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 Sech

 http://functions.wolfram.com/01.24.21.0107.01

 Input Form

 Integrate[Cosh[b z]^m Sech[c z], z] == Binomial[m, m/2] (1 - Mod[m, 2]) ((2^(1 - m) ArcTan[E^(c z)])/c) + 2^(1 - m) Sum[Binomial[m, s] ((1/(c + b (m - 2 s))) (E^((c + b (m - 2 s)) z) Hypergeometric2F1[1, (c + b m - 2 b s)/ (2 c), (3 c + b m - 2 b s)/(2 c), -E^(2 c z)]) + (1/(c - b m + 2 b s)) (E^((c - b m + 2 b s) z) Hypergeometric2F1[1, (c - b m + 2 b s)/(2 c), (3 c - b m + 2 b s)/(2 c), -E^(2 c z)])), {s, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["b", " ", "z"]], "]"]], "m"], " ", RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", RowBox[List["ArcTan", "[", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], "]"]]]], "c"]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", FractionBox[RowBox[List["c", "+", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "b", " ", "s"]]]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "+", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "b", " ", "s"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["c", "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", FractionBox[RowBox[List["c", "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 cosh m ( b z ) sech ( c z ) z ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["m", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] 2 1 - m tan - 1 ( c z ) ( 1 - m mod 2 \$CellContext`m 2 ) c + 2 1 - m s = 0 m - 1 2 ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( c - b m + 2 b s ) z c - b m + 2 b s 2 F 1 ( 1 , c - b m + 2 b s 2 c ; 3 c - b m + 2 b s 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["1", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["c", "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + ( c + b ( m - 2 s ) ) z c + b ( m - 2 s ) 2 F 1 ( 1 , c + b m - 2 b s 2 c ; 3 c + b m - 2 b s 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["1", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["c", "+", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "b", " ", "s"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "+", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "b", " ", "s"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ) /; m + TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] Condition z b z m c z Binomial m m 2 -1 2 1 -1 m c z 1 -1 \$CellContext`m 2 c -1 2 1 -1 m s 0 m -1 2 -1 Binomial m s c -1 b m 2 b s z c -1 b m 2 b s -1 Hypergeometric2F1 1 c -1 b m 2 b s 2 c -1 3 c -1 b m 2 b s 2 c -1 -1 2 c z c b m -1 2 s z c b m -1 2 s -1 Hypergeometric2F1 1 c b m -1 2 b s 2 c -1 3 c b m -1 2 b s 2 c -1 -1 2 c z m [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["b_", " ", "z_"]], "]"]], "m_"], " ", RowBox[List["Sech", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", RowBox[List["ArcTan", "[", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], "]"]]]], ")"]]]], "c"], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", FractionBox[RowBox[List["c", "+", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "b", " ", "s"]]]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "+", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "b", " ", "s"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", FractionBox[RowBox[List["c", "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["c", "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]

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

 2002-12-18