sin^3(5x) (4x - x^2)^(1/3) (1 - tan x)^(-1) - Wolfram|Alpha
... you'll see your only final error is your overall negative sign (where from?). Although earlier (in the SOOOOO line) you're missing the polynomial part of the middle term.
The best chance of minimising tangles in this case is probably ...
Product rule - Wikipedia, the free encyclopedia
... from which you get this overview...
... the bottom line of which is pretty much Wolfram's final version.
(Straight lines differentiate downwards with respect to x.)
If there's a better way I don't see it. Edit: D'oh! Yes of course, thanks Skeeter! Although, funnily enough, the log method doesn't seem any simpler in this case. Though it often is, obviously.
I probably won't zoom in on the chain rules on this occasion - it might just re-aggravate the tangle. Not promising, though!
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!