Integrate[z ArcCsc[a z + b], z] == ((b + a z)/(2 a^2)) Sqrt[(-1 + b^2 + 2 a b z + a^2 z^2)/(b + a z)^2] + (1/2) z^2 ArcCsc[b + a z] - (b^2 ArcCsc[b + a z])/(2 a^2) - (b Log[2 (b + a z) + 2 (b + a z) Sqrt[(-1 + b^2 + 2 a b z + a^2 z^2)/ (b + a z)^2]])/a^2

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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> <mo> &#8747; </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> <mo> - </mo> <mrow> <mfrac> <mi> b </mi> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mfrac> <mo> &#8290; 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2001-10-29