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 Coth

 http://functions.wolfram.com/01.22.21.0182.01

 Input Form

 Integrate[Sin[a z]^m Cosh[c z]^\[Mu] Coth[c z], z] == (-(1/(c (-2 + \[Mu]) \[Mu]))) (((\[Mu] AppellF1[1 - \[Mu]/2, -\[Mu], 1, 2 - \[Mu]/2, -E^(-2 c z), E^(-2 c z)] + E^(2 c z) (-2 + \[Mu]) AppellF1[-(\[Mu]/2), -\[Mu], 1, 1 - \[Mu]/2, -E^(-2 c z), E^(-2 c z)]) Binomial[m, m/2] Cosh[c z]^\[Mu] (-1 + Mod[m, 2]))/(2^m E^(2 c z) (1 + E^(-2 c z))^\[Mu])) + (Cosh[c z]^\[Mu] Sum[(-1)^k ((E^((I m Pi)/2 - I a (-2 k + m) z) AppellF1[-((I a (2 k - m) + c \[Mu])/(2 c)), -\[Mu], 1, (2 c - 2 I a k + I a m - c \[Mu])/(2 c), -E^(-2 c z), E^(-2 c z)])/ (I a (2 k - m) + c \[Mu]) + (E^((I m Pi)/2 + (-2 c - I a (-2 k + m)) z) AppellF1[-((I a (2 k - m) + c (-2 + \[Mu]))/(2 c)), -\[Mu], 1, (I a (-2 k + m) + c (4 - \[Mu]))/(2 c), -E^(-2 c z), E^(-2 c z)])/ ((-I) a (-2 k + m) + c (-2 + \[Mu])) + (E^((-(1/2)) I m Pi + I a (-2 k + m) z) AppellF1[ -((I a (-2 k + m) + c \[Mu])/(2 c)), -\[Mu], 1, ((-I) a (-2 k + m) + c (2 - \[Mu]))/(2 c), -E^(-2 c z), E^(-2 c z)])/(I a (-2 k + m) + c \[Mu]) + (E^((-(1/2)) I m Pi + (-2 c + I a (-2 k + m)) z) AppellF1[-((I a (-2 k + m) + c (\[Mu] - 2))/(2 c)), -\[Mu], 1, ((-I) a (-2 k + m) + c (4 - \[Mu]))/(2 c), -E^(-2 c z), E^(-2 c z)])/(I a (-2 k + m) + c (-2 + \[Mu]))) Binomial[m, k], {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

 sin m ( a z ) cosh μ ( c z ) coth ( c z ) z 2 - m ( 1 + - 2 c z ) - μ cosh μ ( c z ) 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]]]]] ( ( a ( m - 2 k ) - 2 c ) z - m π 2 F 1 AppellF1 ( - ( μ - 2 ) c + a ( m - 2 k ) 2 c ; - μ , 1 ; c ( 4 - μ ) - a ( m - 2 k ) 2 c ; - - 2 c z , - 2 c z ) a ( m - 2 k ) + c ( μ - 2 ) + ( π m 2 + ( - 2 c - a ( m - 2 k ) ) z F 1 AppellF1 ( - a ( 2 k - m ) + c ( μ - 2 ) 2 c ; - μ , 1 ; a ( m - 2 k ) + c ( 4 - μ ) 2 c ; - - 2 c z , - 2 c z ) ) / ( c ( μ - 2 ) - a ( m - 2 k ) ) + m π 2 - a ( m - 2 k ) z F 1 AppellF1 ( - a ( 2 k - m ) + c μ 2 c ; - μ , 1 ; - μ c + 2 c - 2 a k + a m 2 c ; - - 2 c z , - 2 c z ) a ( 2 k - m ) + c μ + a ( m - 2 k ) z - m π 2 F 1 AppellF1 ( - a ( m - 2 k ) + c μ 2 c ; - μ , 1 ; c ( 2 - μ ) - a ( m - 2 k ) 2 c ; - - 2 c z , - 2 c z ) a ( m - 2 k ) + c μ ) - 1 c ( μ - 2 ) μ 2 - m - 2 c z ( 1 + - 2 c z ) - μ ( 2 c z ( μ - 2 ) F 1 AppellF1 ( - μ 2 ; - μ , 1 ; 1 - μ 2 ; - - 2 c z , - 2 c z ) + μ F 1 AppellF1 ( 1 - μ 2 ; - μ , 1 ; 2 - μ 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]]]]] cosh μ ( c z ) ( m mod 2 \$CellContext`m 2 - 1 ) /; m + Condition z a z m c z μ c z 2 -1 m 1 -2 c z -1 μ c z μ k 0 m -1 2 -1 -1 k Binomial m k a m -1 2 k -1 2 c z -1 m 2 -1 AppellF1 -1 μ -2 c a m -1 2 k 2 c -1 -1 μ 1 c 4 -1 μ -1 a m -1 2 k 2 c -1 -1 -2 c z -2 c z a m -1 2 k c μ -2 -1 m 2 -1 -2 c -1 a m -1 2 k z AppellF1 -1 a 2 k -1 m c μ -2 2 c -1 -1 μ 1 a m -1 2 k c 4 -1 μ 2 c -1 -1 -2 c z -2 c z c μ -2 -1 a m -1 2 k -1 m 2 -1 -1 a m -1 2 k z AppellF1 -1 a 2 k -1 m c μ 2 c -1 -1 μ 1 -1 μ c 2 c -1 2 a k a m 2 c -1 -1 -2 c z -2 c z a 2 k -1 m c μ -1 a m -1 2 k z -1 m 2 -1 AppellF1 -1 a m -1 2 k c μ 2 c -1 -1 μ 1 c 2 -1 μ -1 a m -1 2 k 2 c -1 -1 -2 c z -2 c z a m -1 2 k c μ -1 -1 1 c μ -2 μ -1 2 -1 m -2 c z 1 -2 c z -1 μ 2 c z μ -2 AppellF1 -1 μ 2 -1 -1 μ 1 1 -1 μ 2 -1 -1 -2 c z -2 c z μ AppellF1 1 -1 μ 2 -1 -1 μ 1 2 -1 μ 2 -1 -1 -2 c z -2 c z Binomial m m 2 -1 c z μ \$CellContext`m 2 -1 m SuperPlus [/itex]

 Rule Form

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

 2002-12-18