html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Cosh

 http://functions.wolfram.com/01.20.21.4388.01

 Input Form

 Integrate[Cosh[e z]/((c Cosh[e z] + d Sinh[e z]) Sqrt[a + b Cosh[2 e z]]), z] == -((1/e) (I ((I d ArcTanh[(Sqrt[c^2 - d^2] Sqrt[a + b Cosh[2 e z]])/ Sqrt[(a - b) c^2 - (a + b) d^2]])/(Sqrt[c^2 - d^2] Sqrt[(a - b) c^2 - (a + b) d^2]) + (Sqrt[(a + b Cosh[2 e z])/(a + b)] (c^2 EllipticF[I e z, (2 b)/(a + b)] - d^2 EllipticPi[1 - d^2/c^2, I e z, (2 b)/(a + b)]))/((c^3 - c d^2) Sqrt[a + b Cosh[2 e z]]))))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["d", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]], SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List["-", RowBox[List[FractionBox["1", "e"], RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "d", " ", RowBox[List["ArcTanh", "[", FractionBox[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[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox["c", "2"]]], "-", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox["d", "2"]]]]]]], "]"]]]], RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["c", "2"], "-", SuperscriptBox["d", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox["c", "2"]]], "-", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox["d", "2"]]]]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[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[SuperscriptBox["c", "2"], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "e", " ", "z"]], ",", FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["d", "2"], " ", RowBox[List["EllipticPi", "[", RowBox[List[RowBox[List["1", "-", 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[RowBox[List["(", RowBox[List[SuperscriptBox["c", "3"], "-", RowBox[List["c", " ", SuperscriptBox["d", "2"]]]]], ")"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]]]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]]]

 MathML Form

 cosh ( e z ) ( c cosh ( e z ) + d sinh ( e z ) ) a + b cosh ( 2 e z ) z - 1 e ( ( d tanh - 1 ( c 2 - d 2 a + b cosh ( 2 e z ) ( a - b ) c 2 - ( a + b ) d 2 ) c 2 - d 2 ( a - b ) c 2 - ( a + b ) d 2 + ( a + b cosh ( 2 e z ) a + b ( c 2 F ( e z 2 b a + b ) - d 2 Π ( 1 - d 2 c 2 ; e z 2 b a + b ) ) ) / ( ( c 3 - c d 2 ) a + b cosh ( 2 e z ) ) ) ) z e z c e z d e z a b 2 e z 1 2 -1 -1 1 e -1 d c 2 -1 d 2 1 2 a b 2 e z 1 2 a -1 b c 2 -1 a b d 2 1 2 -1 c 2 -1 d 2 1 2 a -1 b c 2 -1 a b d 2 1 2 -1 a b 2 e z a b -1 1 2 c 2 EllipticF e z 2 b a b -1 -1 d 2 EllipticPi 1 -1 d 2 c 2 -1 e z 2 b a b -1 c 3 -1 c d 2 a b 2 e z 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Cosh", "[", RowBox[List["e_", " ", "z_"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["c_", " ", RowBox[List["Cosh", "[", RowBox[List["e_", " ", "z_"]], "]"]]]], "+", RowBox[List["d_", " ", RowBox[List["Sinh", "[", RowBox[List["e_", " ", "z_"]], "]"]]]]]], ")"]], " ", SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "e_", " ", "z_"]], "]"]]]]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "d", " ", RowBox[List["ArcTanh", "[", FractionBox[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[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox["c", "2"]]], "-", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox["d", "2"]]]]]]], "]"]]]], RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["c", "2"], "-", SuperscriptBox["d", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox["c", "2"]]], "-", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SuperscriptBox["d", "2"]]]]]]]]], "+", FractionBox[RowBox[List[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[SuperscriptBox["c", "2"], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "e", " ", "z"]], ",", FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["d", "2"], " ", RowBox[List["EllipticPi", "[", RowBox[List[RowBox[List["1", "-", 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["(", RowBox[List[SuperscriptBox["c", "3"], "-", RowBox[List["c", " ", SuperscriptBox["d", "2"]]]]], ")"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]]]]]]]]]], ")"]]]], "e"]]]]]]]

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

 2002-12-18