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 sinh(d z)(a sinh2(e z)+b sinh(2 e z)+c cosh2(e z))-n





http://functions.wolfram.com/01.20.21.4305.01









  


  








Input Form





Integrate[Sinh[d z]/(a Sinh[e z]^2 + b Sinh[2 e z] + c Cosh[e z]^2), z] == (1/2) (((a + 2 b + c) E^((-d + 2 e) z) ((a - c + 2 Sqrt[b^2 - a c]) Hypergeometric2F1[1 - d/(2 e), 1, 2 - d/(2 e), ((a + 2 b + c) E^(2 e z))/(a - c - 2 Sqrt[b^2 - a c])] + (-a + c + 2 Sqrt[b^2 - a c]) Hypergeometric2F1[1 - d/(2 e), 1, 2 - d/(2 e), ((a + 2 b + c) E^(2 e z))/(a - c + 2 Sqrt[b^2 - a c])]))/ (Sqrt[b^2 - a c] (a - c + 2 Sqrt[b^2 - a c]) (-a + c + 2 Sqrt[b^2 - a c]) (-d + 2 e)) - ((a + 2 b + c) E^((d + 2 e) z) ((a - c + 2 Sqrt[b^2 - a c]) Hypergeometric2F1[1 + d/(2 e), 1, 2 + d/(2 e), ((a + 2 b + c) E^(2 e z))/(a - c - 2 Sqrt[b^2 - a c])] + (-a + c + 2 Sqrt[b^2 - a c]) Hypergeometric2F1[1 + d/(2 e), 1, 2 + d/(2 e), ((a + 2 b + c) E^(2 e z))/(a - c + 2 Sqrt[b^2 - a c])]))/ (Sqrt[b^2 - a c] (a - c + 2 Sqrt[b^2 - a c]) (-a + c + 2 Sqrt[b^2 - a c]) (d + 2 e)))








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Sinh", "[", RowBox[List["d", " ", "z"]], "]"]], RowBox[List[RowBox[List["a", " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]], "2"]]], "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", "e", " ", "z"]], "]"]]]], "+", RowBox[List["c", " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]], "2"]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", RowBox[List["2", " ", "e"]]]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "c", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "-", FractionBox["d", RowBox[List["2", " ", "e"]]]]], ",", "1", ",", RowBox[List["2", "-", FractionBox["d", RowBox[List["2", " ", "e"]]]]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List["a", "-", "c", "-", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "c", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "-", FractionBox["d", RowBox[List["2", " ", "e"]]]]], ",", "1", ",", RowBox[List["2", "-", FractionBox["d", RowBox[List["2", " ", "e"]]]]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List["a", "-", "c", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]]]]], "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]], " ", RowBox[List["(", RowBox[List["a", "-", "c", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "c", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", RowBox[List["2", " ", "e"]]]], ")"]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["d", "+", RowBox[List["2", " ", "e"]]]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "c", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", FractionBox["d", RowBox[List["2", " ", "e"]]]]], ",", "1", ",", RowBox[List["2", "+", FractionBox["d", RowBox[List["2", " ", "e"]]]]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List["a", "-", "c", "-", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "c", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", FractionBox["d", RowBox[List["2", " ", "e"]]]]], ",", "1", ",", RowBox[List["2", "+", FractionBox["d", RowBox[List["2", " ", "e"]]]]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List["a", "-", "c", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]]]]], "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]], " ", RowBox[List["(", RowBox[List["a", "-", "c", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "c", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["2", " ", "e"]]]], ")"]]]], ")"]]]]]], ")"]]]]]]]]








