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 Sech

 http://functions.wolfram.com/01.24.21.0255.01

 Input Form

 Integrate[E^(p z) Sin[a z]^m Tanh[c z]^u Sech[c z], z] == (1/(p + c (1 + u))) (I^u 2^(1 - m) E^((p + c (u + 1)) z) Binomial[m, m/2] Binomial[u, u/2] HypergeometricPFQ[{1/2 + p/(2 c) + u/2, 1 + u}, {3/2 + p/(2 c) + u/2}, -E^(2 c z)] (1 - Mod[m, 2]) (1 - Mod[u, 2])) + I^(m + u) 2^(1 - m) E^(c (u + 1) z) Binomial[u, u/2] (1 - Mod[u, 2]) Sum[(-1)^k Binomial[m, k] ((E^(((-I) a (-2 k + m) + p) z) HypergeometricPFQ[{1/2 + (I a k)/c - (I a m)/(2 c) + p/(2 c) + u/2, 1 + u}, {3/2 + (I a k)/c - (I a m)/(2 c) + p/(2 c) + u/2}, -E^(2 c z)])/(I a (2 k - m) + p + c (1 + u)) + ((-1)^m E^((I a (-2 k + m) + p) z) HypergeometricPFQ[ {1/2 - (I a k)/c + (I a m)/(2 c) + p/(2 c) + u/2, 1 + u}, {3/2 - (I a k)/c + (I a m)/(2 c) + p/(2 c) + u/2}, -E^(2 c z)])/ (I a (-2 k + m) + p + c (1 + u))), {k, 0, Floor[(1/2) (-1 + m)]}] + 2^(1 - m) E^(c (u + 1) z) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(-1)^s Binomial[u, s] ((1/(p + c (1 + 2 s))) ((-1)^u E^((p - c (-2 s + u)) z) HypergeometricPFQ[ {1/2 + p/(2 c) + s, 1 + u}, {3/2 + p/(2 c) + s}, -E^(2 c z)]) + (E^((p + c (-2 s + u)) z) HypergeometricPFQ[ {1 + u, 1/2 + p/(2 c) - s + u}, {3/2 + p/(2 c) - s + u}, -E^(2 c z)])/(p + c (1 - 2 s + 2 u))), {s, 0, Floor[(1/2) (-1 + u)]}] + 2^(1 - m) E^(c (u + 1) z) Sum[(-1)^k Binomial[m, k] Sum[(-1)^s Binomial[u, s] (((-1)^u E^((I m Pi)/2 + ((-I) a (-2 k + m) + p - c (-2 s + u)) z) HypergeometricPFQ[{1/2 + (I a k)/c - (I a m)/(2 c) + p/(2 c) + s, 1 + u}, {3/2 + (I a k)/c - (I a m)/(2 c) + p/(2 c) + s}, -E^(2 c z)])/(c + 2 I a k - I a m + p + 2 c s) + ((-1)^u E^((-(1/2)) I m Pi + (I a (-2 k + m) + p - c (-2 s + u)) z) HypergeometricPFQ[{1/2 - (I a k)/c + (I a m)/(2 c) + p/(2 c) + s, 1 + u}, {3/2 - (I a k)/c + (I a m)/(2 c) + p/(2 c) + s}, -E^(2 c z)])/(c - 2 I a k + I a m + p + 2 c s) + (E^((I m Pi)/2 + ((-I) a (-2 k + m) + p + c (-2 s + u)) z) HypergeometricPFQ[{1 + u, 1/2 + (I a k)/c - (I a m)/(2 c) + p/(2 c) - s + u}, {3/2 + (I a k)/c - (I a m)/(2 c) + p/(2 c) - s + u}, -E^(2 c z)])/(I a (2 k - m) + p + c (1 - 2 s + 2 u)) + (E^((-(1/2)) I m Pi + (I a (-2 k + m) + p + c (-2 s + u)) z) HypergeometricPFQ[{1 + u, 1/2 - (I a k)/c + (I a m)/(2 c) + p/(2 c) - s + u}, {3/2 - (I a k)/c + (I a m)/(2 c) + p/(2 c) - s + u}, -E^(2 c z)])/(I a (-2 k + m) + p + c (1 - 2 s + 2 u))), {s, 0, Floor[(1/2) (-1 + u)]}], {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0 && Element[u, Integers] && u > 0

