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 Sech

 http://functions.wolfram.com/01.24.21.0239.01

 Input Form

 Integrate[(z^n Tanh[c z] Sech[c z])/E^(3 c z), z] == (6 z^(1 + n))/(1 + n) + Gamma[1 + n, 2 c z]/(2^n c^(n + 1)) - 4 n! E^(2 c z) Sum[(((-1)^j (2 c)^(-1 - j) z^(-j + n))/(-j + n)!) HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, j + 2], 3}, {1 + Subscript[a, 1], \[Ellipsis], 1 + Subscript[a, j + 2]}, -E^(2 c z)], {j, 0, n}] + 6 n! E^(2 c z) Sum[(((-2 c)^(-1 - j) z^(-j + n))/(-j + n)!) HypergeometricPFQ[ {Subscript[a, 1], \[Ellipsis], Subscript[a, j + 2], 4}, {1 + Subscript[a, 1], \[Ellipsis], 1 + Subscript[a, j + 1], 3}, -E^(2 c z)], {j, 0, n}] /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 2] == 1 && Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n - 3 c z tanh ( c z ) sech ( c z ) z 6 z n + 1 n + 1 - 4 2 c z n ! j = 0 n ( - 1 ) j ( 2 c ) - j - 1 z n - j ( n - j ) ! j + 3 F j + 2 ( 1 , , 1 , 3 ; 2 , , 2 ; - 2 c z ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "3"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ], ",", TagBox["3", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox["2", HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox["2", 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]]]]] + 6 n ! 2 c z j = 0 n ( - 2 c ) - j - 1 z n - j ( n - j ) ! j + 3 F j + 2 ( 1 , , 1 , 4 ; 2 , , 2 , 3 ; - 2 c z ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "3"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ], ",", TagBox["4", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox["2", HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox["2", HypergeometricPFQ], ",", TagBox["3", 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]]]]] + 2 - n Γ ( n + 1 , 2 c z ) c n + 1 /; n Condition z z n -3 c z c z c z 6 z n 1 n 1 -1 -1 4 2 c z n j 0 n -1 j 2 c -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ 1 1 3 2 2 -1 2 c z 6 n 2 c z j 0 n -2 c -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ 1 1 4 2 2 3 -1 2 c z 2 -1 n Gamma n 1 2 c z c n 1 -1 n [/itex]

 Rule Form

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

 2002-12-18