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 Cosh

 http://functions.wolfram.com/01.20.21.4386.01

 Input Form

 Integrate[Sqrt[a + b Cosh[2 e z]]/(c + d Sinh[e z]), z] == (1/(d e)) ((Sqrt[-2 b c^2 - a d^2 - b d^2] ArcTan[(Sqrt[-2 b c^2 - a d^2 - b d^2] Cosh[e z])/ (Sqrt[c^2 + d^2] Sqrt[a + b Cosh[2 e z]])])/Sqrt[c^2 + d^2] + (I Sqrt[(a + b Cosh[2 e z])/(a + b)] (2 b c^2 EllipticF[I e z, (2 b)/(a + b)] + (-2 b c^2 - (a + b) d^2) EllipticPi[-(d^2/c^2), I e z, (2 b)/(a + b)]))/ (c d Sqrt[a + b Cosh[2 e z]]) + Sqrt[2] Sqrt[b Cosh[e z]^2] Log[Sqrt[a + b Cosh[2 e z]] + Sqrt[b + b Cosh[2 e z]]] Sech[e z])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]]], RowBox[List["c", "+", RowBox[List["d", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["d", " ", "e"]]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", SuperscriptBox["c", "2"]]], "-", RowBox[List["a", " ", SuperscriptBox["d", "2"]]], "-", RowBox[List["b", " ", SuperscriptBox["d", "2"]]]]]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", SuperscriptBox["c", "2"]]], "-", RowBox[List["a", " ", SuperscriptBox["d", "2"]]], "-", RowBox[List["b", " ", SuperscriptBox["d", "2"]]]]]], " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["c", "2"], "+", SuperscriptBox["d", "2"]]]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]]]]]], "]"]]]], SqrtBox[RowBox[List[SuperscriptBox["c", "2"], "+", SuperscriptBox["d", "2"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", SuperscriptBox["c", "2"], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "e", " ", "z"]], ",", FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", SuperscriptBox["c", "2"]]], "-", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox["d", "2"]]]]], ")"]], " ", RowBox[List["EllipticPi", "[", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["d", "2"], SuperscriptBox["c", "2"]]]], ",", RowBox[List["\[ImaginaryI]", " ", "e", " ", "z"]], ",", FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["c", " ", "d", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]]]]], ")"]]]], "+", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["b", " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]], "2"]]]], " ", RowBox[List["Log", "[", RowBox[List[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]]], "+", SqrtBox[RowBox[List["b", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]]]]], "]"]], " ", RowBox[List["Sech", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]]]]]]]]

 MathML Form

 a + b cosh ( 2 e z ) c + d sinh ( e z ) z 1 d e ( - 2 b c 2 - a d 2 - b d 2 tan - 1 ( - 2 b c 2 - a d 2 - b d 2 cosh ( e z ) c 2 + d 2 a + b cosh ( 2 e z ) ) c 2 + d 2 + ( a + b cosh ( 2 e z ) a + b ( 2 b F ( e z 2 b a + b ) c 2 + ( - 2 b c 2 - ( a + b ) d 2 ) Π ( - d 2 c 2 ; e z 2 b a + b ) ) ) / ( c d a + b cosh ( 2 e z ) ) + 2 b cosh 2 ( e z ) log ( a + b cosh ( 2 e z ) + cosh ( 2 e z ) b + b ) sech ( e z ) ) z a b 2 e z 1 2 c d e z -1 1 d e -1 -2 b c 2 -1 a d 2 -1 b d 2 1 2 -2 b c 2 -1 a d 2 -1 b d 2 1 2 e z c 2 d 2 1 2 a b 2 e z 1 2 -1 c 2 d 2 1 2 -1 a b 2 e z a b -1 1 2 2 b EllipticF e z 2 b a b -1 c 2 -2 b c 2 -1 a b d 2 EllipticPi -1 d 2 c 2 -1 e z 2 b a b -1 c d a b 2 e z 1 2 -1 2 1 2 b e z 2 1 2 a b 2 e z 1 2 2 e z b b 1 2 e z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "e_", " ", "z_"]], "]"]]]]]]], RowBox[List["c_", "+", RowBox[List["d_", " ", RowBox[List["Sinh", "[", RowBox[List["e_", " ", "z_"]], "]"]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", SuperscriptBox["c", "2"]]], "-", RowBox[List["a", " ", SuperscriptBox["d", "2"]]], "-", RowBox[List["b", " ", SuperscriptBox["d", "2"]]]]]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", SuperscriptBox["c", "2"]]], "-", RowBox[List["a", " ", SuperscriptBox["d", "2"]]], "-", RowBox[List["b", " ", SuperscriptBox["d", "2"]]]]]], " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["c", "2"], "+", SuperscriptBox["d", "2"]]]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]]]]]], "]"]]]], SqrtBox[RowBox[List[SuperscriptBox["c", "2"], "+", SuperscriptBox["d", "2"]]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", SuperscriptBox["c", "2"], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "e", " ", "z"]], ",", FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", SuperscriptBox["c", "2"]]], "-", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox["d", "2"]]]]], ")"]], " ", RowBox[List["EllipticPi", "[", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["d", "2"], SuperscriptBox["c", "2"]]]], ",", RowBox[List["\[ImaginaryI]", " ", "e", " ", "z"]], ",", FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]]]]]], ")"]]]], RowBox[List["c", " ", "d", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]]]]]], "+", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["b", " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]], "2"]]]], " ", RowBox[List["Log", "[", RowBox[List[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]]], "+", SqrtBox[RowBox[List["b", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]]]]], "]"]], " ", RowBox[List["Sech", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["d", " ", "e"]]]]]]]

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

 2002-12-18