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 Sech

 http://functions.wolfram.com/01.24.21.0316.01

 Input Form

 Integrate[Sech[c z] Sqrt[(a + b Sech[c z])^3], z] == -(16 Cosh[(c z)/2]^2 Cosh[c z]^2 Coth[(c z)/2] Sqrt[1 - Sech[c z]] Sqrt[(a + b Sech[c z])^3] (b (-1 + Sech[c z]) (a + b Sech[c z]) + (Sqrt[1 - Sech[c z]] ((4 a ((a + b) EllipticE[ArcSin[Sqrt[(a + b Sech[c z])/(a - b)]], (a - b)/(a + b)] - b EllipticF[ArcSin[Sqrt[(a + b Sech[c z])/ (a - b)]], (a - b)/(a + b)]) (1 + Sech[c z]) Sqrt[(b - b Sech[c z])/(a + b)] Sqrt[(a + b Sech[c z])/(a - b)])/ Sqrt[(b (1 + Sech[c z]))/(-a + b)] - 3 a^2 EllipticF[ArcSin[Sqrt[1 - Sech[c z]]/Sqrt[2]], (2 b)/(a + b)] Sqrt[(a + b Sech[c z])/(a + b)] Sqrt[Tanh[c z]^2] - b^2 EllipticF[ArcSin[Sqrt[1 - Sech[c z]]/Sqrt[2]], (2 b)/(a + b)] Sqrt[(a + b Sech[c z])/(a + b)] Sqrt[Tanh[c z]^2]))/ (Sqrt[1 + Sech[c z]] Sqrt[Tanh[c z]^2])))/ (c (4 Cosh[(c z)/2]^2 (4 a b Cosh[c z] + 3 (a^2 + 2 b^2 + a^2 Cosh[2 c z])) Sqrt[1 - Sech[c z]] + 16 a b Cosh[c z]^2 Sqrt[1 + Sech[c z]] Sqrt[Tanh[c z]^2]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "3"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["16", " ", SuperscriptBox[RowBox[List["Cosh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], " ", RowBox[List["Coth", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]]]]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "3"]], " ", RowBox[List["(", 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"2"], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], "+", RowBox[List["16", " ", "a", " ", "b", " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], " ", SqrtBox[RowBox[List["1", "+", RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]]]]], " ", SqrtBox[SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 sech ( c z ) ( a + b sech ( c z ) ) 3 z - ( 16 cosh 2 ( c z 2 ) cosh 2 ( c z ) coth ( c z 2 ) 1 - sech ( c z ) ( a + b sech ( c z ) ) 3 ( b ( sech ( c z ) - 1 ) ( a + b sech ( c z ) ) + 1 sech ( c z ) + 1 tanh 2 ( c z ) ( 1 - sech ( c z ) ( - 3 F ( sin - 1 ( 1 - sech ( c z ) 2 ) 2 b a + b ) a + b sech ( c z ) a + b tanh 2 ( c z ) a 2 - b 2 F ( sin - 1 ( 1 - sech ( c z ) 2 ) 2 b a + b ) a + b sech ( c z ) a + b tanh 2 ( c z ) + 1 b ( sech ( c z ) + 1 ) b - a ( 4 a ( ( a + b ) E ( sin - 1 ( a + b sech ( c z ) a - b ) a - b a + b ) - b F ( sin - 1 ( a + b sech ( c z ) a - b ) a - b a + b ) ) ( sech ( c z ) + 1 ) b - b sech ( c z ) a + b a + b sech ( c z ) a - b ) ) ) ) ) / ( c ( 4 ( 4 a b cosh ( c z ) + 3 ( cosh ( 2 c z ) a 2 + a 2 + 2 b 2 ) ) 1 - sech ( c z ) cosh 2 ( c z 2 ) + 16 a b cosh 2 ( c z ) sech ( c z ) + 1 tanh 2 ( c z ) ) ) z c z a b c z 3 1 2 -1 16 c z 2 -1 2 c z 2 c z 2 -1 1 -1 c z 1 2 a b c z 3 1 2 b c z -1 a b c z 1 c z 1 1 2 c z 2 1 2 -1 1 -1 c z 1 2 -3 EllipticF 1 -1 c z 1 2 2 1 2 -1 2 b a b -1 a b c z a b -1 1 2 c z 2 1 2 a 2 -1 b 2 EllipticF 1 -1 c z 1 2 2 1 2 -1 2 b a b -1 a b c z a b -1 1 2 c z 2 1 2 1 b c z 1 b -1 a -1 1 2 -1 4 a a b EllipticE a b c z a -1 b -1 1 2 a -1 b a b -1 -1 b EllipticF a b c z a -1 b -1 1 2 a -1 b a b -1 c z 1 b -1 b c z a b -1 1 2 a b c z a -1 b -1 1 2 c 4 4 a b c z 3 2 c z a 2 a 2 2 b 2 1 -1 c z 1 2 c z 2 -1 2 16 a b c z 2 c z 1 1 2 c z 2 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Sech", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Sech", "[", RowBox[List["c_", " ", "z_"]], "]"]]]]]], ")"]], "3"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List["16", " ", SuperscriptBox[RowBox[List["Cosh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], " ", RowBox[List["Coth", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]]]]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], 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 Date Added to functions.wolfram.com (modification date)

 2002-12-18