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 ArcCsc

 http://functions.wolfram.com/01.17.21.0007.01

 Input Form

 Integrate[ArcCsc[a z + b]/z, z] == (1/8) (I (Pi - 2 ArcCsc[b + a z])^2 + 32 I ArcSin[Sqrt[1 - 1/b]/Sqrt[2]] ArcTan[((1 + b) Tan[(1/4) (Pi - 2 ArcCsc[b + a z])])/Sqrt[1 - b^2]] - 4 (Pi - 2 ArcCsc[b + a z] - 4 ArcSin[Sqrt[1 - 1/b]/Sqrt[2]]) Log[1 + (I (-1 + Sqrt[1 - b^2]))/(b E^(I ArcCsc[b + a z]))] - 4 (Pi - 2 ArcCsc[b + a z] + 4 ArcSin[Sqrt[1 - 1/b]/Sqrt[2]]) Log[1 - (I (1 + Sqrt[1 - b^2]))/(b E^(I ArcCsc[b + a z]))] - 8 ArcCsc[b + a z] Log[1 - E^(2 I ArcCsc[b + a z])] + 4 (Pi - 2 ArcCsc[b + a z]) Log[-((a z)/(b + a z))] + 8 ArcCsc[b + a z] Log[-((a z)/(b + a z))] + 8 I (PolyLog[2, -((I (-1 + Sqrt[1 - b^2]))/(b E^(I ArcCsc[b + a z])))] + PolyLog[2, (I (1 + Sqrt[1 - b^2]))/(b E^(I ArcCsc[b + a z]))]) + 4 I (ArcCsc[b + a z]^2 + PolyLog[2, E^(2 I ArcCsc[b + a z])]))

 Standard Form

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 MathML Form

 csc - 1 ( a z + b ) z z 1 8 ( ( π - 2 csc - 1 ( b + a z ) ) 2 + 4 log ( - a z b + a z ) ( π - 2 csc - 1 ( b + a z ) ) + 32 sin - 1 ( 1 2 1 - 1 b ) tan - 1 ( b + 1 1 - b 2 tan ( 1 4 ( π - 2 csc - 1 ( b + a z ) ) ) ) - 8 csc - 1 ( b + a z ) log ( 1 - 2 csc - 1 ( b + a z ) ) - 4 ( - 2 csc - 1 ( b + a z ) - 4 sin - 1 ( 1 2 1 - 1 b ) + π ) log ( 1 + ( 1 - b 2 - 1 ) b csc - 1 ( b + a z ) ) - 4 ( - 2 csc - 1 ( b + a z ) + 4 sin - 1 ( 1 2 1 - 1 b ) + π ) log ( 1 - ( 1 - b 2 + 1 ) b csc - 1 ( b + a z ) ) + 8 csc - 1 ( b + a z ) log ( - a z b + a z ) + 4 ( csc - 1 ( b + a z ) 2 + Li PolyLog 2 ( 2 csc - 1 ( b + a z ) ) ) + 8 ( Li PolyLog 2 ( ( 1 - b 2 + 1 ) b csc - 1 ( b + a z ) ) + Li PolyLog 2 ( - ( 1 - b 2 - 1 ) b csc - 1 ( b + a z ) ) ) ) z a z b z -1 1 8 -1 2 b a z 2 4 -1 a z b a z -1 -1 2 b a z 32 1 2 1 2 -1 1 -1 1 b -1 1 2 b 1 1 -1 b 2 1 2 -1 1 4 -1 2 b a z -1 8 b a z 1 -1 2 b a z -1 4 -2 b a z -1 4 1 2 1 2 -1 1 -1 1 b -1 1 2 1 1 -1 b 2 1 2 -1 b b a z -1 -1 4 -2 b a z 4 1 2 1 2 -1 1 -1 1 b -1 1 2 1 -1 1 -1 b 2 1 2 1 b b a z -1 8 b a z -1 a z b a z -1 4 b a z 2 PolyLog 2 2 b a z 8 PolyLog 2 1 -1 b 2 1 2 1 b b a z -1 PolyLog 2 -1 1 -1 b 2 1 2 -1 b b a z -1 [/itex]

 Rule Form

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 Date Added to functions.wolfram.com (modification date)

 2001-10-29