Hyperbolic cosine integral: Integration (formula 01.20.21.4408)

Cosh

$Mathematica Notation$

Elementary Functions

Cosh[z]

Integration

Indefinite integration

Involving functions of the direct function and hyperbolic functions

Involving algebraic functions of the direct function and hyperbolic functions

Involving algebraic functions of sinh

Involving (a+b sinh(e z)+c cosh(e z))^beta

http://functions.wolfram.com/01.20.21.4408.01

Input Form

Integrate[1/(a + b Sinh[e z] + c Cosh[e z])^(3/2), z] == ((2 (a c + (-b^2 + c^2) Cosh[e z]))/b + (b ((-(b^2 - 2 c^2)) Cosh[e z] + c (2 a + b Sinh[e z])))/(-b^2 + c^2) + (b^2 AppellF1[-(1/2), -(1/2), -(1/2), 1/2, (a + c Cosh[e z] + b Sinh[e z])/ (a + Sqrt[1 - b^2/c^2] c), (a + c Cosh[e z] + b Sinh[e z])/ (a - Sqrt[1 - b^2/c^2] c)] Sinh[e z + ArcTanh[b/c]])/ (Sqrt[1 - b^2/c^2] c Sqrt[(Sqrt[1 - b^2/c^2] c (1 + Cosh[e z + ArcTanh[b/c]]))/(-a + Sqrt[1 - b^2/c^2] c)] Sqrt[(Sqrt[1 - b^2/c^2] c - c Cosh[e z] - b Sinh[e z])/ (a + Sqrt[1 - b^2/c^2] c)]) - (c AppellF1[-(1/2), -(1/2), -(1/2), 1/2, (a + c Cosh[e z] + b Sinh[e z])/ (a + Sqrt[1 - b^2/c^2] c), (a + c Cosh[e z] + b Sinh[e z])/ (a - Sqrt[1 - b^2/c^2] c)] Sinh[e z + ArcTanh[b/c]])/ (Sqrt[1 - b^2/c^2] Sqrt[(Sqrt[1 - b^2/c^2] c (1 + Cosh[e z + ArcTanh[b/c]]))/(-a + Sqrt[1 - b^2/c^2] c)] Sqrt[(Sqrt[1 - b^2/c^2] c - c Cosh[e z] - b Sinh[e z])/ (a + Sqrt[1 - b^2/c^2] c)]) - (1/(b Sqrt[1 - b^2/c^2] (-b^2 + c^2))) (c (2 a Sqrt[1 - b^2/c^2] c^2 + 2 c (-b^2 + c^2) Cosh[e z + ArcTanh[b/c]] + b (b^2 - c^2) Sinh[e z + ArcTanh[b/c]])) + (1/(b Sqrt[1 - c^2/b^2])) (2 a AppellF1[1/2, 1/2, 1/2, 3/2, (a + c Cosh[e z] + b Sinh[e z])/(a + I b Sqrt[1 - c^2/b^2]), (a + c Cosh[e z] + b Sinh[e z])/(a - I b Sqrt[1 - c^2/b^2])] Sech[e z + ArcTanh[c/b]] (a + c Cosh[e z] + b Sinh[e z]) Sqrt[-((c Cosh[e z] + b ((-I) Sqrt[1 - c^2/b^2] + Sinh[e z]))/ (a + I b Sqrt[1 - c^2/b^2]))] Sqrt[-((b Sqrt[1 - c^2/b^2] (I + Sinh[e z + ArcTanh[c/b]]))/ (a - I b Sqrt[1 - c^2/b^2]))]))/((a^2 + b^2 - c^2) e Sqrt[a + c Cosh[e z] + b Sinh[e z]])

Standard Form

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MathML Form

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<msup> <mi> c </mi> <mn> 2 </mn> </msup> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mrow> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </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> sinh </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> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sech </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> tanh </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> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </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> sinh </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> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </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> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mrow> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> + </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> tanh </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> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> tanh </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> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> tanh </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> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mrow> <mrow> <mo> - </mo> <mrow> <mi> cosh </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> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> sinh </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> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> </mrow> </mrow> </mfrac> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <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> cosh </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> sinh </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> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </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> sinh </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> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> </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> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> </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> c </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> tanh </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> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> tanh </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> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> tanh </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> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> tanh </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> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mrow> <mrow> <mo> - </mo> <mrow> <mi> cosh </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> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> sinh </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> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> </mrow> </mrow> </mfrac> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <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> cosh </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> sinh </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> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </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> sinh </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> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <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> c </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cosh </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> 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Rule Form

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

2002-12-18