Cosine: Integration (formula 01.07.21.2386)

Cos

$Mathematica Notation$

Elementary Functions

Cos[z]

Integration

Indefinite integration

Involving functions of the direct function and trigonometric functions

Involving algebraic functions of the direct function and trigonometric functions

Involving algebraic functions of sin

Involving (a+b sin(e z)+c cos(e z))^beta

http://functions.wolfram.com/01.07.21.2386.01

Input Form

Integrate[1/(a + b Sin[e z] + c Cos[e z])^(3/2), z] == ((2 (a c + (b^2 + c^2) Cos[e z]))/b + (1/(b Sqrt[1 + c^2/b^2])) (2 a AppellF1[1/2, 1/2, 1/2, 3/2, (a + c Cos[e z] + b Sin[e z])/ (a + b Sqrt[1 + c^2/b^2]), (a + c Cos[e z] + b Sin[e z])/ (a - b Sqrt[1 + c^2/b^2])] Sec[e z + ArcTan[c/b]] Sqrt[((-c) Cos[e z] + b (Sqrt[1 + c^2/b^2] - Sin[e z]))/ (a + b Sqrt[1 + c^2/b^2])] (a + c Cos[e z] + b Sin[e z]) Sqrt[(c Cos[e z] + b (Sqrt[1 + c^2/b^2] + Sin[e z]))/ (-a + b Sqrt[1 + c^2/b^2])]) - (b ((b^2 + 2 c^2) Cos[e z] + c (2 a + b Sin[e z])))/(b^2 + c^2) - (b^2 AppellF1[-(1/2), -(1/2), -(1/2), 1/2, (a + c Cos[e z] + b Sin[e z])/ (a + Sqrt[1 + b^2/c^2] c), (a + c Cos[e z] + b Sin[e z])/ (a - Sqrt[1 + b^2/c^2] c)] Sin[e z - ArcTan[b/c]])/ (Sqrt[1 + b^2/c^2] c Sqrt[(Sqrt[1 + b^2/c^2] c - c Cos[e z] - b Sin[e z])/ (a + Sqrt[1 + b^2/c^2] c)] Sqrt[(Sqrt[1 + b^2/c^2] c + c Cos[e z] + b Sin[e z])/(-a + Sqrt[1 + b^2/c^2] c)]) - (c AppellF1[-(1/2), -(1/2), -(1/2), 1/2, (a + c Cos[e z] + b Sin[e z])/ (a + Sqrt[1 + b^2/c^2] c), (a + c Cos[e z] + b Sin[e z])/ (a - Sqrt[1 + b^2/c^2] c)] Sin[e z - ArcTan[b/c]])/ (Sqrt[1 + b^2/c^2] Sqrt[(Sqrt[1 + b^2/c^2] c - c Cos[e z] - b Sin[e z])/ (a + Sqrt[1 + b^2/c^2] c)] Sqrt[(Sqrt[1 + b^2/c^2] c + c Cos[e z] + b Sin[e z])/(-a + Sqrt[1 + b^2/c^2] c)]) - (c (2 a Sqrt[1 + b^2/c^2] c^2 + 2 c (b^2 + c^2) Cos[e z - ArcTan[b/c]] - b (b^2 + c^2) Sin[e z - ArcTan[b/c]]))/(b Sqrt[1 + b^2/c^2] (b^2 + c^2)))/((a^2 - b^2 - c^2) e Sqrt[a + c Cos[e z] + b Sin[e z]])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", "c"]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]]]], "b"], "+", RowBox[List[FractionBox["1", RowBox[List["b", " ", SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]]]]], RowBox[List["(", RowBox[List["2", " ", "a", " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["3", "2"], ",", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]]]]]]], ",", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", RowBox[List["b", " ", SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]]]]]]]]], "]"]], " ", RowBox[List["Sec", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTan", "[", FractionBox["c", "b"], "]"]]]], "]"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]], "-", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]], ")"]]]]]], RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]]]]]]]], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]], "+", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]], ")"]]]]]], RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["b", " ", SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]]]]]]]]]], ")"]]]], "-", FractionBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "+", RowBox[List["2", " ", SuperscriptBox["c", "2"]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List[SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", FractionBox["1", "2"], ",", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]], ",", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["e", " ", "z"]], "-", RowBox[List["ArcTan", "[", FractionBox["b", "c"], "]"]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c", " ", SqrtBox[FractionBox[RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]], "-", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "-", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]], "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["c", " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", FractionBox["1", "2"], ",", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]], ",", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["e", " ", "z"]], "-", RowBox[List["ArcTan", "[", FractionBox["b", "c"], "]"]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]], "-", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "-", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]], "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a", " ", SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", SuperscriptBox["c", "2"]]], "+", RowBox[List["2", " ", "c", " ", RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["e", " ", "z"]], "-", RowBox[List["ArcTan", "[", FractionBox["b", "c"], "]"]]]], "]"]]]], "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["e", " ", "z"]], "-", RowBox[List["ArcTan", "[", FractionBox["b", "c"], "]"]]]], "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["b", " ", SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]], ")"]]]], ")"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"], "-", SuperscriptBox["c", "2"]]], ")"]], " ", "e", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]]]]], ")"]]]]]]]]