 Standard Form

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 MathML Form

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a k c + p 2 c + s + 3 2 - a 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[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "k"]], "c"], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", "s", "+", FractionBox["1", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox[RowBox[List["u", "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "k"]], "c"], "+", FractionBox["p", RowBox[List["2", " ", "c"]]], "+", "s", "+", FractionBox["3", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]]]], 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] ) / ( 2 s c + c + 2 a k - a m + p ) + ( ( a ( m - 2 k ) + p + c ( u - 2 s ) ) z - m π 2 2 F 1 ( u + 1 , - a k c + a m 2 c + p 2 c - s + u + 1 2 ; 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m + u + Condition z p z a z m c z u c z 1 p c u 1 -1 u 2 1 -1 m u 1 c p z Binomial m m 2 -1 Binomial u u 2 -1 HypergeometricPFQ p 2 c -1 u 2 -1 1 2 u 1 p 2 c -1 u 2 -1 3 2 -1 2 c z 1 -1 \$CellContext`m 2 1 -1 \$CellContext`u 2 m u 2 1 -1 m c u 1 z Binomial u u 2 -1 1 -1 \$CellContext`u 2 k 0 m -1 2 -1 -1 k Binomial m k -1 m a m -1 2 k p z HypergeometricPFQ -1 a k c -1 a m 2 c -1 p 2 c -1 u 2 -1 1 2 u 1 -1 a k c -1 a m 2 c -1 p 2 c -1 u 2 -1 3 2 -1 2 c z a m -1 2 k p c u 1 -1 p -1 a m -1 2 k z HypergeometricPFQ a k c -1 p 2 c -1 u 2 -1 1 2 -1 a m 2 c -1 u 1 a k c -1 p 2 c -1 u 2 -1 3 2 -1 a m 2 c -1 -1 2 c z a 2 k -1 m p c u 1 -1 2 1 -1 m c u 1 z Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 u -1 2 -1 -1 s Binomial u s -1 u p -1 c u -1 2 s z HypergeometricPFQ p 2 c -1 s 1 2 u 1 p 2 c -1 s 3 2 -1 2 c z p c 2 s 1 -1 p c u -1 2 s z HypergeometricPFQ u 1 p 2 c -1 -1 s u 1 2 p 2 c -1 -1 s u 3 2 -1 2 c z p c -2 s 2 u 1 -1 2 1 -1 m c u 1 z k 0 m -1 2 -1 -1 k Binomial m k s 0 u -1 2 -1 -1 s Binomial u s -1 u a m -1 2 k p -1 c u -1 2 s z -1 m 2 -1 HypergeometricPFQ -1 a k c -1 a m 2 c -1 p 2 c -1 s 1 2 u 1 -1 a k c -1 a m 2 c -1 p 2 c -1 s 3 2 -1 2 c z 2 s c c -1 2 a k a m p -1 -1 u m 2 -1 -1 a m -1 2 k p -1 c u -1 2 s z HypergeometricPFQ a k c -1 p 2 c -1 s 1 2 -1 a m 2 c -1 u 1 a k c -1 p 2 c -1 s 3 2 -1 a m 2 c -1 -1 2 c z 2 s c c 2 a k -1 a m p -1 a m -1 2 k p c u -1 2 s z -1 m 2 -1 HypergeometricPFQ u 1 -1 a k c -1 a m 2 c -1 p 2 c -1 -1 s u 1 2 -1 a k c -1 a m 2 c -1 p 2 c -1 -1 s u 3 2 -1 2 c z a m -1 2 k p c -2 s 2 u 1 -1 m 2 -1 -1 a m -1 2 k p c u -1 2 s z HypergeometricPFQ u 1 a k c -1 p 2 c -1 -1 s u 1 2 -1 a m 2 c -1 a k c -1 p 2 c -1 -1 s u 3 2 -1 a m 2 c -1 -1 2 c z a 2 k -1 m p c -2 s 2 u 1 -1 m SuperPlus u SuperPlus [/itex]

 Rule Form

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

 2002-12-18