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> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sinh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> cosh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> </mrow> <mo> - </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> c </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> d </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> </mrow> </mfrac> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mfrac> <mi> d </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> </mrow> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, FractionBox[&quot;d&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;e&quot;]]]]], Hypergeometric2F1], &quot;,&quot;, TagBox[&quot;1&quot;, Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[&quot;2&quot;, &quot;-&quot;, FractionBox[&quot;d&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;e&quot;]]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[FractionBox[RowBox[List[RowBox[List[&quot;(&quot;, RowBox[List[&quot;a&quot;, &quot;+&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;b&quot;]], &quot;+&quot;, &quot;c&quot;]], &quot;)&quot;]], &quot; &quot;, SuperscriptBox[&quot;\[ExponentialE]&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;e&quot;, &quot; &quot;, &quot;z&quot;]]]]], RowBox[List[&quot;a&quot;, &quot;-&quot;, &quot;c&quot;, &quot;+&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, SqrtBox[RowBox[List[SuperscriptBox[&quot;b&quot;, &quot;2&quot;], &quot;-&quot;, RowBox[List[&quot;a&quot;, &quot; &quot;, &quot;c&quot;]]]]]]]]]], Hypergeometric2F1]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> d </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> </mrow> </mfrac> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mfrac> <mi> d </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> </mrow> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, FractionBox[&quot;d&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;e&quot;]]]]], Hypergeometric2F1], &quot;,&quot;, TagBox[&quot;1&quot;, Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[&quot;2&quot;, &quot;-&quot;, FractionBox[&quot;d&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;e&quot;]]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[FractionBox[RowBox[List[RowBox[List[&quot;(&quot;, RowBox[List[&quot;a&quot;, &quot;+&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;b&quot;]], &quot;+&quot;, &quot;c&quot;]], &quot;)&quot;]], &quot; &quot;, SuperscriptBox[&quot;\[ExponentialE]&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;e&quot;, &quot; &quot;, &quot;z&quot;]]]]], RowBox[List[&quot;a&quot;, &quot;-&quot;, &quot;c&quot;, &quot;-&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, SqrtBox[RowBox[List[SuperscriptBox[&quot;b&quot;, &quot;2&quot;], &quot;-&quot;, RowBox[List[&quot;a&quot;, &quot; &quot;, &quot;c&quot;]]]]]]]]]], Hypergeometric2F1]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> c </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> </mrow> <mo> - </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> c </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mfrac> <mi> d </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mfrac> <mi> d </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> </mrow> </mfrac> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ; </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[&quot;d&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;e&quot;]]], &quot;+&quot;, &quot;1&quot;]], Hypergeometric2F1], &quot;,&quot;, TagBox[&quot;1&quot;, Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[FractionBox[&quot;d&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;e&quot;]]], &quot;+&quot;, &quot;2&quot;]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[FractionBox[RowBox[List[RowBox[List[&quot;(&quot;, RowBox[List[&quot;a&quot;, &quot;+&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;b&quot;]], &quot;+&quot;, &quot;c&quot;]], &quot;)&quot;]], &quot; &quot;, SuperscriptBox[&quot;\[ExponentialE]&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;e&quot;, &quot; &quot;, &quot;z&quot;]]]]], RowBox[List[&quot;a&quot;, &quot;-&quot;, &quot;c&quot;, &quot;+&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, SqrtBox[RowBox[List[SuperscriptBox[&quot;b&quot;, &quot;2&quot;], &quot;-&quot;, RowBox[List[&quot;a&quot;, &quot; &quot;, &quot;c&quot;]]]]]]]]]], Hypergeometric2F1]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mfrac> <mi> d </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mfrac> <mi> d </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> </mrow> </mfrac> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ; </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[&quot;d&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;e&quot;]]], &quot;+&quot;, &quot;1&quot;]], Hypergeometric2F1], &quot;,&quot;, TagBox[&quot;1&quot;, Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[FractionBox[&quot;d&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;e&quot;]]], &quot;+&quot;, &quot;2&quot;]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[FractionBox[RowBox[List[RowBox[List[&quot;(&quot;, RowBox[List[&quot;a&quot;, &quot;+&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;b&quot;]], &quot;+&quot;, &quot;c&quot;]], &quot;)&quot;]], &quot; &quot;, SuperscriptBox[&quot;\[ExponentialE]&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;e&quot;, &quot; &quot;, &quot;z&quot;]]]]], RowBox[List[&quot;a&quot;, &quot;-&quot;, &quot;c&quot;, &quot;-&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, SqrtBox[RowBox[List[SuperscriptBox[&quot;b&quot;, &quot;2&quot;], &quot;-&quot;, RowBox[List[&quot;a&quot;, &quot; &quot;, &quot;c&quot;]]]]]]]]]], Hypergeometric2F1]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> c </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <sinh /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <sinh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sinh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> c </ci> <apply> <power /> <apply> <cosh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <ci> c </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> c </ci> <apply> <times /> <cn type='integer'> 2 </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 /> <ci> a </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> d </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> d </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <ci> c </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> 2 </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 /> <ci> a </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> 2 </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 /> <ci> a </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> d </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> d </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <ci> c </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </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 /> <ci> a </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <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 /> <ci> a </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> 2 </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 /> <ci> a </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> c </ci> <apply> <times /> <cn type='integer'> 2 </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 /> <ci> a </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <ci> c </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> c </ci> <apply> <times /> <cn type='integer'> 2 </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 /> <ci> a </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <apply> <times /> <ci> d </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> <apply> <plus /> <apply> <times /> <ci> d </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <ci> c </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> 2 </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 /> <ci> a </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> 2 </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 /> <ci> a </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <apply> <times /> <ci> d </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> <apply> <plus /> <apply> <times /> <ci> d </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <ci> c </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </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 /> <ci> a </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <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 /> <ci> a </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> 2 </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 /> <ci> a </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> c </ci> <apply> <times /> <cn type='integer'> 2 </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 /> <ci> a </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </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[RowBox[List["Sinh", "[", RowBox[List["d_", " ", "z_"]], "]"]], RowBox[List[RowBox[List["a_", " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["e_", " ", "z_"]], "]"]], "2"]]], "+", RowBox[List["b_", " ", RowBox[List["Sinh", "[", RowBox[List["2", " ", "e_", " ", "z_"]], "]"]]]], "+", RowBox[List["c_", " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["e_", " ", "z_"]], "]"]], "2"]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", RowBox[List["2", " ", "e"]]]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "c", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "-", FractionBox["d", RowBox[List["2", " ", "e"]]]]], ",", "1", ",", RowBox[List["2", "-", FractionBox["d", RowBox[List["2", " ", "e"]]]]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List["a", "-", "c", "-", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "c", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "-", FractionBox["d", RowBox[List["2", " ", "e"]]]]], ",", "1", ",", RowBox[List["2", "-", FractionBox["d", RowBox[List["2", " ", "e"]]]]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List["a", "-", "c", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]]]]], "]"]]]]]], ")"]]]], RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]], " ", RowBox[List["(", RowBox[List["a", "-", "c", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "c", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", RowBox[List["2", " ", "e"]]]], ")"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["d", "+", RowBox[List["2", " ", "e"]]]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "c", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", FractionBox["d", RowBox[List["2", " ", "e"]]]]], ",", "1", ",", RowBox[List["2", "+", FractionBox["d", RowBox[List["2", " ", "e"]]]]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List["a", "-", "c", "-", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "c", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", FractionBox["d", RowBox[List["2", " ", "e"]]]]], ",", "1", ",", RowBox[List["2", "+", FractionBox["d", RowBox[List["2", " ", "e"]]]]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "z"]]]]], RowBox[List["a", "-", "c", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]]]]], "]"]]]]]], ")"]]]], RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]], " ", RowBox[List["(", RowBox[List["a", "-", "c", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "c", "+", RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["a", " ", "c"]]]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["2", " ", "e"]]]], ")"]]]]]]], ")"]]]]]]]]


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





2002-12-18