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 Cosh

 http://functions.wolfram.com/01.20.21.3720.01

 Input Form

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

 Rule Form

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

 2002-12-18