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 Sech

 http://functions.wolfram.com/01.24.21.0257.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18