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 Sech

 http://functions.wolfram.com/01.24.21.0181.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18