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 Coth

 http://functions.wolfram.com/01.22.21.0205.01

 Input Form

 Integrate[E^(p z) Cos[a z]^m Sinh[c z]^\[Mu] Coth[c z], z] == ((-2^(-\[Mu] - m)) Binomial[m, m/2] (-E^((-c) z) + E^(c z))^\[Mu] ((1/(p - c \[Mu])) E^(p z) Hypergeometric2F1[(p - c \[Mu])/(2 c), 1 - \[Mu], (2 c + p - c \[Mu])/(2 c), E^(2 c z)] + (1/(2 c + p - c \[Mu])) E^((2 c + p) z) Hypergeometric2F1[ (2 c + p - c \[Mu])/(2 c), 1 - \[Mu], (4 c + p - c \[Mu])/(2 c), E^(2 c z)]) (1 - Mod[m, 2]))/(1 - E^(2 c z))^\[Mu] - (2^(-\[Mu] - m) (-E^((-c) z) + E^(c z))^\[Mu] Sum[Binomial[m, k] ((E^(((-I) a (-2 k + m) + p) z) Hypergeometric2F1[ ((-I) a (-2 k + m) + p - c \[Mu])/(2 c), 1 - \[Mu], (2 c - I a (-2 k + m) + p - c \[Mu])/(2 c), E^(2 c z)])/ ((-I) a (-2 k + m) + p - c \[Mu]) + (E^((2 c - I a (-2 k + m) + p) z) Hypergeometric2F1[ (2 c - I a (-2 k + m) + p - c \[Mu])/(2 c), 1 - \[Mu], (4 c - I a (-2 k + m) + p - c \[Mu])/(2 c), E^(2 c z)])/ (2 c - I a (-2 k + m) + p - c \[Mu]) + (E^((I a (-2 k + m) + p) z) Hypergeometric2F1[ (I a (-2 k + m) + p - c \[Mu])/(2 c), 1 - \[Mu], (2 c + I a (-2 k + m) + p - c \[Mu])/(2 c), E^(2 c z)])/ (I a (-2 k + m) + p - c \[Mu]) + (E^((2 c + I a (-2 k + m) + p) z) Hypergeometric2F1[(2 c + I a (-2 k + m) + p - c \[Mu])/(2 c), 1 - \[Mu], (4 c + I a (-2 k + m) + p - c \[Mu])/(2 c), E^(2 c z)])/ (2 c + I a (-2 k + m) + p - c \[Mu])), {k, 0, Floor[(1/2) (-1 + m)]}])/(1 - E^(2 c z))^\[Mu] /; Element[m, Integers] && m > 0

 Standard Form

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 MathML Form

 p z cos m ( a z ) sinh μ ( c z ) coth ( c z ) z - 2 - m - μ ( - - c z + c z ) μ ( 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]]]]] ( p z 2 F 1 ( p - c μ 2 c , 1 - μ ; - μ c + 2 c + p 2 c ; 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["p", "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox[RowBox[List["1", "-", "\[Mu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[Mu]"]], " ", "c"]], "+", RowBox[List["2", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] p - c μ + ( 2 c + p ) z 2 F 1 ( - μ c + 2 c + p 2 c , 1 - μ ; 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m + Condition z p z a z m c z μ c z -1 2 -1 m -1 μ -1 -1 c z c z μ 1 -1 2 c z -1 μ Binomial m m 2 -1 p z Hypergeometric2F1 p -1 c μ 2 c -1 1 -1 μ -1 μ c 2 c p 2 c -1 2 c z p -1 c μ -1 2 c p z Hypergeometric2F1 -1 μ c 2 c p 2 c -1 1 -1 μ -1 μ c 4 c p 2 c -1 2 c z -1 μ c 2 c p -1 1 -1 \$CellContext`m 2 -1 2 -1 m -1 μ -1 -1 c z c z μ 1 -1 2 c z -1 μ k 0 m -1 2 -1 Binomial m k p -1 a m -1 2 k z Hypergeometric2F1 -1 a m -1 2 k p -1 c μ 2 c -1 1 -1 μ -1 μ c 2 c -1 a m -1 2 k p 2 c -1 2 c z -1 a m -1 2 k p -1 c μ -1 a m -1 2 k p z Hypergeometric2F1 a m -1 2 k p -1 c μ 2 c -1 1 -1 μ -1 μ c 2 c a m -1 2 k p 2 c -1 2 c z a m -1 2 k p -1 c μ -1 2 c a m -1 2 k p z Hypergeometric2F1 -1 μ c 2 c a m -1 2 k p 2 c -1 1 -1 μ -1 μ c 4 c a m -1 2 k p 2 c -1 2 c z -1 μ c 2 c a m -1 2 k p -1 2 c -1 a m -1 2 k p z Hypergeometric2F1 -1 μ c 2 c -1 a m -1 2 k p 2 c -1 1 -1 μ -1 μ c 4 c -1 a m -1 2 k p 2 c -1 2 c z -1 μ c 2 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