|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/01.19.21.2266.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Integrate[E^(c z)/(a + b Sinh[c z]^2)^2, z] ==
-(Sqrt[2 a - 2 Sqrt[a] Sqrt[a - b] - b] (2 a + Sqrt[a] Sqrt[a - b] - b)
ArcTan[(Sqrt[b] E^(c z))/Sqrt[2 a - 2 Sqrt[a] Sqrt[a - b] - b]] +
(-2 Sqrt[a] Sqrt[a - b] Sqrt[b] E^(c z) (b + 2 a E^(2 c z) -
b E^(2 c z)) + Sqrt[2 a + 2 Sqrt[a] Sqrt[a - b] - b]
(-2 a + Sqrt[a] Sqrt[a - b] + b) (4 a E^(2 c z) +
b (-1 + E^(2 c z))^2) ArcTan[(Sqrt[b] E^(c z))/
Sqrt[2 a + 2 Sqrt[a] Sqrt[a - b] - b]])/(4 a E^(2 c z) +
b (-1 + E^(2 c z))^2))/(2 a^(3/2) (a - b)^(3/2) Sqrt[b] c)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "a"]], "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "-", "b"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List[SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "-", "b"]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], SqrtBox[RowBox[List[RowBox[List["2", " ", "a"]], "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "-", "b"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]], " ", SqrtBox["b"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "-", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], ")"]]]], "+", RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "-", "b"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", RowBox[List[SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "+", "b"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], "2"]]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], SqrtBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "-", "b"]]]], "]"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], "2"]]]]], ")"]]]]]], ")"]]]], "/", RowBox[List["(", RowBox[List["2", " ", SuperscriptBox["a", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], RowBox[List["3", "/", "2"]]], " ", SqrtBox["b"], " ", "c"]], ")"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mfrac> <msup> <mi> ⅇ </mi> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> sinh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mi> b </mi> </msqrt> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mi> a </mi> </msqrt> </mrow> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <msqrt> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mi> a </mi> </msqrt> </mrow> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <msqrt> <mi> b </mi> </msqrt> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mi> a </mi> </msqrt> </mrow> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> a </mi> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mi> b </mi> </msqrt> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <msqrt> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mi> a </mi> </msqrt> </mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mi> a </mi> </msqrt> </mrow> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <msqrt> <mi> b </mi> </msqrt> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mi> a </mi> </msqrt> </mrow> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> </mfrac> <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> <power /> <exponentiale /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <sinh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </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> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </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 /> <cn type='integer'> -2 </cn> <ci> a </ci> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> b </ci> </apply> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <arctan /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <ci> b </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arctan /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["c_", " ", "z_"]]], SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]]]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "a"]], "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "-", "b"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List[SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "-", "b"]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], SqrtBox[RowBox[List[RowBox[List["2", " ", "a"]], "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "-", "b"]]]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]], " ", SqrtBox["b"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "-", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], ")"]]]], "+", RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "-", "b"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", RowBox[List[SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "+", "b"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], "2"]]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], SqrtBox[RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "-", "b"]]]]], "-", "b"]]]], "]"]]]]]], RowBox[List[RowBox[List["4", " ", "a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], "2"]]]]]]]], RowBox[List["2", " ", SuperscriptBox["a", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], RowBox[List["3", "/", "2"]]], " ", SqrtBox["b"], " ", "c"]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
){kind=small}
