Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

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
 











Sech






Mathematica Notation

Traditional Notation









Elementary Functions > Sech[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 coth > Involving (a coth(z)+b sech(z))-n





http://functions.wolfram.com/01.24.21.0458.01









  


  










Input Form





Integrate[1/(a Coth[z] + b Sech[z])^2, z] == (Csch[z]^2 Sech[z]^2 (a + a Cosh[2 z] + 2 b Sinh[z])^2 (-((Sqrt[2] Sqrt[(-b) (b + Sqrt[-4 a^2 + b^2])] (-4 a^4 + a^2 b (7 b - 5 Sqrt[-4 a^2 + b^2]) + b^3 (-b + Sqrt[-4 a^2 + b^2])) ArcTan[(2 a + (-b + Sqrt[-4 a^2 + b^2]) Tanh[z/2])/ (Sqrt[2] Sqrt[b] Sqrt[-b + Sqrt[-4 a^2 + b^2]])] + Sqrt[b] Sqrt[-b + Sqrt[-4 a^2 + b^2]] ((-(-4 a^2 + b^2)^(3/2)) Sqrt[(-b) (b + Sqrt[-4 a^2 + b^2])] z + Sqrt[2] (4 a^4 + b^3 (b + Sqrt[-4 a^2 + b^2]) - a^2 b (7 b + 5 Sqrt[-4 a^2 + b^2])) ArcTan[(2 a - (b + Sqrt[-4 a^2 + b^2]) Tanh[z/2])/ (Sqrt[2] Sqrt[(-b) (b + Sqrt[-4 a^2 + b^2])])]))/ (Sqrt[b] (-4 a^2 + b^2)^(3/2) Sqrt[-b + Sqrt[-4 a^2 + b^2]] Sqrt[(-b) (b + Sqrt[-4 a^2 + b^2])])) - (2 a Cosh[z] ((-a) b + (2 a^2 - b^2) Sinh[z]))/ ((4 a^2 - b^2) (a + a Cosh[2 z] + 2 b Sinh[z]))))/ (4 a^2 (a Coth[z] + b Sech[z])^2)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["Coth", "[", "z", "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sech", "[", "z", "]"]]]]]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Csch", "[", "z", "]"]], "2"], " ", SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["a", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List["2", " ", "b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], ")"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "4"]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["7", " ", "b"]], "-", RowBox[List["5", " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["b", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]], " ", RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]]]]]], RowBox[List[SqrtBox["2"], " ", SqrtBox["b"], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]]]]]], "]"]]]], "+", RowBox[List[SqrtBox["b"], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], RowBox[List["3", "/", "2"]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]]]]], " ", "z"]], "+", RowBox[List[SqrtBox["2"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "4"]]], "+", RowBox[List[SuperscriptBox["b", "3"], " ", RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["7", " ", "b"]], "+", RowBox[List["5", " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "-", RowBox[List[RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]], " ", RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]]]]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]]]]]]]], "]"]]]]]], ")"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox["b"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], RowBox[List["3", "/", "2"]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]]]]]]], ")"]]]]]], "-", FractionBox[RowBox[List["2", " ", "a", " ", RowBox[List["Cosh", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], " ", "b"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "2"]]], "-", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "-", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["a", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List["2", " ", "b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], ")"]]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["4", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["Coth", "[", "z", "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sech", "[", "z", "]"]]]]]], ")"]], "2"]]], ")"]]]]]]]]










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> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mi> coth </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sech </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mi> coth </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sech </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> csch </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> sech </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> a </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <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> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> b </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </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> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> <mo> - </mo> <mi> b </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> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <msqrt> <mi> b </mi> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msqrt> <mi> b </mi> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> b </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </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> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> </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> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mi> b </mi> </msqrt> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mrow> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> a </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <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> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <coth /> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sech /> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <coth /> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sech /> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <csch /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <sech /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <ci> a </ci> </apply> <ci> a </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <apply> <sinh /> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <plus /> <ci> b </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 7 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> </apply> <apply> <arctan /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </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> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 7 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <ci> b </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <arctan /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </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> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <plus /> <ci> b </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <plus /> <ci> b </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <plus /> <ci> b </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <apply> <cosh /> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </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> <apply> <sinh /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> b </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </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> <apply> <plus /> <apply> <times /> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <ci> a </ci> </apply> <ci> a </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <apply> <sinh /> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a_", " ", RowBox[List["Coth", "[", "z_", "]"]]]], "+", RowBox[List["b_", " ", RowBox[List["Sech", "[", "z_", "]"]]]]]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Csch", "[", "z", "]"]], "2"], " ", SuperscriptBox[RowBox[List["Sech", "[", "z", "]"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["a", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List["2", " ", "b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], ")"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "4"]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["7", " ", "b"]], "-", RowBox[List["5", " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["b", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]], " ", RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]]]]]], RowBox[List[SqrtBox["2"], " ", SqrtBox["b"], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]]]]]], "]"]]]], "+", RowBox[List[SqrtBox["b"], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], RowBox[List["3", "/", "2"]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]]]]], " ", "z"]], "+", RowBox[List[SqrtBox["2"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "4"]]], "+", RowBox[List[SuperscriptBox["b", "3"], " ", RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["7", " ", "b"]], "+", RowBox[List["5", " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "a"]], "-", RowBox[List[RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]], " ", RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]]]]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]]]]]]]], "]"]]]]]], ")"]]]]]], RowBox[List[SqrtBox["b"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], RowBox[List["3", "/", "2"]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]], ")"]]]]]]]]]], "-", FractionBox[RowBox[List["2", " ", "a", " ", RowBox[List["Cosh", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], " ", "b"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "2"]]], "-", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["a", "2"]]], "-", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["a", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List["2", " ", "b", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]], ")"]]]]]]], ")"]]]], RowBox[List["4", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["Coth", "[", "z", "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sech", "[", "z", "]"]]]]]], ")"]], "2"]]]]]]]]










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





2002-12-18





© 1998-2014 Wolfram Research, Inc.