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> ∫ </mo> <mrow> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mfrac> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mrow> <semantics> <msub> <mi> F </mi> <mn> 1 </mn> </msub> <annotation-xml encoding='MathML-Content'> <ci> AppellF1 </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mfrac> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mfrac> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sec </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> c </mi> <mi> b </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mfrac> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> - </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mfrac> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <msqrt> <mrow> <mfrac> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mfrac> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> b </mi> <mi> c </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mrow> <mrow> <mo> - </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mrow> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <semantics> <msub> <mi> F </mi> <mn> 1 </mn> </msub> <annotation-xml encoding='MathML-Content'> <ci> AppellF1 </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> b </mi> <mi> c </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mrow> <mrow> <mo> - </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mrow> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <semantics> <msub> <mi> F </mi> <mn> 1 </mn> </msub> <annotation-xml encoding='MathML-Content'> <ci> AppellF1 </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> b </mi> <mi> c </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> b </mi> <mi> c </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> b </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> e </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </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 /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> c </ci> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> c </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <ci> b </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <apply> <ci> AppellF1 </ci> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> c </ci> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> c </ci> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sec /> <apply> <plus /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> <apply> <arctan /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> c </ci> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> c </ci> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> c </ci> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <plus /> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <sin /> <apply> <plus /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arctan /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <ci> c </ci> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> c </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <ci> c </ci> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> AppellF1 </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> c </ci> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> c </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> c </ci> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <ci> c </ci> <apply> <sin /> <apply> <plus /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arctan /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <ci> c </ci> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> c </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <ci> c </ci> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> AppellF1 </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> c </ci> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> c </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> c </ci> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <ci> c </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <cos /> <apply> <plus /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arctan /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <sin /> <apply> <plus /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arctan /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> c </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> e </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> c </ci> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </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["a_", "+", RowBox[List["b_", " ", RowBox[List["Sin", "[", RowBox[List["e_", " ", "z_"]], "]"]]]], "+", RowBox[List["c_", " ", RowBox[List["Cos", "[", RowBox[List["e_", " ", "z_"]], "]"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", "c"]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]]]], "b"], "+", FractionBox[RowBox[List["2", " ", "a", " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["3", "2"], ",", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]]]]]]], ",", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", RowBox[List["b", " ", SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]]]]]]]]], "]"]], " ", RowBox[List["Sec", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTan", "[", FractionBox["c", "b"], "]"]]]], "]"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]], "-", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]], ")"]]]]]], RowBox[List["a", "+", RowBox[List["b", " ", SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]]]]]]]], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]], "+", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]], ")"]]]]]], RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["b", " ", SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]]]]]]]]]], RowBox[List["b", " ", SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["c", "2"], SuperscriptBox["b", "2"]]]]]]]], "-", FractionBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "+", RowBox[List["2", " ", SuperscriptBox["c", "2"]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List[SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", FractionBox["1", "2"], ",", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]], ",", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["e", " ", "z"]], "-", RowBox[List["ArcTan", "[", FractionBox["b", "c"], "]"]]]], "]"]]]], RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c", " ", SqrtBox[FractionBox[RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]], "-", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "-", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]], "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]]]]]], "-", FractionBox[RowBox[List["c", " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", FractionBox["1", "2"], ",", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]], ",", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["e", " ", "z"]], "-", RowBox[List["ArcTan", "[", FractionBox["b", "c"], "]"]]]], "]"]]]], RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]], "-", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "-", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]], "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]]]]]], "-", FractionBox[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a", " ", SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", SuperscriptBox["c", "2"]]], "+", RowBox[List["2", " ", "c", " ", RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["e", " ", "z"]], "-", RowBox[List["ArcTan", "[", FractionBox["b", "c"], "]"]]]], "]"]]]], "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["e", " ", "z"]], "-", RowBox[List["ArcTan", "[", FractionBox["b", "c"], "]"]]]], "]"]]]]]], ")"]]]], RowBox[List["b", " ", SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]], ")"]]]]]]], RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"], "-", SuperscriptBox["c", "2"]]], ")"]], " ", "e", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]]]]]]]]]]

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

2002-12-18