 2 f 2 s -1 v -1 1 2 v -1 2 s Gamma 1 2 h k 2 b m -1 2 u 2 f v -1 2 s z 1 2 2 f 8 s -1 4 v -1 4 g s h 0 k k 0 n -1 k -1 h 4 k b m -1 2 u -1 h -1 k 2 n b m -1 2 u 2 f 2 s -1 v z 1 2 h k -1 b m -1 2 u 2 f 2 s -1 v z 1 2 2 f 2 s -1 v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b m -1 2 u b m -1 2 u 2 f 2 s -1 v z 1 2 Gamma 1 2 h k 1 -1 b m -1 2 u 2 f 2 s -1 v z 1 2 2 f 8 s -1 4 v -1 2 f 2 s -1 v -1 b m -1 2 u 2 f 2 s -1 v z 1 2 2 f 2 s -1 v -1 1 2 Gamma 1 2 h k 2 -1 b m -1 2 u 2 f 2 s -1 v z 1 2 2 f 8 s -1 4 v -1 -1 m 4 g s h 0 k k 0 n -1 k -1 h 4 k -1 b m -1 2 u -1 h -1 k 2 n 2 f 2 s -1 v z 1 2 -1 b m -1 2 u h k -1 b m -1 2 u 2 f v -1 2 s z 1 2 2 f 2 s -1 v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b m -1 2 u b m -1 2 u 2 f v -1 2 s z 1 2 Gamma 1 2 h k 1 -1 b m -1 2 u 2 f v -1 2 s z 1 2 2 f 8 s -1 4 v -1 -1 2 f v -1 2 s -1 b m -1 2 u 2 f v -1 2 s z 1 2 2 f 2 s -1 v -1 1 2 Gamma 1 2 h k 2 -1 b m -1 2 u 2 f v -1 2 s z 1 2 2 f 8 s -1 4 v -1 n m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18