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 Cosh

 http://functions.wolfram.com/01.20.21.1012.01

 Input Form

 Integrate[z^n Sin[b z^2 + e]^m Cosh[c z^2 + f z], z] == (-2^(-2 - m)) Binomial[m, m/2] (1 - Mod[m, 2]) ((-c)^(-1 - n) E^(f^2/(4 c)) Sum[2^(j - n) f^(-j + n) (-f - 2 c z)^(1 + j) ((-f - 2 c z)^2/c)^((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, (-f - 2 c z)^2/(4 c)], {j, 0, n}] + (c^(-1 - n) Sum[2^(j - n) (-f)^(-j + n) (f + 2 c z)^(1 + j) (-((f + 2 c z)^2/c))^((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -((f + 2 c z)^2/(4 c))], {j, 0, n}])/ E^(f^2/(4 c))) - 2^(-2 - m) Sum[(-1)^k Binomial[m, k] (((-c - I b (-2 k + m))^(-1 - n) Sum[2^(j - n) f^(-j + n) (-f + 2 (-c - I b (-2 k + m)) z)^(1 + j) (-((-f + 2 (-c - I b (-2 k + m)) z)^2/(-c - I b (-2 k + m))))^ ((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -((-f + 2 (-c - I b (-2 k + m)) z)^2/(4 (-c - I b (-2 k + m))))], {j, 0, n}])/E^((1/4) I (4 e (-2 k + m) + f^2/((-I) c - 2 b k + b m) - 2 m Pi)) + ((c - I b (-2 k + m))^(-1 - n) Sum[2^(j - n) (-f)^(-j + n) (f + 2 (c - I b (-2 k + m)) z)^(1 + j) (-((f + 2 (c - I b (-2 k + m)) z)^2/(c - I b (-2 k + m))))^ ((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -((f + 2 (c - I b (-2 k + m)) z)^2/(4 (c - I b (-2 k + m))))], {j, 0, n}])/E^((1/4) I (4 e (-2 k + m) - f^2/((-I) c + 2 b k - b m) - 2 m Pi)) + E^((1/4) I (4 e (-2 k + m) - f^2/((-I) c + 2 b k - b m) - 2 m Pi)) (-c + I b (-2 k + m))^(-1 - n) Sum[2^(j - n) f^(-j + n) (-f + 2 (-c + I b (-2 k + m)) z)^(1 + j) (-((-f + 2 (-c + I b (-2 k + m)) z)^2/(-c + I b (-2 k + m))))^ ((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -((-f + 2 (-c + I b (-2 k + m)) z)^2/(4 (-c + I b (-2 k + m))))], {j, 0, n}] + E^((1/4) I (4 e (-2 k + m) + f^2/((-I) c - 2 b k + b m) - 2 m Pi)) (c + I b (-2 k + m))^ (-1 - n) Sum[2^(j - n) (-f)^(-j + n) (f + 2 (c + I b (-2 k + m)) z)^ (1 + j) (-((f + 2 (c + I b (-2 k + m)) z)^2/(c + I b (-2 k + m))))^ ((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -((f + 2 (c + I b (-2 k + m)) z)^2/(4 (c + I b (-2 k + m))))], {j, 0, n}]), {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0

