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 Sech

 http://functions.wolfram.com/01.24.21.0237.01

 Input Form

 Integrate[z^n E^(p z) Tanh[c z] Sech[c z], z] == 2 E^(2 c z) n! ((-E^((-c + p) z)) Sum[(1/(-j + n)!) (-1)^j z^(-j + n) (p + c)^(-1 - j) HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, j + 1], 2}, {1 + Subscript[a, 1], \[Ellipsis], 1 + Subscript[a, j + 1]}, -E^(2 c z)], {j, 0, n}] + E^((c + p) z) Sum[(1/(-j + n)!) (-1)^j z^(-j + n) (p + 3 c)^(-1 - j) HypergeometricPFQ[{Subscript[b, 1], \[Ellipsis], Subscript[b, j + 1], 2}, {1 + Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, j + 1]}, -E^(2 c z)], {j, 0, n}]) /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 1] == (p + c)/(2 c) && Subscript[b, 1] == Subscript[b, 2] == \[Ellipsis] == Subscript[b, n + 1] == (p + 3 c)/(2 c) && Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n p z tanh ( c z ) sech ( c z ) z 2 2 c z n ! ( - ( - c + p ) z j = 0 n ( - 1 ) j z n - j ( p + c ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( p + c 2 c , , p + c 2 c , 2 ; p + 3 c 2 c , , p + 3 c 2 c ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["p", "+", "c"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["p", "+", "c"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["2", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["p", "+", RowBox[List["3", "c"]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["p", "+", RowBox[List["3", "c"]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + ( c + p ) z j = 0 n ( - 1 ) j z n - j ( p + 3 c ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( p + 3 c 2 c , , p + 3 c 2 c , 2 ; p + 5 c 2 c , , p + 5 c 2 c ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["p", "+", RowBox[List["3", "c"]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["p", "+", RowBox[List["3", "c"]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["2", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["p", "+", RowBox[List["5", "c"]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["p", "+", RowBox[List["5", "c"]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) /; n Condition z z n p z c z c z 2 2 c z n -1 -1 c p z j 0 n -1 j z n -1 j p c -1 j -1 n -1 j -1 HypergeometricPFQ p c 2 c -1 p c 2 c -1 2 p 3 c 2 c -1 p 3 c 2 c -1 -1 2 c z c p z j 0 n -1 j z n -1 j p 3 c -1 j -1 n -1 j -1 HypergeometricPFQ p 3 c 2 c -1 p 3 c 2 c -1 2 p 5 c 2 c -1 p 5 c 2 c -1 -1 2 c z n [/itex]

 Rule Form

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

 2002-12-18