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 Sech

 http://functions.wolfram.com/01.24.21.0086.01

 Input Form

 Integrate[Sinh[b z]^m Sech[c z], z] == I^m Binomial[m, m/2] (1 - Mod[m, 2]) (ArcTan[Tanh[(c z)/2]]/ (2^(m - 1) c)) + 2^(1 - m) Sum[(-1)^k Binomial[m, k] (((-1)^m/(c + 2 b k - b m)) (E^((c + 2 b k - b m) z) Hypergeometric2F1[1, (c + 2 b k - b m)/(2 c), (3 c + 2 b k - b m)/ (2 c), -E^(2 c z)]) + (1/(c + b (-2 k + m))) (E^((c + b (-2 k + m)) z) Hypergeometric2F1[1, (c - 2 b k + b m)/ (2 c), (3 c - 2 b k + b m)/(2 c), -E^(2 c z)])), {k, 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["Sinh", "[", RowBox[List["b", " ", "z"]], "]"]], "m"], " ", RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["\[ImaginaryI]", "m"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], FractionBox[RowBox[List["ArcTan", "[", RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "]"]], RowBox[List[SuperscriptBox["2", RowBox[List["m", "-", "1"]]], "c"]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], RowBox[List["(", " ", RowBox[List[RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], RowBox[List["c", "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", FractionBox[RowBox[List["c", "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", FractionBox[RowBox[List["c", "-", RowBox[List["2", " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]]]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "-", RowBox[List["2", " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]]]], 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

 sinh 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]] m ( 1 - m mod 2 \$CellContext`m 2 ) tan - 1 ( tanh ( c z 2 ) ) 2 m - 1 c + 2 1 - m k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( c + b ( m - 2 k ) ) z c + b ( m - 2 k ) 2 F 1 ( 1 , c - 2 b k + b m 2 c ; 3 c - 2 b k + b m 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["2", " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]]]], 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["2", " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]]]], 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] + ( - 1 ) m ( c + 2 b k - b m ) z c + 2 b k - b m 2 F 1 ( 1 , c + 2 b k - b m 2 c ; 3 c + 2 b k - b m 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["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], 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["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], 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 m 1 -1 \$CellContext`m 2 c z 2 -1 2 m -1 c -1 2 1 -1 m k 0 m -1 2 -1 -1 k Binomial m k c b m -1 2 k z c b m -1 2 k -1 Hypergeometric2F1 1 c -1 2 b k b m 2 c -1 3 c -1 2 b k b m 2 c -1 -1 2 c z -1 m c 2 b k -1 b m z c 2 b k -1 b m -1 Hypergeometric2F1 1 c 2 b k -1 b m 2 c -1 3 c 2 b k -1 b m 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["Sinh", "[", RowBox[List["b_", " ", "z_"]], "]"]], "m_"], " ", RowBox[List["Sech", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ImaginaryI]", "m"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["ArcTan", "[", RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "]"]]]], RowBox[List[SuperscriptBox["2", RowBox[List["m", "-", "1"]]], " ", "c"]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", FractionBox[RowBox[List["c", "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]]]], RowBox[List["c", "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", FractionBox[RowBox[List["c", "-", RowBox[List["2", " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]]]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "-", RowBox[List["2", " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]

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

 2002-12-18