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 Sech

 http://functions.wolfram.com/01.24.21.0222.01

 Input Form

 Integrate[Cos[a z]^m Csch[c z]^\[Mu] Sech[c z], z] == (-(1/(c (1 + \[Mu])))) ((2^(1 - m) (1 - E^(-2 c z))^\[Mu] AppellF1[(1 + \[Mu])/2, 1, \[Mu], (3 + \[Mu])/2, -E^(-2 c z), E^(-2 c z)] Binomial[m, m/2] Csch[c z]^\[Mu] (1 - Mod[m, 2]))/ E^(c z)) + (2^(1 - m) (1 - E^(-2 c z))^\[Mu] Csch[c z]^\[Mu] Sum[((E^(I a (-2 k + m) z) AppellF1[((-I) a (-2 k + m) + c (1 + \[Mu]))/ (2 c), 1, \[Mu], ((-I) a (-2 k + m) + c (3 + \[Mu]))/(2 c), -E^(-2 c z), E^(-2 c z)])/(I a (-2 k + m) - c (1 + \[Mu])) + AppellF1[(I a (-2 k + m) + c (1 + \[Mu]))/(2 c), 1, \[Mu], (I a (-2 k + m) + c (3 + \[Mu]))/(2 c), -E^(-2 c z), E^(-2 c z)]/ E^(I a (-2 k + m) z)/((-I) a (-2 k + m) - c (1 + \[Mu]))) Binomial[m, k], {k, 0, Floor[(1/2) (-1 + m)]}])/E^(c z) /; Element[m, Integers] && m > 0

 Standard Form

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 MathML Form

 cos m ( a z ) csch μ ( c z ) sech ( c z ) z 2 1 - m - c z ( 1 - - 2 c z ) μ csch μ ( c z ) k = 0 m - 1 2 ( a ( m - 2 k ) z F 1 AppellF1 ( c ( μ + 1 ) - a ( m - 2 k ) 2 c ; 1 , μ ; c ( μ + 3 ) - a ( m - 2 k ) 2 c ; - - 2 c z , - 2 c z ) a ( m - 2 k ) - c ( μ + 1 ) + - a ( m - 2 k ) z F 1 AppellF1 ( a ( m - 2 k ) + c ( μ + 1 ) 2 c ; 1 , μ ; a ( m - 2 k ) + c ( μ + 3 ) 2 c ; - - 2 c z , - 2 c z ) - a ( m - 2 k ) - c ( μ + 1 ) ) ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] - 2 1 - m - c z ( 1 - - 2 c z ) μ csch μ ( c z ) ( 1 - m mod 2 \$CellContext`m 2 ) c ( μ + 1 ) F 1 AppellF1 ( μ + 1 2 ; 1 , μ ; μ + 3 2 ; - - 2 c z , - 2 c z ) ( 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 + Condition z a z m c z μ c z 2 1 -1 m -1 c z 1 -1 -2 c z μ c z μ k 0 m -1 2 -1 a m -1 2 k z AppellF1 c μ 1 -1 a m -1 2 k 2 c -1 1 μ c μ 3 -1 a m -1 2 k 2 c -1 -1 -2 c z -2 c z a m -1 2 k -1 c μ 1 -1 -1 a m -1 2 k z AppellF1 a m -1 2 k c μ 1 2 c -1 1 μ a m -1 2 k c μ 3 2 c -1 -1 -2 c z -2 c z -1 a m -1 2 k -1 c μ 1 -1 Binomial m k -1 2 1 -1 m -1 c z 1 -1 -2 c z μ c z μ 1 -1 \$CellContext`m 2 c μ 1 -1 AppellF1 μ 1 2 -1 1 μ μ 3 2 -1 -1 -2 c z -2 c z Binomial m m 2 -1 m SuperPlus [/itex]

 Rule Form

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

 2002-12-18

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