Integrate[1/(a Coth[z] + b Sech[z])^2, z] == (Csch[z]^2 Sech[z]^2 (a + a Cosh[2 z] + 2 b Sinh[z])^2 (-((Sqrt[2] Sqrt[(-b) (b + Sqrt[-4 a^2 + b^2])] (-4 a^4 + a^2 b (7 b - 5 Sqrt[-4 a^2 + b^2]) + b^3 (-b + Sqrt[-4 a^2 + b^2])) ArcTan[(2 a + (-b + Sqrt[-4 a^2 + b^2]) Tanh[z/2])/ (Sqrt[2] Sqrt[b] Sqrt[-b + Sqrt[-4 a^2 + b^2]])] + Sqrt[b] Sqrt[-b + Sqrt[-4 a^2 + b^2]] ((-(-4 a^2 + b^2)^(3/2)) Sqrt[(-b) (b + Sqrt[-4 a^2 + b^2])] z + Sqrt[2] (4 a^4 + b^3 (b + Sqrt[-4 a^2 + b^2]) - a^2 b (7 b + 5 Sqrt[-4 a^2 + b^2])) ArcTan[(2 a - (b + Sqrt[-4 a^2 + b^2]) Tanh[z/2])/ (Sqrt[2] Sqrt[(-b) (b + Sqrt[-4 a^2 + b^2])])]))/ (Sqrt[b] (-4 a^2 + b^2)^(3/2) Sqrt[-b + Sqrt[-4 a^2 + b^2]] Sqrt[(-b) (b + Sqrt[-4 a^2 + b^2])])) - (2 a Cosh[z] ((-a) b + (2 a^2 - b^2) Sinh[z]))/ ((4 a^2 - b^2) (a + a Cosh[2 z] + 2 b Sinh[z]))))/ (4 a^2 (a Coth[z] + b Sech[z])^2)

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<mrow> <mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> </mrow> 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2002-12-18