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 Cosh

 http://functions.wolfram.com/01.20.21.3691.01

 Input Form

 Integrate[z^n Cos[d z + e]^m Cosh[c Sqrt[z] + g]^v, z] == (1/(1 + n)) (2^(-m - v) z^(1 + n) Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2])) + (-1)^n 2^(-m - v) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[Binomial[m, k] (E^(I e (-2 k + m)) (I d (-2 k + m))^(-1 - n) Gamma[1 + n, (-I) d (-2 k + m) z] + (((-I) d (-2 k + m))^(-1 - n) Gamma[1 + n, I d (-2 k + m) z])/E^(I e (-2 k + m))), {k, 0, Floor[(1/2) (-1 + m)]}] - 2^(1 - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(Binomial[v, k] (E^(g (-2 k + v)) Gamma[2 (1 + n), (-c) (-2 k + v) Sqrt[z]] + Gamma[2 (1 + n), c (-2 k + v) Sqrt[z]]/ E^(g (-2 k + v))))/(c (-2 k + v))^(2 (1 + n)), {k, 0, Floor[(1/2) (-1 + v)]}] + 2^(-1 - m - 2 n - v) Sum[Binomial[m, k] Sum[Binomial[v, s] ((E^((-I) e (-2 k + m) - g (-2 s + v) - (I c^2 (-2 s + v)^2)/ (4 d (-2 k + m))) Sum[(-1)^(-h + j) 4^j ((-c) (-2 s + v))^ (-h - j + 2 n) ((-c) (-2 s + v) - 2 I d (-2 k + m) Sqrt[z])^ (h + j) (-((I ((-c) (-2 s + v) - 2 I d (-2 k + m) Sqrt[z])^2)/ (d (-2 k + m))))^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((-c) (-2 s + v) ((-c) (-2 s + v) - 2 I d (-2 k + m) Sqrt[z]) Gamma[(1/2) (1 + h + j), -((I ((-c) (-2 s + v) - 2 I d (-2 k + m) Sqrt[z])^2)/ (4 d (-2 k + m)))] - 2 I d (-2 k + m) Sqrt[ -((I ((-c) (-2 s + v) - 2 I d (-2 k + m) Sqrt[z])^2)/ (d (-2 k + m)))] Gamma[(1/2) (2 + h + j), -((I ((-c) (-2 s + v) - 2 I d (-2 k + m) Sqrt[z])^2)/ (4 d (-2 k + m)))]), {j, 0, n}, {h, 0, j}])/ ((-I) d (-2 k + m))^(2 (1 + n)) + (E^((-I) e (-2 k + m) + g (-2 s + v) - (I c^2 (-2 s + v)^2)/ (4 d (-2 k + m))) Sum[(-1)^(-h + j) 4^j (c (-2 s + v))^ (-h - j + 2 n) (c (-2 s + v) - 2 I d (-2 k + m) Sqrt[z])^ (h + j) (-((I (c (-2 s + v) - 2 I d (-2 k + m) Sqrt[z])^2)/ (d (-2 k + m))))^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (c (-2 s + v) (c (-2 s + v) - 2 I d (-2 k + m) Sqrt[z]) Gamma[(1/2) (1 + h + j), -((I (c (-2 s + v) - 2 I d (-2 k + m) Sqrt[z])^2)/ (4 d (-2 k + m)))] - 2 I d (-2 k + m) Sqrt[ -((I (c (-2 s + v) - 2 I d (-2 k + m) Sqrt[z])^2)/ (d (-2 k + m)))] Gamma[(1/2) (2 + h + j), -((I (c (-2 s + v) - 2 I d (-2 k + m) Sqrt[z])^2)/ (4 d (-2 k + m)))]), {j, 0, n}, {h, 0, j}])/ ((-I) d (-2 k + m))^(2 (1 + n)) + (E^(I e (-2 k + m) - g (-2 s + v) + (I c^2 (-2 s + v)^2)/ (4 d (-2 k + m))) Sum[(-1)^(-h + j) 4^j ((-c) (-2 s + v))^ (-h - j + 2 n) ((-c) (-2 s + v) + 2 I d (-2 k + m) Sqrt[z])^ (h + j) ((I ((-c) (-2 s + v) + 