Integrate[ArcCsch[a^z], z] == z ArcCsch[a^z] + (1/(8 Sqrt[1 + a^(2 z)] Log[a])) (a^z Sqrt[1 + a^(-2 z)] (Log[-a^(2 z)]^2 + ArcTanh[Sqrt[1 + a^(2 z)]] (-8 z Log[a] + 4 Log[-a^(2 z)]) - 4 Log[-a^(2 z)] Log[(1/2) (1 + Sqrt[1 + a^(2 z)])] + 2 Log[(1/2) (1 + Sqrt[1 + a^(2 z)])]^2 - 4 PolyLog[2, 1/2 - (1/2) Sqrt[1 + a^(2 z)]]))

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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> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msup> <mi> a </mi> <mi> z </mi> </msup> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> a </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <msup> <mi> a </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; 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2001-10-29