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 Cosh

 http://functions.wolfram.com/01.20.21.1387.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18