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 Sech

 http://functions.wolfram.com/01.24.21.0216.01

 Input Form

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

 Standard Form

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

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

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["a_", " ", "z_"]], "]"]], "m_"], " ", SuperscriptBox[RowBox[List["Coth", "[", RowBox[List["c_", " ", "z_"]], "]"]], "\[Mu]_"], " ", RowBox[List["Sech", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "c"]], " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]]]], ")"]], "\[Mu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]]]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", 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 Date Added to functions.wolfram.com (modification date)

 2002-12-18