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http://functions.wolfram.com/01.12.16.0218.01
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a ArcSin[x] + b ArcSech[y] ==
I Pi
(1 - (-1)^(Floor[-(Arg[(Sqrt[1/y - 1] Sqrt[1 + 1/y] + 1/y)^b/
(I x + Sqrt[1 - x^2])^(I a) + 1]/(2 Pi))] -
Floor[-(Arg[(Sqrt[1/y - 1] Sqrt[1 + 1/y] + 1/y)^b/(I x + Sqrt[1 - x^2])^
(I a)]/(2 Pi))])) +
(-1)^(-Floor[-(Arg[(Sqrt[1/y - 1] Sqrt[1 + 1/y] + 1/y)^b/
(I x + Sqrt[1 - x^2])^(I a) - 1]/(2 Pi))] +
Floor[-(Arg[(Sqrt[1/y - 1] Sqrt[1 + 1/y] + 1/y)^b/(I x + Sqrt[1 - x^2])^
(I a) + 1]/(2 Pi))] +
Floor[-(Arg[(Sqrt[1/y - 1] Sqrt[1 + 1/y] + 1/y)^b/(I x + Sqrt[1 - x^2])^
(I a)]/Pi)] +
Floor[Arg[(Sqrt[1/y - 1] Sqrt[1 + 1/y] + 1/y)^b/(I x + Sqrt[1 - x^2])^
(I a)]/Pi - (2 Arg[(Sqrt[1/y - 1] Sqrt[1 + 1/y] + 1/y)^b/
(I x + Sqrt[1 - x^2])^(I a) - 1])/Pi])
ArcSech[(2 (Sqrt[1/y - 1] Sqrt[1 + 1/y] + 1/y)^b)/
((I x + Sqrt[1 - x^2])^(I a) ((Sqrt[1/y - 1] Sqrt[1 + 1/y] + 1/y)^(2 b)/
(I x + Sqrt[1 - x^2])^(2 I a) + 1))] -
2 I Pi (Floor[(-Arg[(I x + Sqrt[1 - x^2])^((-I) a)] -
Arg[(Sqrt[1/y - 1] Sqrt[1 + 1/y] + 1/y)^b] + Pi)/(2 Pi)] +
Floor[(Pi - Im[b Log[Sqrt[1/y - 1] Sqrt[1 + 1/y] + 1/y]])/(2 Pi)] +
Floor[(Re[a Log[I x + Sqrt[1 - x^2]]] + Pi)/(2 Pi)])
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⁢ </mo> <mi> a </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> b </mi> </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> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> a </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> b </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> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> b </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> a </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> </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> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> a </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> b </mi> </msup> </mrow> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <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> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> a </mi> </mrow> </msup> <mo> ) </mo> </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> y </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> b </mi> </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> a </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </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> b </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> y </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> <arcsin /> <ci> x </ci> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <arcsech /> <ci> y </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <pi /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn 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<apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <pi /> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <floor /> <apply> <times /> <apply> <plus /> <pi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <imaginary /> <apply> <times /> <ci> b </ci> <apply> <ln /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> 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Date Added to functions.wolfram.com (modification date)
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