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 Tanh

 http://functions.wolfram.com/01.21.21.0157.01

 Input Form

 Integrate[E^(p z) Cosh[b z]^u Tanh[c z], z] == (Binomial[u, u/2] ((-(1/p)) (E^(p z) Hypergeometric2F1[p/(2 c), 1, (2 c + p)/(2 c), -E^(2 c z)]) + (1/(2 c + p)) (E^((2 c + p) z) Hypergeometric2F1[(2 c + p)/(2 c), 1, (4 c + p)/(2 c), -E^(2 c z)])) (1 - Mod[u, 2]))/2^u + Sum[Binomial[u, k] (-(E^((p - b (-2 k + u)) z) Hypergeometric2F1[ (p - b (-2 k + u))/(2 c), 1, (2 c + p - b (-2 k + u))/(2 c), -E^(2 c z)])/(p - b (-2 k + u)) + (E^((2 c + p - b (-2 k + u)) z) Hypergeometric2F1[(2 c + p - b (-2 k + u))/(2 c), 1, (4 c + p - b (-2 k + u))/(2 c), -E^(2 c z)])/ (2 c + p - b (-2 k + u)) - (E^((p + b (-2 k + u)) z) Hypergeometric2F1[(p + b (-2 k + u))/(2 c), 1, (2 c + p + b (-2 k + u))/(2 c), -E^(2 c z)])/(p + b (-2 k + u)) + (E^((2 c + p + b (-2 k + u)) z) Hypergeometric2F1[ (2 c + p + b (-2 k + u))/(2 c), 1, (4 c + p + b (-2 k + u))/(2 c), -E^(2 c z)])/(2 c + p + b (-2 k + u))), {k, 0, Floor[(1/2) (-1 + u)]}]/2^u /; Element[u, Integers] && u > 0

 Standard Form

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"c"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["4", " ", "c"]], "+", "p", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c"]], "+", "p", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "u"]], ")"]]]]]], ")"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["u", "\[Element]", "Integers"]], "\[And]", RowBox[List["u", ">", "0"]]]]]]]]

 MathML Form

 p z cosh u ( b z ) tanh ( c z ) z 2 - u ( 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 ) z 2 F 1 ( 2 c + p 2 c , 1 ; 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["2", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["4", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] 2 c + p - p z 2 F 1 ( p 2 c , 1 ; 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["p", RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["2", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] p ) ( 1 - u mod 2 \$CellContext`u 2 ) + 2 - u k = 0 u - 1 2 ( u k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( 2 c + p + b ( u - 2 k ) ) z 2 F 1 ( 2 c + p + b ( u - 2 k ) 2 c , 1 ; 4 c + p + b ( u - 2 k ) 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["2", " ", "c"]], "+", "p", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["4", " ", "c"]], "+", "p", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] 2 c + p + b ( u - 2 k ) - ( p - b ( u - 2 k ) ) z 2 F 1 ( p - b ( u - 2 k ) 2 c , 1 ; 2 c + p - b ( u - 2 k ) 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["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["2", " ", "c"]], "+", "p", "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] p - b ( u - 2 k ) + ( 2 c + p - b ( u - 2 k ) ) z 2 F 1 ( 2 c + p - b ( u - 2 k ) 2 c , 1 ; 4 c + p - b ( u - 2 k ) 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["2", " ", "c"]], "+", "p", "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["4", " ", "c"]], "+", "p", "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] 2 c + p - b ( u - 2 k ) - ( p + b ( u - 2 k ) ) z 2 F 1 ( p + b ( u - 2 k ) 2 c , 1 ; 2 c + p + b ( u - 2 k ) 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["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["2", " ", "c"]], "+", "p", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] p + b ( u - 2 k ) ) /; u + Condition z p z b z u c z 2 -1 u Binomial u u 2 -1 2 c p z Hypergeometric2F1 2 c p 2 c -1 1 4 c p 2 c -1 -1 2 c z 2 c p -1 -1 p z Hypergeometric2F1 p 2 c -1 1 2 c p 2 c -1 -1 2 c z p -1 1 -1 \$CellContext`u 2 2 -1 u k 0 u -1 2 -1 Binomial u k 2 c p b u -1 2 k z Hypergeometric2F1 2 c p b u -1 2 k 2 c -1 1 4 c p b u -1 2 k 2 c -1 -1 2 c z 2 c p b u -1 2 k -1 -1 p -1 b u -1 2 k z Hypergeometric2F1 p -1 b u -1 2 k 2 c -1 1 2 c p -1 b u -1 2 k 2 c -1 -1 2 c z p -1 b u -1 2 k -1 2 c p -1 b u -1 2 k z Hypergeometric2F1 2 c p -1 b u -1 2 k 2 c -1 1 4 c p -1 b u -1 2 k 2 c -1 -1 2 c z 2 c p -1 b u -1 2 k -1 -1 p b u -1 2 k z Hypergeometric2F1 p b u -1 2 k 2 c -1 1 2 c p b u -1 2 k 2 c -1 -1 2 c z p b u -1 2 k -1 u SuperPlus [/itex]

 Rule Form

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

 2002-12-18