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 Sech

 http://functions.wolfram.com/01.24.21.0264.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["a", " ", "z"]], "]"]], "m"], SuperscriptBox[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["p", "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "+", "\[Mu]"]], ")"]]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["p", "-", "c"]], ")"]], " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]]]], ")"]], "\[Mu]"], " ", RowBox[List["AppellF1", "[", 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 MathML Form

 p z sin m ( a z ) csch μ ( c z ) sech ( c z ) z 2 1 - m - c z ( 1 - - 2 c z ) μ ( k = 0 m - 1 2 ( - 1 ) k ( ( ( a ( m - 2 k ) + p ) z - m π 2 F 1 AppellF1 ( - a ( m - 2 k ) - p + c ( μ + 1 ) 2 c ; 1 , μ ; - a ( m - 2 k ) - p + c ( μ + 3 ) 2 c ; - - 2 c z , - 2 c z ) ) / ( a ( m - 2 k ) + p - c ( μ + 1 ) ) + ( π m 2 + ( p - a ( m - 2 k ) ) z F 1 AppellF1 ( a ( m - 2 k ) - p + c ( μ + 1 ) 2 c ; 1 , μ ; a ( m - 2 k ) - p + c ( μ + 3 ) 2 c ; - - 2 c z , - 2 c z ) ) / ( - a ( m - 2 k ) + p - 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]]]]] ) csch μ ( c z ) + 1 p - c ( μ + 1 ) ( 2 1 - m ( p - c ) z ( 1 - - 2 c z ) μ F 1 AppellF1 ( c ( μ + 1 ) - p 2 c ; 1 , μ ; c ( μ + 3 ) - p 2 c ; - - 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]]]]] csch μ ( c z ) ( 1 - m mod 2 \$CellContext`m 2 ) ) /; m + Condition z p z a z m c z μ c z 2 1 -1 m -1 c z 1 -1 -2 c z μ k 0 m -1 2 -1 -1 k a m -1 2 k p z -1 m 2 -1 AppellF1 -1 a m -1 2 k -1 p c μ 1 2 c -1 1 μ -1 a m -1 2 k -1 p c μ 3 2 c -1 -1 -2 c z -2 c z a m -1 2 k p -1 c μ 1 -1 m 2 -1 p -1 a m -1 2 k z AppellF1 a m -1 2 k -1 p c μ 1 2 c -1 1 μ a m -1 2 k -1 p c μ 3 2 c -1 -1 -2 c z -2 c z -1 a m -1 2 k p -1 c μ 1 -1 Binomial m k c z μ 1 p -1 c μ 1 -1 2 1 -1 m p -1 c z 1 -1 -2 c z μ AppellF1 c μ 1 -1 p 2 c -1 1 μ c μ 3 -1 p 2 c -1 -1 -2 c z -2 c z Binomial m m 2 -1 c z μ 1 -1 \$CellContext`m 2 m SuperPlus [/itex]

 Rule Form

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

 2002-12-18