Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Cosh






Mathematica Notation

Traditional Notation









Elementary Functions > Cosh[z] > Integration > Indefinite integration > Involving functions of the direct function and hyperbolic functions > Involving rational functions of the direct function and hyperbolic functions > Involving rational functions of sinh > Involving (a+b sinh(e z)+c cosh(e z))-n





http://functions.wolfram.com/01.20.21.4282.01









  


  










Input Form





Integrate[(A + B Sinh[z] + C Cosh[z])/((a + b Sinh[z] + c Cosh[z]) (\[Alpha] + \[Beta] Sinh[z] + \[Gamma] Cosh[z])), z] == (2 (((-a) b B + A (b^2 - c^2) + a c C) \[Alpha] + ((-a) A b + a^2 B - B c^2 + b c C) \[Beta] + (a A c + b B c - a^2 C - b^2 C) \[Gamma]) ArcTan[(b + (-a + c) Tanh[z/2])/Sqrt[-a^2 - b^2 + c^2]])/ (Sqrt[-a^2 - b^2 + c^2] ((b^2 - c^2) \[Alpha]^2 + (a^2 - c^2) \[Beta]^2 + 2 b c \[Beta] \[Gamma] - (a^2 + b^2) \[Gamma]^2 + \[Alpha] (-2 a b \[Beta] + 2 a c \[Gamma]))) + (2 ((b B - c C) \[Alpha]^2 + (a A - c C) \[Beta]^2 + (B c + b C) \[Beta] \[Gamma] - (a A + b B) \[Gamma]^2 + \[Alpha] (((-A) b - a B) \[Beta] + (A c + a C) \[Gamma])) ArcTan[(\[Beta] + (-\[Alpha] + \[Gamma]) Tanh[z/2])/ Sqrt[-\[Alpha]^2 - \[Beta]^2 + \[Gamma]^2]])/ (Sqrt[-\[Alpha]^2 - \[Beta]^2 + \[Gamma]^2] ((b^2 - c^2) \[Alpha]^2 + (a^2 - c^2) \[Beta]^2 + 2 b c \[Beta] \[Gamma] - (a^2 + b^2) \[Gamma]^2 + \[Alpha] (-2 a b \[Beta] + 2 a c \[Gamma]))) + (((B c - b C) \[Alpha] + ((-A) c + a C) \[Beta] + (A b - a B) \[Gamma]) Log[a + c Cosh[z] + b Sinh[z]])/((-b^2 + c^2) \[Alpha]^2 + (-a^2 + c^2) \[Beta]^2 - 2 b c \[Beta] \[Gamma] + (a^2 + b^2) \[Gamma]^2 + 2 a \[Alpha] (b \[Beta] - c \[Gamma])) - (((B c - b C) \[Alpha] + ((-A) c + a C) \[Beta] + (A b - a B) \[Gamma]) Log[\[Alpha] + \[Gamma] Cosh[z] + \[Beta] Sinh[z]])/ ((-b^2 + c^2) \[Alpha]^2 + (-a^2 + c^2) \[Beta]^2 - 2 b c \[Beta] \[Gamma] + (a^2 + b^2) \[Gamma]^2 + 2 a \[Alpha] (b \[Beta] - c \[Gamma]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["A", "+", RowBox[List["B", " ", RowBox[List["Sinh", "[", "z", "]"]]]], "+", RowBox[List["C", " ", RowBox[List["Cosh", "[", "z", "]"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", "z", "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Cosh", "[", "z", "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", RowBox[List["\[Beta]", " ", RowBox[List["Sinh", "[", "z", "]"]]]], "+", RowBox[List["\[Gamma]", " ", RowBox[List["Cosh", "[", "z", "]"]]]]]], ")"]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], " ", "b", " ", "B"]], "+", RowBox[List["A", " ", RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "-", SuperscriptBox["c", "2"]]], ")"]]]], "+", RowBox[List["a", " ", "c", " ", "C"]]]], ")"]], " ", "\[Alpha]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], " ", "A", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "B"]], "-", RowBox[List["B", " ", SuperscriptBox["c", "2"]]], "+", RowBox[List["b", " ", "c", " ", "C"]]]], ")"]], " ", "\[Beta]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["a", " ", "A", " ", "c"]], "+", RowBox[List["b", " ", "B", " ", "c"]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", "C"]], "-", RowBox[List[SuperscriptBox["b", "2"], " ", "C"]]]], ")"]], " ", "\[Gamma]"]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List["b", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "c"]], ")"]], " ", RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "-", SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "-", SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "-", SuperscriptBox["c", "2"]]], ")"]], " ", SuperscriptBox["\[Alpha]", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["c", "2"]]], ")"]], " ", SuperscriptBox["\[Beta]", "2"]]], "+", RowBox[List["2", " ", "b", " ", "c", " ", "\[Beta]", " ", "\[Gamma]"]], "-", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], " ", SuperscriptBox["\[Gamma]", "2"]]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a", " ", "b", " ", "\[Beta]"]], "+", RowBox[List["2", " ", "a", " ", "c", " ", "\[Gamma]"]]]], ")"]]]]]], ")"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["b", " ", "B"]], "-", RowBox[List["c", " ", "C"]]]], ")"]], " ", SuperscriptBox["\[Alpha]", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["a", " ", "A"]], "-", RowBox[List["c", " ", "C"]]]], ")"]], " ", SuperscriptBox["\[Beta]", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["B", " ", "c"]], "+", RowBox[List["b", " ", "C"]]]], ")"]], " ", "\[Beta]", " ", "\[Gamma]"]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["a", " ", "A"]], "+", RowBox[List["b", " ", "B"]]]], ")"]], " ", SuperscriptBox["\[Gamma]", "2"]]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "A"]], " ", "b"]], "-", RowBox[List["a", " ", "B"]]]], ")"]], " ", "\[Beta]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["A", " ", "c"]], "+", RowBox[List["a", " ", "C"]]]], ")"]], " ", "\[Gamma]"]]]], ")"]]]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List["\[Beta]", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[Alpha]"]], "+", "\[Gamma]"]], ")"]], " ", RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["\[Alpha]", "2"]]], "-", SuperscriptBox["\[Beta]", "2"], "+", SuperscriptBox["\[Gamma]", "2"]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["\[Alpha