ArcCosh[x] - ArcSec[y] == -2 I Pi (Floor[(-Arg[x + Sqrt[x - 1] Sqrt[x + 1]] - Arg[(Sqrt[1 - 1/y^2] + I/y)^(-I)] + Pi)/(2 Pi)] + Floor[(Pi - Im[Log[x + Sqrt[x - 1] Sqrt[x + 1]]])/(2 Pi)] + Floor[(Re[Log[Sqrt[1 - 1/y^2] + I/y]] + Pi)/(2 Pi)]) + I Pi (1 - (-1)^(Floor[-(Arg[(x + Sqrt[x - 1] Sqrt[x + 1])/(Sqrt[1 - 1/y^2] + I/y)^ I + 1]/(2 Pi))] - Floor[-(Arg[(x + Sqrt[x - 1] Sqrt[x + 1])/(Sqrt[1 - 1/y^2] + I/y)^I]/ (2 Pi))])) + (-1)^(-Floor[-(Arg[(x + Sqrt[x - 1] Sqrt[x + 1])/(Sqrt[1 - 1/y^2] + I/y)^ I - 1]/(2 Pi))] + Floor[-(Arg[(x + Sqrt[x - 1] Sqrt[x + 1])/(Sqrt[1 - 1/y^2] + I/y)^I + 1]/ (2 Pi))] + Floor[-(Arg[(x + Sqrt[x - 1] Sqrt[x + 1])/ (Sqrt[1 - 1/y^2] + I/y)^I]/Pi)] + Floor[Arg[(x + Sqrt[x - 1] Sqrt[x + 1])/(Sqrt[1 - 1/y^2] + I/y)^I]/Pi - (2 Arg[(x + Sqrt[x - 1] Sqrt[x + 1])/(Sqrt[1 - 1/y^2] + I/y)^I - 1])/ Pi]) ArcCosh[(((x + Sqrt[x - 1] Sqrt[x + 1])^2/ (Sqrt[1 - 1/y^2] + I/y)^(2 I) + 1) (Sqrt[1 - 1/y^2] + I/y)^I)/ (2 (x + Sqrt[x - 1] Sqrt[x + 1]))] - Pi/2

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<mrow> <mi> x </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </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> &#8520; </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> </mrow> <mo> &#8971; </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> &#8970; </mo> <mrow> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <msqrt> <mrow> <mi> x </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </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> &#8520; </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mi> &#960; </mi> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <msqrt> <mrow> <mi> x </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( 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2007-05-02