I just give the final answer and the important step :

The final answer is - arccosh| cosec { x + arctan [(4-x^2 )/ ( 4x )] } | + c

And the important step is

4x sinx + ( 4 - x^2 ) cosx =

sqrt [ (4x)^2 + (4-x^2)^2 ] * sin { x + arctan[ (4-x^2)/(4x) ] }

= sqrt [ x^4 + 8x^2 + 16 ] * sin { x + arctan[ (4-x^2)/(4x) ] }

= ( x^2 + 4 ) * sin { x + arctan[ (4-x^2)/(4x) ] }