]", "2"]]], "-", SuperscriptBox["\[Beta]", "2"], "+", SuperscriptBox["\[Gamma]", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "-", SuperscriptBox["c", "2"]]], ")"]], " ", SuperscriptBox["\[Alpha]", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["c", "2"]]], ")"]], " ", SuperscriptBox["\[Beta]", "2"]]], "+", RowBox[List["2", " ", "b", " ", "c", " ", "\[Beta]", " ", "\[Gamma]"]], "-", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], " ", SuperscriptBox["\[Gamma]", "2"]]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a", " ", "b", " ", "\[Beta]"]], "+", RowBox[List["2", " ", "a", " ", "c", " ", "\[Gamma]"]]]], ")"]]]]]], ")"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["B", " ", "c"]], "-", RowBox[List["b", " ", "C"]]]], ")"]], " ", "\[Alpha]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "A"]], " ", "c"]], "+", RowBox[List["a", " ", "C"]]]], ")"]], " ", "\[Beta]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["A", " ", "b"]], "-", RowBox[List["a", " ", "B"]]]], ")"]], " ", "\[Gamma]"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], "+", SuperscriptBox["c", "2"]]], ")"]], " ", SuperscriptBox["\[Alpha]", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["c", "2"]]], ")"]], " ", SuperscriptBox["\[Beta]", "2"]]], "-", RowBox[List["2", " ", "b", " ", "c", " ", "\[Beta]", " ", "\[Gamma]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], " ", SuperscriptBox["\[Gamma]", "2"]]], "+", RowBox[List["2", " ", "a", " ", "\[Alpha]", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Beta]"]], "-", RowBox[List["c", " ", "\[Gamma]"]]]], ")"]]]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["B", " ", "c"]], "-", RowBox[List["b", " ", "C"]]]], ")"]], " ", "\[Alpha]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "A"]], " ", "c"]], "+", RowBox[List["a", " ", "C"]]]], ")"]], " ", "\[Beta]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["A", " ", "b"]], "-", RowBox[List["a", " ", "B"]]]], ")"]], " ", "\[Gamma]"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["\[Alpha]", "+", RowBox[List["\[Gamma]", " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List["\[Beta]", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], "+", SuperscriptBox["c", "2"]]], ")"]], " ", SuperscriptBox["\[Alpha]", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["c", "2"]]], ")"]], " ", SuperscriptBox["\[Beta]", "2"]]], "-", RowBox[List["2", " ", "b", " ", "c", " ", "\[Beta]", " ", "\[Gamma]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], " ", SuperscriptBox["\[Gamma]", "2"]]], "+", RowBox[List["2", " ", "a", " ", "\[Alpha]", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Beta]"]], "-", RowBox[List["c", " ", "\[Gamma]"]]]], ")"]]]]]], ")"]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8747; </mo> <mrow> <mfrac> <mrow> <mi> A </mi> <mo> + </mo> <mrow> <mi> C </mi> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> B </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#945; </mi> <mo> + </mo> <mrow> <mi> &#946; </mi> <mo> &#8290; </mo> <mrow> <mi> Sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> &#947; </mi> <mo> &#8290; </mo> <mrow> <mi> Cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mi> B </mi> </mrow> <mo> + </mo> <mrow> <mi> A </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> C </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#945; </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> B </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mi> B </mi> <mo> &#8290; </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> A </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> C </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#946; </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> C </mi> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> A </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> B </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> C </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#947; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#945; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> &#947; </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mi> &#946; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#945; </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#946; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> &#946; </mi> <mo> &#8290; </mo> <mi> &#947; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> B </mi> </mrow> <mo> - </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> C </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#945; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> A </mi> </mrow> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> B </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#946; </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> A </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> C </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#947; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#945; </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> A </mi> </mrow> <mo> - </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> C </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#946; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> A </mi> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> B </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> B </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> C </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#946; </mi> <mo> &#8290; </mo> <mi> &#947; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> &#946; </mi> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> &#947; </mi> <mo> - </mo> <mi> &#945; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <msup> <mi> &#945; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <msup> <mi> &#946; </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <msup> <mi> &#945; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <msup> <mi> &#946; </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#945; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> &#947; </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mi> &#946; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#945; </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#946; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> &#946; </mi> <mo> &#8290; </mo> <mi> &#947; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> B </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> C </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#945; </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> C </mi> </mrow> <mo> - </mo> <mrow> <mi> A </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#946; </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> A </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> B </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#947; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#945; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#946; </mi> </mrow> <mo> - </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> &#947; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#945; </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#946; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> &#946; </mi> <mo> &#8290; </mo> <mi> &#947; </mi> </mrow> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> B </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> C </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#945; </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> C </mi> </mrow> <mo> - </mo> <mrow> <mi> A </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#946; </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> A </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> B </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#947; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#945; </mi> <mo> + </mo> <mrow> <mi> &#947; </mi> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> &#946; </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#945; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#946; </mi> </mrow> <mo> - </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> &#947; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#945; </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#946; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> &#946; </mi> <mo> &#8290; </mo> <mi> &#947; </mi> </mrow> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <plus /> <ci> A </ci> <apply> <times /> <ci> C </ci> <apply> <cosh /> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> B </ci> <apply> <sinh /> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> c </ci> <apply> <cosh /> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sinh /> <ci> z </ci> </apply> </apply> </apply> <apply> <plus /> <ci> &#945; </ci> <apply> <times /> <ci> &#946; </ci> <apply> <sinh /> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> &#947; </ci> <apply> <cosh /> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> b </ci> <ci> B </ci> </apply> <apply> <times /> <ci> A </ci> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> c </ci> <ci> C </ci> </apply> </apply> <ci> &#945; </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> B </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> B </ci> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> A </ci> <ci> b </ci> </apply> </apply> <apply> <times /> <ci> b </ci> <ci> c </ci> <ci> C </ci> </apply> </apply> <ci> &#946; </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> C </ci> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> A </ci> <ci> c </ci> <ci> a </ci> </apply> <apply> <times /> <ci> b </ci> <ci> B </ci> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <ci> C </ci> </apply> </apply> </apply> <ci> &#947; </ci> </apply> </apply> <apply> <arctan /> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <tanh /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> &#945; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> c </ci> <ci> &#947; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> b </ci> <ci> &#946; </ci> </apply> </apply> </apply> <ci> &#945; </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> &#946; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> &#947; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> c </ci> <ci> &#946; </ci> <ci> &#947; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> b </ci> <ci> B </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> c </ci> <ci> C </ci> </apply> </apply> </apply> <apply> <power /> <ci> &#945; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> A </ci> </apply> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> B </ci> </apply> </apply> </apply> <ci> &#946; </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> A </ci> <ci> c </ci> </apply> <apply> <times /> <ci> a </ci> <ci> C </ci> </apply> </apply> <ci> &#947; </ci> </apply> </apply> <ci> &#945; </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> A </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> c </ci> <ci> C </ci> </apply> </apply> </apply> <apply> <power /> <ci> &#946; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> A </ci> </apply> <apply> <times /> <ci> b </ci> <ci> B </ci> </apply> </apply> <apply> <power /> <ci> &#947; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> B </ci> <ci> c </ci> </apply> <apply> <times /> <ci> b </ci> <ci> C </ci> </apply> </apply> <ci> &#946; </ci> <ci> &#947; </ci> </apply> </apply> <apply> <arctan /> <apply> <times /> <apply> <plus /> <ci> &#946; </ci> <apply> <times /> <apply> <plus /> <ci> &#947; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> <apply> <tanh /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> &#945; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> &#946; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> &#947; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> &#945; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> &#946; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> &#947; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> &#945; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> c </ci> <ci> &#947; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> b </ci> <ci> &#946; </ci> </apply> </apply> </apply> <ci> &#945; </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> &#946; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> &#947; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> c </ci> <ci> &#946; </ci> <ci> &#947; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> B </ci> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <ci> C </ci> </apply> </apply> </apply> <ci> &#945; </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> C </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> A </ci> <ci> c </ci> </apply> </apply> </apply> <ci> &#946; </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> A </ci> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> B </ci> </apply> </apply> </apply> <ci> &#947; </ci> </apply> </apply> <apply> <ln /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> c </ci> <apply> <cosh /> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sinh /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> &#945; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <apply> <plus /> <apply> <times /> <ci> b </ci> <ci> &#946; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> c </ci> <ci> &#947; </ci> </apply> </apply> </apply> <ci> &#945; </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> &#946; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> &#947; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> c </ci> <ci> &#946; </ci> <ci> &#947; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> B </ci> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <ci> C </ci> </apply> </apply> </apply> <ci> &#945; </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> C </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> A </ci> <ci> c </ci> </apply> </apply> </apply> <ci> &#946; </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> A </ci> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> B </ci> </apply> </apply> </apply> <ci> &#947; </ci> </apply> </apply> <apply> <ln /> <apply> <plus /> <ci> &#945; </ci> <apply> <times /> <ci> &#947; </ci> <apply> <cosh /> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> &#946; </ci> <apply> <sinh /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> &#945; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <apply> <plus /> <apply> <times /> <ci> b </ci> <ci> &#946; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> c </ci> <ci> &#947; </ci> </apply> </apply> </apply> <ci> &#945; </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> &#946; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> &#947; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> c </ci> <ci> &#946; </ci> <ci> &#947; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["A_", "+", RowBox[List["B_", " ", RowBox[List["Sinh", "[", "z_", "]"]]]], "+", RowBox[List["C", " ", RowBox[List["Cosh", "[", "z_", "]"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Sinh", "[", "z_", "]"]]]], "+", RowBox[List["c_", " ", RowBox[List["Cosh", "[", "z_", "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["\[Alpha]_", "+", RowBox[List["\[Beta]_", " ", RowBox[List["Sinh", "[", "z_", "]"]]]], "+", RowBox[List["\[Gamma]_", " ", RowBox[List["Cosh", "[", "z_", "]"]]]]]], ")"]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], " ", "b", " ", "B"]], "+", RowBox[List["A", " ", RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "-", SuperscriptBox["c", "2"]]], ")"]]]], "+", RowBox[List["a", " ", "c", " ", "C"]]]], ")"]], " ", "\[Alpha]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], " ", "A", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "B"]], "-", RowBox[List