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 Coth

 http://functions.wolfram.com/01.22.21.0166.01

 Input Form

 Integrate[E^(p z) Sinh[c z]^\[Mu] Coth[c z], z] == ((-2^(-\[Mu])) (-E^((-c) z) + E^(c z))^\[Mu] ((E^(p z) Hypergeometric2F1[(p - c \[Mu])/(2 c), 1 - \[Mu], (2 c + p - c \[Mu])/(2 c), E^(2 c z)])/(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)])/(2 c + p - c \[Mu])))/ (1 - E^(2 c z))^\[Mu]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", SuperscriptBox["2", RowBox[List["-", "\[Mu]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "c"]], " ", "z"]]]]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], ")"]], "\[Mu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["p", "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["1", "-", "\[Mu]"]], ",", FractionBox[RowBox[List[RowBox[List["2", " ", "c"]], "+", "p", "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List["p", "-", RowBox[List["c", " ", "\[Mu]"]]]]], "+", FractionBox[RowBox[List[" ", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "c"]], "+", "p", "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["1", "-", "\[Mu]"]], ",", FractionBox[RowBox[List[RowBox[List["4", " ", "c"]], "+", "p", "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]]]], RowBox[List[RowBox[List["2", " ", "c"]], "+", "p", "-", RowBox[List["c", " ", "\[Mu]"]]]]]]], ")"]]]]]]]]

 MathML Form

 p z sinh μ ( c z ) coth ( c z ) z - 2 - μ ( - - c z + c z ) μ ( 1 - 2 c z ) - μ ( 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 ; - μ c + 4 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[RowBox[List[RowBox[List["-", "\[Mu]"]], " ", "c"]], "+", RowBox[List["2", " ", "c"]], "+", "p"]], 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["4", " ", "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] - μ c + 2 c + p ) z p z c z μ c z -1 2 -1 μ -1 -1 c z c z μ 1 -1 2 c z -1 μ 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 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", "z_"]]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "\[Mu]_"], " ", RowBox[List["Coth", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", SuperscriptBox["2", RowBox[List["-", "\[Mu]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "c"]], " ", "z"]]]]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], ")"]], "\[Mu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["p", "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["1", "-", "\[Mu]"]], ",", FractionBox[RowBox[List[RowBox[List["2", " ", "c"]], "+", "p", "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List["p", "-", RowBox[List["c", " ", "\[Mu]"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "c"]], "+", "p", "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["1", "-", "\[Mu]"]], ",", FractionBox[RowBox[List[RowBox[List["4", " ", "c"]], "+", "p", "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List["2", " ", "c"]], "+", "p", "-", RowBox[List["c", " ", "\[Mu]"]]]]]]], ")"]]]]]]]]

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

 2002-12-18