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 Coth

 http://functions.wolfram.com/01.22.21.0171.01

 Input Form

 Integrate[E^(p z) Cosh[b z]^u Coth[c z], z] == -((1/(p (2 c + p))) ((E^(p z) Binomial[u, u/2] ((2 c + p) Hypergeometric2F1[p/(2 c), 1, 1 + p/(2 c), E^(2 c z)] + E^(2 c z) p Hypergeometric2F1[1 + p/(2 c), 1, 2 + p/(2 c), E^(2 c z)]) (1 - Mod[u, 2]))/2^u)) - Sum[Binomial[u, s] ((E^((p - b (-2 s + u)) z) ((2 c + p - b (-2 s + u)) Hypergeometric2F1[(p - b (-2 s + u))/ (2 c), 1, 1 + (p - b (-2 s + u))/(2 c), E^(2 c z)] + E^(2 c z) (p - b (-2 s + u)) Hypergeometric2F1[ 1 + (p - b (-2 s + u))/(2 c), 1, 2 + (p - b (-2 s + u))/(2 c), E^(2 c z)]))/((p - b (-2 s + u)) (2 c + p - b (-2 s + u))) + (E^((p + b (-2 s + u)) z) ((2 c + p + b (-2 s + u)) Hypergeometric2F1[(p + b (-2 s + u))/(2 c), 1, 1 + (p + b (-2 s + u))/(2 c), E^(2 c z)] + E^(2 c z) (p + b (-2 s + u)) Hypergeometric2F1[ 1 + (p + b (-2 s + u))/(2 c), 1, 2 + (p + b (-2 s + u))/(2 c), E^(2 c z)]))/((p + b (-2 s + u)) (2 c + p + b (-2 s + u)))), {s, 0, Floor[(1/2) (-1 + u)]}]/2^u /; Element[u, Integers] && u > 0

 Standard Form

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 MathML Form

 p z cosh u ( b z ) coth ( c z ) z - 1 p ( 2 c + p ) ( 2 - u p z ( u u 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox[FractionBox["u", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( 2 c + p ) 2 F 1 ( p 2 c , 1 ; p 2 c + 1 ; 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["p", RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["p", RowBox[List["2", " ", "c"]]], "+", "1"]], 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] + 2 c z p 2 F 1 ( p 2 c + 1 , 1 ; p 2 c + 2 ; 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[RowBox[List[FractionBox["p", RowBox[List["2", " ", "c"]]], "+", "1"]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["p", RowBox[List["2", " ", "c"]]], "+", "2"]], 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] ) ( 1 - u mod 2 \$CellContext`u 2 ) ) - 2 - u s = 0 u - 1 2 ( u s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( ( p + b ( u - 2 s ) ) z ( ( 2 c + p + b ( u - 2 s ) ) 2 F 1 ( p + b ( u - 2 s ) 2 c , 1 ; p + b ( u - 2 s ) 2 c + 1 ; 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["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["p", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], 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] + 2 c z ( p + b ( u - 2 s ) ) 2 F 1 ( p + b ( u - 2 s ) 2 c + 1 , 1 ; p + b ( u - 2 s ) 2 c + 2 ; 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[RowBox[List[FractionBox[RowBox[List["p", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["p", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "2"]], 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 + b ( u - 2 s ) ) ( 2 c + p + b ( u - 2 s ) ) ) + ( ( p - b ( u - 2 s ) ) z ( ( 2 c + p - b ( u - 2 s ) ) 2 F 1 ( p - b ( u - 2 s ) 2 c , 1 ; p - b ( u - 2 s ) 2 c + 1 ; 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["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["p", "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], 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] + 2 c z ( p - b ( u - 2 s ) ) 2 F 1 ( p - b ( u - 2 s ) 2 c + 1 , 1 ; p - b ( u - 2 s ) 2 c + 2 ; 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[RowBox[List[FractionBox[RowBox[List["p", "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["p", "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "2"]], 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 - b ( u - 2 s ) ) ( 2 c + p - b ( u - 2 s ) ) ) ) /; u + Condition z p z b z u c z -1 1 p 2 c p -1 2 -1 u p z Binomial u u 2 -1 2 c p Hypergeometric2F1 p 2 c -1 1 p 2 c -1 1 2 c z 2 c z p Hypergeometric2F1 p 2 c -1 1 1 p 2 c -1 2 2 c z 1 -1 \$CellContext`u 2 -1 2 -1 u s 0 u -1 2 -1 Binomial u s p b u -1 2 s z 2 c p b u -1 2 s Hypergeometric2F1 p b u -1 2 s 2 c -1 1 p b u -1 2 s 2 c -1 1 2 c z 2 c z p b u -1 2 s Hypergeometric2F1 p b u -1 2 s 2 c -1 1 1 p b u -1 2 s 2 c -1 2 2 c z p b u -1 2 s 2 c p b u -1 2 s -1 p -1 b u -1 2 s z 2 c p -1 b u -1 2 s Hypergeometric2F1 p -1 b u -1 2 s 2 c -1 1 p -1 b u -1 2 s 2 c -1 1 2 c z 2 c z p -1 b u -1 2 s Hypergeometric2F1 p -1 b u -1 2 s 2 c -1 1 1 p -1 b u -1 2 s 2 c -1 2 2 c z p -1 b u -1 2 s 2 c p -1 b u -1 2 s -1 u SuperPlus [/itex]

 Rule Form

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

 2002-12-18