["B", " ", SuperscriptBox["c", "2"]]], "+", RowBox[List["b", " ", "c", " ", "C"]]]], ")"]], " ", "\[Beta]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["a", " ", "A", " ", "c"]], "+", RowBox[List["b", " ", "B", " ", "c"]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", "C"]], "-", RowBox[List[SuperscriptBox["b", "2"], " ", "C"]]]], ")"]], " ", "\[Gamma]"]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List["b", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "c"]], ")"]], " ", RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "-", SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]]]], "]"]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "-", SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "-", SuperscriptBox["c", "2"]]], ")"]], " ", SuperscriptBox["\[Alpha]", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["c", "2"]]], ")"]], " ", SuperscriptBox["\[Beta]", "2"]]], "+", RowBox[List["2", " ", "b", " ", "c", " ", "\[Beta]", " ", "\[Gamma]"]], "-", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], " ", SuperscriptBox["\[Gamma]", "2"]]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a", " ", "b", " ", "\[Beta]"]], "+", RowBox[List["2", " ", "a", " ", "c", " ", "\[Gamma]"]]]], ")"]]]]]], ")"]]]]], "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["b", " ", "B"]], "-", RowBox[List["c", " ", "C"]]]], ")"]], " ", SuperscriptBox["\[Alpha]", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["a", " ", "A"]], "-", RowBox[List["c", " ", "C"]]]], ")"]], " ", SuperscriptBox["\[Beta]", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["B", " ", "c"]], "+", RowBox[List["b", " ", "C"]]]], ")"]], " ", "\[Beta]", " ", "\[Gamma]"]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["a", " ", "A"]], "+", RowBox[List["b", " ", "B"]]]], ")"]], " ", SuperscriptBox["\[Gamma]", "2"]]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "A"]], " ", "b"]], "-", RowBox[List["a", " ", "B"]]]], ")"]], " ", "\[Beta]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["A", " ", "c"]], "+", RowBox[List["a", " ", "C"]]]], ")"]], " ", "\[Gamma]"]]]], ")"]]]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List["\[Beta]", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[Alpha]"]], "+", "\[Gamma]"]], ")"]], " ", RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["\[Alpha]", "2"]]], "-", SuperscriptBox["\[Beta]", "2"], "+", SuperscriptBox["\[Gamma]", "2"]]]]], "]"]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["\[Alpha]", "2"]]], "-", SuperscriptBox["\[Beta]", "2"], "+", SuperscriptBox["\[Gamma]", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "-", SuperscriptBox["c", "2"]]], ")"]], " ", SuperscriptBox["\[Alpha]", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["c", "2"]]], ")"]], " ", SuperscriptBox["\[Beta]", "2"]]], "+", RowBox[List["2", " ", "b", " ", "c", " ", "\[Beta]", " ", "\[Gamma]"]], "-", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], " ", SuperscriptBox["\[Gamma]", "2"]]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a", " ", "b", " ", "\[Beta]"]], "+", RowBox[List["2", " ", "a", " ", "c", " ", "\[Gamma]"]]]], ")"]]]]]], ")"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["B", " ", "c"]], "-", RowBox[List["b", " ", "C"]]]], ")"]], " ", "\[Alpha]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "A"]], " ", "c"]], "+", RowBox[List["a", " ", "C"]]]], ")"]], " ", "\[Beta]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["A", " ", "b"]], "-", RowBox[List["a", " ", "B"]]]], ")"]], " ", "\[Gamma]"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], "+", SuperscriptBox["c", "2"]]], ")"]], " ", SuperscriptBox["\[Alpha]", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["c", "2"]]], ")"]], " ", SuperscriptBox["\[Beta]", "2"]]], "-", RowBox[List["2", " ", "b", " ", "c", " ", "\[Beta]", " ", "\[Gamma]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], " ", SuperscriptBox["\[Gamma]", "2"]]], "+", RowBox[List["2", " ", "a", " ", "\[Alpha]", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Beta]"]], "-", RowBox[List["c", " ", "\[Gamma]"]]]], ")"]]]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["B", " ", "c"]], "-", RowBox[List["b", " ", "C"]]]], ")"]], " ", "\[Alpha]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "A"]], " ", "c"]], "+", RowBox[List["a", " ", "C"]]]], ")"]], " ", "\[Beta]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["A", " ", "b"]], "-", RowBox[List["a", " ", "B"]]]], ")"]], " ", "\[Gamma]"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["\[Alpha]", "+", RowBox[List["\[Gamma]", " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List["\[Beta]", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], "+", SuperscriptBox["c", "2"]]], ")"]], " ", SuperscriptBox["\[Alpha]", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["c", "2"]]], ")"]], " ", SuperscriptBox["\[Beta]", "2"]]], "-", RowBox[List["2", " ", "b", " ", "c", " ", "\[Beta]", " ", "\[Gamma]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], " ", SuperscriptBox["\[Gamma]", "2"]]], "+", RowBox[List["2", " ", "a", " ", "\[Alpha]", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Beta]"]], "-", RowBox[List["c", " ", "\[Gamma]"]]]], ")"]]]]]]]]]]]]]










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





2002-12-18