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 Coth

 http://functions.wolfram.com/01.22.21.0208.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18