2 I d (-2 k + m) Sqrt[z])^2)/(d (-2 k + m)))^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((-c) (-2 s + v) ((-c) (-2 s + v) + 2 I d (-2 k + m) Sqrt[z]) Gamma[(1/2) (1 + h + j), (I ((-c) (-2 s + v) + 2 I d (-2 k + m) Sqrt[z])^2)/ (4 d (-2 k + m))] + 2 I d (-2 k + m) Sqrt[ (I ((-c) (-2 s + v) + 2 I d (-2 k + m) Sqrt[z])^2)/ (d (-2 k + m))] Gamma[(1/2) (2 + h + j), (I ((-c) (-2 s + v) + 2 I d (-2 k + m) Sqrt[z])^2)/ (4 d (-2 k + m))]), {j, 0, n}, {h, 0, j}])/ (I d (-2 k + m))^(2 (1 + n)) + (E^(I e (-2 k + m) + g (-2 s + v) + (I c^2 (-2 s + v)^2)/ (4 d (-2 k + m))) Sum[(-1)^(-h + j) 4^j (c (-2 s + v))^ (-h - j + 2 n) (c (-2 s + v) + 2 I d (-2 k + m) Sqrt[z])^ (h + j) ((I (c (-2 s + v) + 2 I d (-2 k + m) Sqrt[z])^2)/(d (-2 k + m)))^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (c (-2 s + v) (c (-2 s + v) + 2 I d (-2 k + m) Sqrt[z]) Gamma[(1/2) (1 + h + j), (I (c (-2 s + v) + 2 I d (-2 k + m) Sqrt[z])^2)/(4 d (-2 k + m))] + 2 I d (-2 k + m) Sqrt[(I (c (-2 s + v) + 2 I d (-2 k + m) Sqrt[z])^2)/(d (-2 k + m))] Gamma[(1/2) (2 + h + j), (I (c (-2 s + v) + 2 I d (-2 k + m) Sqrt[z])^2)/ (4 d (-2 k + m))]), {j, 0, n}, {h, 0, j}])/ (I d (-2 k + m))^(2 (1 + n))), {s, 0, Floor[(1/2) (-1 + v)]}], {k, 0, Floor[(1/2) (-1 + m)]}] /; 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 cos m ( e + d z ) cosh v ( z c + g ) z 2 - m - v 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]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) n + 1 + ( - 1 ) n 2 - m - v ( 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]]]]] ( 1 - v mod 2 \$CellContext`v 2 ) k = 0 m - 1 2 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - e ( m - 2 k ) Γ ( n + 1 , d ( m - 2 k ) z ) ( - d ( m - 2 k ) ) - n - 1 + e ( m - 2 k ) ( d ( m - 2 k ) ) - n - 1 Γ ( n + 1 , - d ( m - 2 k ) z ) ) - 2 - m - v + 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]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) k = 0 v - 1 2 ( c ( v - 2 k ) ) - 2 ( n + 1 ) ( v k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( g ( v - 2 k ) Γ ( 2 ( n + 1 ) , - c ( v - 2 k ) z ) + - g ( v - 2 k ) Γ ( 2 ( n + 1 ) , c ( v - 2 k ) z ) ) + 2 - m - 2 n - v - 1 k = 0 m - 1 2 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] 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]]]]] ( - c 2 ( v - 2 s ) 2 4 d ( m - 2 k ) - g ( v - 2 s ) - e ( m - 2 k ) ( j = 0 n h = 0 j ( - 1 ) j - h 4 j ( - c ( v - 2 s ) ) - h - j + 2 n ( - 2 d z ( m - 2 k ) - c ( v - 2 s ) ) h + j ( - ( - 2 d z ( m - 2 k ) - c ( v - 2 s ) ) 2 d ( m - 2 k ) ) 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]]]]] ( - c ( v - 2 s ) ( - 2 d z ( m - 2 k ) - c ( v - 2 s ) ) Γ ( 1 2 ( h + j + 1 ) , - ( - 2 d z ( m - 2 k ) - c ( v - 2 s ) ) 2 4 d ( m - 2 k ) ) - 2 d ( m - 2 k ) - ( - 2 d z ( m - 2 k ) - c ( v - 2 s ) ) 2 d ( m - 2 k ) Γ ( 1 2 ( h + j + 2 ) , - ( - 2 d z ( m - 2 k ) - c ( v - 2 s ) ) 2 4 d ( m - 2 k ) ) ) ) ( - d ( m - 2 k ) ) - 2 ( n + 1 ) + - c 2 ( v - 2 s ) 2 4 d ( m - 2 k ) + g ( v - 2 s ) - e ( m - 2 k ) ( j = 0 n h = 0 j ( - 1 ) j - h 4 j ( c ( v - 2 s ) ) - h - j + 2 n ( c ( v - 2 s ) - 2 d ( m - 2 k ) z ) h + j ( - ( c ( v - 2 s ) - 2 d ( m - 2 k ) z ) 2 d ( m - 2 k ) ) 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]]]]] ( c ( v - 2 s ) ( c ( v - 2 s ) - 2 d ( m - 2 k ) z ) Γ ( 1 2 ( h + j + 1 ) , - ( c ( v - 2 s ) - 2 d ( m - 2 k ) z ) 2 4 d ( m - 2 k ) ) - 2 d ( m - 2 k ) - ( c ( v - 2 s ) - 2 d ( m - 2 k ) z ) 2 d ( m - 2 k ) Γ ( 1 2 ( h + j + 2 ) , - ( c ( v - 2 s ) - 2 d ( m - 2 k ) z ) 2 4 d ( m - 2 k ) ) ) ) ( - d ( m - 2 k ) ) - 2 ( n + 1 ) + c 2 ( v - 2 s ) 2 4 d ( m - 2 k ) - g ( v - 2 s ) + e ( m - 2 k ) ( d ( m - 2 k ) ) - 2 ( n + 1 ) j = 0 n h = 0 j ( - 1 ) j - h 4 j ( - c ( v - 2 s ) ) - h - j + 2 n ( 2 d ( m - 2 k ) z - c ( v - 2 s ) ) h + j ( ( 2 d ( m - 2 k ) z - c ( v - 2 s ) ) 2 d ( m - 2 k ) ) 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 d ( m - 2 k ) ( 2 d ( m - 2 k ) z - c ( v - 2 s ) ) 2 d ( m - 2 k ) Γ ( 1 2 ( h + j + 2 ) , ( 2 d ( m - 2 k ) z - c ( v - 2 s ) ) 2 4 d ( m - 2 k ) ) - c ( v - 2 s ) ( 2 d ( m - 2 k ) z - c ( v - 2 s ) ) Γ ( 1 2 ( h + j + 1 ) , ( 2 d ( m - 2 k ) z - c ( v - 2 s ) ) 2 4 d ( m - 2 k ) ) ) + c 2 ( v - 2 s ) 2 4 d ( m - 2 k ) + g ( v - 2 s ) + e ( m - 2 k ) ( d ( m - 2 k ) ) - 2 ( n + 1 ) j = 0 n h = 0 j ( - 1 ) j - h 4 j ( c ( v - 2 s ) ) - h - j + 2 n ( 2 d z ( m - 2 k ) + c ( v - 2 s ) ) h + j ( ( 2 d z ( m - 2 k ) + c ( v - 2 s ) ) 2 d ( m - 2 k ) ) 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]]]]] ( c ( v - 2 s ) ( 2 d z ( m - 2 k ) + c ( v - 2 s ) ) Γ ( 1 2 ( h + j + 1 ) , ( 2 d z ( m - 2 k ) + c ( v - 2 s ) ) 2 4 d ( m - 2 k ) ) + 2 d ( m - 2 k ) Γ ( 1 2 ( h + j + 2 ) , ( 2 d z ( m - 2 k ) + c ( v - 2 s ) ) 2 4 d ( m - 2 k ) ) ( 2 d z ( m - 2 k ) + c ( v - 2 s ) ) 2 d ( m - 2 k ) ) ) /; n m + v + Condition z z n e d z m z 1 2 c g v 2 -1 m -1 v z n 1 Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 n 1 -1 -1 n 2 -1 m -1 v Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 Binomial m k -1 e m -1 