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 Cosh

 http://functions.wolfram.com/01.20.21.1384.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18