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 Cosh

 http://functions.wolfram.com/01.20.21.1135.01

 Input Form

 Integrate[z^(\[Alpha] - 1) Cos[b z^r + e]^m Cosh[c z^r], z] == -((1/r) (2^(-1 - m) z^\[Alpha] Binomial[m, m/2] (Gamma[\[Alpha]/r, (-c) z^r]/((-c) z^r)^(\[Alpha]/r) + Gamma[\[Alpha]/r, c z^r]/(c z^r)^(\[Alpha]/r)) (1 - Mod[m, 2]))) - (1/r) (2^(-1 - m) z^\[Alpha] Sum[Binomial[m, k] ((E^(-2 I e k + I e m) Gamma[\[Alpha]/r, (-c + 2 I b k - I b m) z^r])/ ((-c + 2 I b k - I b m) z^r)^(\[Alpha]/r) + (E^(-2 I e k + I e m) Gamma[\[Alpha]/r, (c + 2 I b k - I b m) z^r])/ ((c + 2 I b k - I b m) z^r)^(\[Alpha]/r) + (E^(2 I e k - I e m) Gamma[\[Alpha]/r, (-c - 2 I b k + I b m) z^r])/ ((-c - 2 I b k + I b m) z^r)^(\[Alpha]/r) + (E^(2 I e k - I e m) Gamma[\[Alpha]/r, (c - 2 I b k + I b m) z^r])/ ((c - 2 I b k + I b m) z^r)^(\[Alpha]/r)), {k, 0, Floor[(1/2) (-1 + m)]}]) /; Element[m, Integers] && m > 0

 Standard Form

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 MathML Form

 z α - 1 cos m ( b z r + e ) cosh ( c z r ) z - 2 - m - 1 z α ( 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 ( α r , - c z r ) ( - c z r ) - α r + ( c z r ) - α r Γ ( α r , c z r ) ) ( 1 - m mod 2 \$CellContext`m 2 ) r - 1 r 2 - m - 1 z α 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]]]]] ( 2 e k - e m Γ ( α r , ( - c - 2 b k + b m ) z r ) ( ( - c - 2 b k + b m ) z r ) - α r + 2 e k - e m ( ( c - 2 b k + b m ) z r ) - α r Γ ( α r , ( c - 2 b k + b m ) z r ) + - 2 e k + e m ( ( - c + 2 b k - b m ) z r ) - α r Γ ( α r , ( - c + 2 b k - b m ) z r ) + - 2 e k + e m ( ( c + 2 b k - b m ) z r ) - α r Γ ( α r , ( c + 2 b k - b m ) z r ) ) /; m + Condition z z α -1 b z r e m c z r -1 2 -1 m -1 z α Binomial m m 2 -1 Gamma α r -1 -1 c z r -1 c z r -1 α r -1 c z r -1 α r -1 Gamma α r -1 c z r 1 -1 \$CellContext`m 2 r -1 -1 1 r -1 2 -1 m -1 z α k 0 m -1 2 -1 Binomial m k 2 e k -1 e m Gamma α r -1 -1 c -1 2 b k b m z r -1 c -1 2 b k b m z r -1 α r -1 2 e k -1 e m c -1 2 b k b m z r -1 α r -1 Gamma α r -1 c -1 2 b k b m z r -2 e k e m -1 c 2 b k -1 b m z r -1 α r -1 Gamma α r -1 -1 c 2 b k -1 b m z r -2 e k e m c 2 b k -1 b m z r -1 α r -1 Gamma α r -1 c 2 b k -1 b m z r m SuperPlus [/itex]

 Rule Form

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

 2002-12-18