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 Sech

 http://functions.wolfram.com/01.24.21.0223.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], RowBox[List["Sinh", "[", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], "]"]], RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], " ", RowBox[List["n", "!"]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", "+", RowBox[List[RowBox[List["(", RowBox[List["p", "+", "b"]], ")"]], " ", "z"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "j"]], ")"]], "!"]], SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", "p", "+", "b"]], ")"]], RowBox[List["j", "+", "1"]]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["j", "+", "1"]]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List[RowBox[List["(", RowBox[List["p", "-", "b"]], ")"]], " ", "z"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "j"]], ")"]], "!"]], SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", "p", "-", "b"]], ")"]], RowBox[List["j", "+", "1"]]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", RowBox[List["j", "+", "1"]]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", RowBox[List["j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["n", "+", "1"]]], "\[Equal]", FractionBox[RowBox[List["b", "+", "p", "+", "c"]], RowBox[List["2", " ", "c"]]]]], "\[And]", RowBox[List[SubscriptBox["b", "1"], "\[Equal]", SubscriptBox["b", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["b", RowBox[List["n", "+", "1"]]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "p", "+", "c"]], RowBox[List["2", " ", "c"]]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[RowBox[List["p", "+", "b"]], "\[NotEqual]", RowBox[List["-", "c"]]]], "\[And]", RowBox[List[RowBox[List["p", "-", "b"]], "\[NotEqual]", RowBox[List["-", "c"]]]]]]]]]]

 MathML Form

 z n p z sinh ( a + b z ) sech ( c z ) z c z n ! ( a + ( b + p ) z j = 0 n ( ( - 1 ) j z n - j ) ( n - j ) ! ( b + c + p ) j + 1 j + 2 F j + 1 ( b + c + p 2 c , , b + c + p 2 c , 1 ; b + c + p 2 c + 1 , , b + c + p 2 c + 1 ; - 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["b", "+", "c", "+", "p"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["b", "+", "c", "+", "p"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["b", "+", "c", "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["b", "+", "c", "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], 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] - ( p - b ) z - a j = 0 n ( ( - 1 ) j z n - j ) ( n - j ) ! ( - b + c + p ) j + 1 j + 2 F j + 1 ( - b + c + p 2 c , , - b + c + p 2 c , 1 ; - b + c + p 2 c + 1 , , - b + c + p 2 c + 1 ; - 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[RowBox[List["-", "b"]], "+", "c", "+", "p"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "c", "+", "p"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "c", "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "c", "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], 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 b + p - c p - b - c Condition z z n p z a b z c z c z n a b p z j 0 n -1 j z n -1 j n -1 j b c p j 1 -1 HypergeometricPFQ b c p 2 c -1 b c p 2 c -1 1 b c p 2 c -1 1 b c p 2 c -1 1 -1 2 c z -1 p -1 b z -1 a j 0 n -1 j z n -1 j n -1 j -1 b c p j 1 -1 HypergeometricPFQ -1 b c p 2 c -1 -1 b c p 2 c -1 1 -1 b c p 2 c -1 1 -1 b c p 2 c -1 1 -1 2 c z n b p -1 c p -1 b -1 c [/itex]

 Rule Form

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

 2002-12-18