2 k Gamma n 1 d m -1 2 k z -1 d m -1 2 k -1 n -1 e m -1 2 k d m -1 2 k -1 n -1 Gamma n 1 -1 d m -1 2 k z -1 2 -1 m -1 v 1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 k 0 v -1 2 -1 c v -1 2 k -2 n 1 Binomial v k g v -1 2 k Gamma 2 n 1 -1 c v -1 2 k z 1 2 -1 g v -1 2 k Gamma 2 n 1 c v -1 2 k z 1 2 2 -1 m -1 2 n -1 v -1 k 0 m -1 2 -1 Binomial m k s 0 v -1 2 -1 Binomial v s -1 c 2 v -1 2 s 2 4 d m -1 2 k -1 -1 g v -1 2 s -1 e m -1 2 k h 0 j j 0 n -1 j -1 h 4 j -1 c v -1 2 s -1 h -1 j 2 n -2 d z 1 2 m -1 2 k -1 c v -1 2 s h j -1 -2 d z 1 2 m -1 2 k -1 c v -1 2 s 2 d m -1 2 k -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j -1 c v -1 2 s -2 d z 1 2 m -1 2 k -1 c v -1 2 s Gamma 1 2 h j 1 -1 -2 d z 1 2 m -1 2 k -1 c v -1 2 s 2 4 d m -1 2 k -1 -1 2 d m -1 2 k -1 -2 d z 1 2 m -1 2 k -1 c v -1 2 s 2 d m -1 2 k -1 1 2 Gamma 1 2 h j 2 -1 -2 d z 1 2 m -1 2 k -1 c v -1 2 s 2 4 d m -1 2 k -1 -1 d m -1 2 k -2 n 1 -1 c 2 v -1 2 s 2 4 d m -1 2 k -1 g v -1 2 s -1 e m -1 2 k h 0 j j 0 n -1 j -1 h 4 j c v -1 2 s -1 h -1 j 2 n c v -1 2 s -1 2 d m -1 2 k z 1 2 h j -1 c v -1 2 s -1 2 d m -1 2 k z 1 2 2 d m -1 2 k -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j c v -1 2 s c v -1 2 s -1 2 d m -1 2 k z 1 2 Gamma 1 2 h j 1 -1 c v -1 2 s -1 2 d m -1 2 k z 1 2 2 4 d m -1 2 k -1 -1 2 d m -1 2 k -1 c v -1 2 s -1 2 d m -1 2 k z 1 2 2 d m -1 2 k -1 1 2 Gamma 1 2 h j 2 -1 c v -1 2 s -1 2 d m -1 2 k z 1 2 2 4 d m -1 2 k -1 -1 d m -1 2 k -2 n 1 c 2 v -1 2 s 2 4 d m -1 2 k -1 -1 g v -1 2 s e m -1 2 k d m -1 2 k -2 n 1 h 0 j j 0 n -1 j -1 h 4 j -1 c v -1 2 s -1 h -1 j 2 n 2 d m -1 2 k z 1 2 -1 c v -1 2 s h j 2 d m -1 2 k z 1 2 -1 c v -1 2 s 2 d m -1 2 k -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j 2 d m -1 2 k 2 d m -1 2 k z 1 2 -1 c v -1 2 s 2 d m -1 2 k -1 1 2 Gamma 1 2 h j 2 2 d m -1 2 k z 1 2 -1 c v -1 2 s 2 4 d m -1 2 k -1 -1 c v -1 2 s 2 d m -1 2 k z 1 2 -1 c v -1 2 s Gamma 1 2 h j 1 2 d m -1 2 k z 1 2 -1 c v -1 2 s 2 4 d m -1 2 k -1 c 2 v -1 2 s 2 4 d m -1 2 k -1 g v -1 2 s e m -1 2 k d m -1 2 k -2 n 1 h 0 j j 0 n -1 j -1 h 4 j c v -1 2 s -1 h -1 j 2 n 2 d z 1 2 m -1 2 k c v -1 2 s h j 2 d z 1 2 m -1 2 k c v -1 2 s 2 d m -1 2 k -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j c v -1 2 s 2 d z 1 2 m -1 2 k c v -1 2 s Gamma 1 2 h j 1 2 d z 1 2 m -1 2 k c v -1 2 s 2 4 d m -1 2 k -1 2 d m -1 2 k Gamma 1 2 h j 2 2 d z 1 2 m -1 2 k c v -1 2 s 2 4 d m -1 2 k -1 2 d z 1 2 m -1 2 k c v -1 2 s 2 d m -1 2 k -1 1 2 n m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18