 Standard Form

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RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], ")"]]]]]]]]], "]"]]]]]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 z n sin m ( b z 2 + e ) cosh ( c z 2 + f z ) z - 2 - m - 2 ( 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 ) ( f 2 4 c ( j = 0 n 2 j - n f n - j ( - f - 2 c z ) j + 1 ( ( - f - 2 c z ) 2 c ) 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 - 2 c z ) 2 4 c ) ) ( - c ) - n - 1 + c - n - 1 - f 2 4 c j = 0 n 2 j - n ( - f ) n - j ( f + 2 c z ) j + 1 ( - ( f + 2 c z ) 2 c ) 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 + 2 c z ) 2 4 c ) ) - 2 - m - 2 k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 4 ( - f 2 - c + 2 b k - b m + 4 e ( m - 2 k ) - 2 m π ) ( j = 0 n 2 j - n f n - j ( 2 ( b ( m - 2 k ) - c ) z - f ) j + 1 ( - ( 2 ( b ( m - 2 k ) - c ) z - f ) 2 b ( m - 2 k ) - c ) 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 ) z - f ) 2 4 ( b ( m - 2 k ) - c ) ) ) ( b ( m - 2 k ) - c ) - n - 1 + 1 4 ( f 2 - c - 2 b k + b m + 4 e ( m - 2 k ) - 2 m π ) ( c + b ( m - 2 k ) ) - n - 1 j = 0 n 2 j - n ( - f ) n - j ( f + 2 ( c + b ( m - 2 k ) ) z ) j + 1 ( - ( f + 2 ( c + b ( m - 2 k ) ) z ) 2 c + 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 + 2 ( c + b ( m - 2 k ) ) z ) 2 4 ( c + b ( m - 2 k ) ) ) + - 1 4 ( f 2 - c - 2 b k + b m + 4 e ( m - 2 k ) - 2 m π ) ( - c - b ( m - 2 k ) ) - n - 1 j = 0 n 2 j - n f n - j ( 2 ( - c - b ( m - 2 k ) ) z - f ) j + 1 ( - ( 2 ( - c - b ( m - 2 k ) ) z - f ) 2 - c - 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 , - ( 2 ( - c - b ( m - 2 k ) ) z - f ) 2 4 ( - c - b ( m - 2 k ) ) ) + - 1 4 ( - f 2 - c + 2 b k - b m + 4 e ( m - 2 k ) - 2 m π ) ( c - b ( m - 2 k ) ) - n - 1 j = 0 n 2 j - n ( - f ) n - j ( f + 2 ( c - b ( m - 2 k ) ) z ) j + 1 ( - ( f + 2 ( c - b ( m - 2 k ) ) z ) 2 c - 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 + 2 ( c - b ( m - 2 k ) ) z ) 2 4 ( c - b ( m - 2 k ) ) ) ) /; n m + Condition z z n b z 2 e m c z 2 f z -1 2 -1 m -2 Binomial m m 2 -1 1 -1 \$CellContext`m 2 f 2 4 c -1 j 0 n 2 j -1 n f n -1 j -1 f -1 2 c z j 1 -1 f -1 2 c z 2 c -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 f -1 2 c z 2 4 c -1 -1 c -1 n -1 c -1 n -1 -1 f 2 4 c -1 j 0 n 2 j -1 n -1 f n -1 j f 2 c z j 1 -1 f 2 c z 2 c -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 f 2 c z 2 4 c -1 -1 2 -1 m -2 k 0 m -1 2 -1 -1 k Binomial m k 1 4 -1 f 2 -1 c 2 b k -1 b m -1 4 e m -1 2 k -1 2 m j 0 n 2 j -1 n f n -1 j 2 b m -1 2 k -1 c z -1 f j 1 -1 2 b m -1 2 k -1 c z -1 f 2 b m -1 2 k -1 c -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 2 b m -1 2 k -1 c z -1 f 2 4 b m -1 2 k -1 c -1 b m -1 2 k -1 c -1 n -1 1 4 f 2 -1 c -1 2 b k b m -1 4 e m -1 2 k -1 2 m c b m -1 2 k -1 n -1 j 0 n 2 j -1 n -1 f n -1 j f 2 c b m -1 2 k z j 1 -1 f 2 c b m -1 2 k z 2 c b m -1 2 k -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 f 2 c b m -1 2 k z 2 4 c b m -1 2 k -1 -1 1 4 f 2 -1 c -1 2 b k b m -1 4 e m -1 2 k -1 2 m -1 c -1 b m -1 2 k -1 n -1 j 0 n 2 j -1 n f n -1 j 2 -1 c -1 b m -1 2 k z -1 f j 1 -1 2 -1 c -1 b m -1 2 k z -1 f 2 -1 c -1 b m -1 2 k -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 2 -1 c -1 b m -1 2 k z -1 f 2 4 -1 c -1 b m -1 2 k -1 -1 1 4 -1 f 2 -1 c 2 b k -1 b m -1 4 e m -1 2 k -1 2 m c -1 b m -1 2 k -1 n -1 j 0 n 2 j -1 n -1 f n -1 j f 2 c -1 b m -1 2 k z j 1 -1 f 2 c -1 b m -1 2 k z 2 c -1 b m -1 2 k -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 f 2 c -1 b m -1 2 k z 2 4 c -1 b m -1 2 k -1 n m SuperPlus [/itex]

 Rule Form

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

 2002-12-18