Integrate[(A + B Sin[z] + C Cos[z])/((a + b Sin[z] + c Cos[z]) (d + e Sin[z] + f Cos[z])), z] == (-((1/Sqrt[-a^2 + b^2 + c^2]) (2 ((-a) (b B + c C) d - (B c - b C) (c e - b f) + a^2 (B e + C f) + A (b^2 d - a b e + c (c d - a f))) ArcTanh[(b + (a - c) Tan[z/2])/ Sqrt[-a^2 + b^2 + c^2]])) - (1/Sqrt[-d^2 + e^2 + f^2]) (2 (c (C (d^2 - e^2) - A d f + B e f) + b ((-A) d e + C e f + B (d^2 - f^2)) + a ((-B) d e - C d f + A (e^2 + f^2))) ArcTanh[(e + (d - f) Tan[z/2])/Sqrt[-d^2 + e^2 + f^2]]) - (B c d - b C d - A c e + a C e + A b f - a B f) Log[a + c Cos[z] + b Sin[z]] + (B c d - b C d - A c e + a C e + A b f - a B f) Log[d + f Cos[z] + e Sin[z]])/(c^2 (d^2 - e^2) - 2 a c d f + b (-2 a d e + 2 c e f) + b^2 (d^2 - f^2) + a^2 (e^2 + f^2))

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<mi> A </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> e </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> C </mi> <mo> &#8290; </mo> <mi> e </mi> </mrow> <mo> + </mo> <mrow> <mi> A </mi> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mi> f </mi> </mrow> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> B </mi> <mo> &#8290; </mo> <mi> f </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> B </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mo> - 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