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 Cos

 http://functions.wolfram.com/01.07.21.2577.01

 Input Form

 Integrate[(z Sin[2 c z])/(a Sin[c z] + b Cos[c z])^2, z] == (a z^2)/(a^2 b + b^3) + (2 a (a c z - b Log[b Cos[c z] + a Sin[c z]]))/ ((a^2 + b^2)^2 c^2) - (1/(b (a^2 + b^2)^2 c^2)) (a^2 ((-I) b c Pi z + a Sqrt[1 + b^2/a^2] c^2 E^(I ArcTan[b/a]) z^2 - b Pi Log[1 + E^(-2 I c z)] - 2 b c z Log[1 - E^(2 I (c z + ArcTan[b/a]))] + b Pi Log[Cos[c z]] + 2 b ArcTan[b/a] (I c z - Log[1 - E^(2 I (c z + ArcTan[b/a]))] + Log[Sin[c z + ArcTan[b/a]]]) + I b PolyLog[2, E^(2 I (c z + ArcTan[b/a]))])) + (1/((a^2 + b^2)^2 c^2)) (b ((-I) b c Pi z + a Sqrt[1 + b^2/a^2] c^2 E^(I ArcTan[b/a]) z^2 - b Pi Log[1 + E^(-2 I c z)] - 2 b c z Log[1 - E^(2 I (c z + ArcTan[b/a]))] + b Pi Log[Cos[c z]] + 2 b ArcTan[b/a] (I c z - Log[1 - E^(2 I (c z + ArcTan[b/a]))] + Log[Sin[c z + ArcTan[b/a]]]) + I b PolyLog[2, E^(2 I (c z + ArcTan[b/a]))])) - (2 a z Sin[c z])/((a - I b) (a + I b) c (b Cos[c z] + a Sin[c z]))

 Standard Form

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 MathML Form

 z sin ( 2 c z ) ( a sin ( c z ) + b cos ( c z ) ) 2 z z 2 a b 3 + a 2 b + 2 ( a c z - b log ( b cos ( c z ) + a sin ( c z ) ) ) a ( a 2 + b 2 ) 2 c 2 - 2 z sin ( c z ) a ( a - b ) ( a + b ) c ( b cos ( c z ) + a sin ( c z ) ) - 1 b ( a 2 + b 2 ) 2 c 2 ( a 2 ( a b 2 a 2 + 1 c 2 tan - 1 ( b a ) z 2 - 2 b c log ( 1 - 2 ( c z + tan - 1 ( b a ) ) ) z - b c π z - b π log ( 1 + - 2 c z ) + b π log ( cos ( c z ) ) + 2 b tan - 1 ( b a ) ( c z - log ( 1 - 2 ( c z + tan - 1 ( b a ) ) ) + log ( sin ( c z + tan - 1 ( b a ) ) ) ) + b Li PolyLog 2 ( 2 ( c z + tan - 1 ( b a ) ) ) ) ) + 1 ( a 2 + b 2 ) 2 c 2 ( b ( a b 2 a 2 + 1 c 2 tan - 1 ( b a ) z 2 - 2 b c log ( 1 - 2 ( c z + tan - 1 ( b a ) ) ) z - b c π z - b π log ( 1 + - 2 c z ) + b π log ( cos ( c z ) ) + 2 b tan - 1 ( b a ) ( c z - log ( 1 - 2 ( c z + tan - 1 ( b a ) ) ) + log ( sin ( c z + tan - 1 ( b a ) ) ) ) + b Li PolyLog 2 ( 2 ( c z + tan - 1 ( b a ) ) ) ) ) z z 2 c z a c z b c z 2 -1 z 2 a b 3 a 2 b -1 2 a c z -1 b b c z a c z a a 2 b 2 2 c 2 -1 -1 2 z c z a a -1 b a b c b c z a c z -1 -1 1 b a 2 b 2 2 c 2 -1 a 2 a b 2 a 2 -1 1 1 2 c 2 b a -1 z 2 -1 2 b c 1 -1 2 c z b a -1 z -1 b c z -1 b 1 -2 c z b c z 2 b b a -1 c z -1 1 -1 2 c z b a -1 c z b a -1 b PolyLog 2 2 c z b a -1 1 a 2 b 2 2 c 2 -1 b a b 2 a 2 -1 1 1 2 c 2 b a -1 z 2 -1 2 b c 1 -1 2 c z b a -1 z -1 b c z -1 b 1 -2 c z b c z 2 b b a -1 c z -1 1 -1 2 c z b a -1 c z b a -1 b PolyLog 2 2 c z b a -1 [/itex]

 Rule Form

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

 2002-12-18