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 Coth

 http://functions.wolfram.com/01.22.21.0213.01

 Input Form

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

 Standard Form

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 MathML Form

 p z cos m ( a z ) tanh μ ( c z ) coth ( c z ) z 2 - m ( 1 + - 2 c z ) μ - 1 tanh μ - 1 ( c z ) ( 1 - - 2 c z ) 1 - μ 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]]]]] ( ( p - a ( m - 2 k ) ) z F 1 AppellF1 ( - p - a ( m - 2 k ) 2 c ; μ - 1 , 1 - μ ; 1 - p - a ( m - 2 k ) 2 c ; - - 2 c z , - 2 c z ) p - a ( m - 2 k ) + ( a ( m - 2 k ) + p ) z F 1 AppellF1 ( - a ( m - 2 k ) + p 2 c ; μ - 1 , 1 - μ ; 1 - a ( m - 2 k ) + p 2 c ; - - 2 c z , - 2 c z ) a ( m - 2 k ) + p ) + 1 p 2 - m p z ( 1 + - 2 c z ) μ ( 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]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) tanh μ ( c z ) ( 1 - - 2 c z ) - μ F 1 AppellF1 ( - p 2 c ; μ - 1 , 1 - μ ; 1 - p 2 c ; - - 2 c z , - 2 c z ) /; m + Condition z p z a z m c z μ c z 2 -1 m 1 -2 c z μ -1 c z μ -1 1 -1 -2 c z 1 -1 μ k 0 m -1 2 -1 Binomial m k p -1 a m -1 2 k z AppellF1 -1 p -1 a m -1 2 k 2 c -1 μ -1 1 -1 μ 1 -1 p -1 a m -1 2 k 2 c -1 -1 -2 c z -2 c z p -1 a m -1 2 k -1 a m -1 2 k p z AppellF1 -1 a m -1 2 k p 2 c -1 μ -1 1 -1 μ 1 -1 a m -1 2 k p 2 c -1 -1 -2 c z -2 c z a m -1 2 k p -1 1 p -1 2 -1 m p z 1 -2 c z μ Binomial m m 2 -1 1 -1 \$CellContext`m 2 c z μ 1 -1 -2 c z -1 μ AppellF1 -1 p 2 c -1 μ -1 1 -1 μ 1 -1 p 2 c -1 -1 -2 c z -2 c z m SuperPlus [/itex]

 Rule Form

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

 2002-12-18