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http://functions.wolfram.com/01.30.16.0244.01
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a ArcSech[x] + b ArcCsc[y] ==
I Pi
(1 - (-1)^(Floor[-(Arg[(Sqrt[1/x - 1] Sqrt[1 + 1/x] + 1/x)^a/
(Sqrt[1 - 1/y^2] + I/y)^(I b) + 1]/(2 Pi))] -
Floor[-(Arg[(Sqrt[1/x - 1] Sqrt[1 + 1/x] + 1/x)^a/
(Sqrt[1 - 1/y^2] + I/y)^(I b)]/(2 Pi))])) +
(-1)^(-Floor[-(Arg[(Sqrt[1/x - 1] Sqrt[1 + 1/x] + 1/x)^a/
(Sqrt[1 - 1/y^2] + I/y)^(I b) - 1]/(2 Pi))] +
Floor[-(Arg[(Sqrt[1/x - 1] Sqrt[1 + 1/x] + 1/x)^a/
(Sqrt[1 - 1/y^2] + I/y)^(I b) + 1]/(2 Pi))] +
Floor[-(Arg[(Sqrt[1/x - 1] Sqrt[1 + 1/x] + 1/x)^a/
(Sqrt[1 - 1/y^2] + I/y)^(I b)]/Pi)] +
Floor[Arg[(Sqrt[1/x - 1] Sqrt[1 + 1/x] + 1/x)^a/(Sqrt[1 - 1/y^2] + I/y)^
(I b)]/Pi - (2 Arg[(Sqrt[1/x - 1] Sqrt[1 + 1/x] + 1/x)^a/
(Sqrt[1 - 1/y^2] + I/y)^(I b) - 1])/Pi])
ArcSech[(2 (Sqrt[1/x - 1] Sqrt[1 + 1/x] + 1/x)^a)/
((Sqrt[1 - 1/y^2] + I/y)^(I b) ((Sqrt[1/x - 1] Sqrt[1 + 1/x] + 1/x)^
(2 a)/(Sqrt[1 - 1/y^2] + I/y)^(2 I b) + 1))] -
2 I Pi (Floor[(-Arg[(Sqrt[1/x - 1] Sqrt[1 + 1/x] + 1/x)^a] -
Arg[(Sqrt[1 - 1/y^2] + I/y)^((-I) b)] + Pi)/(2 Pi)] +
Floor[(Pi - Im[a Log[Sqrt[1/x - 1] Sqrt[1 + 1/x] + 1/x]])/(2 Pi)] +
Floor[(Re[b Log[Sqrt[1 - 1/y^2] + I/y]] + Pi)/(2 Pi)])
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo>  </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> ⌊ </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> </mrow> <mo> ⌋ </mo> </mrow> <mo> - </mo> <mrow> <mo> ⌊ </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> </mrow> <mo> ⌋ </mo> </mrow> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> ⌊ </mo> <mrow> <mfrac> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mi> π </mi> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> π </mi> </mfrac> </mrow> <mo> ⌋ </mo> </mrow> <mo> + </mo> <mrow> <mo> ⌊ </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mi> π </mi> </mfrac> </mrow> <mo> ⌋ </mo> </mrow> <mo> - </mo> <mrow> <mo> ⌊ </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> </mrow> <mo> ⌋ </mo> </mrow> <mo> + </mo> <mrow> <mo> ⌊ </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> </mrow> <mo> ⌋ </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msup> </mrow> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mrow> <mo> - </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msup> <mo> ) </mo> </mrow> <mo> + </mo> <mi> π </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> <mo> + </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mi> π </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> <mo> + </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> π </mi> <mo> - </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <arcsech /> <ci> x </ci> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <arccsc 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Date Added to functions.wolfram.com (modification date)
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