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/01.20.21.1125.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18