1-x² / (1+x²)² where x= tan θ

(1-tan^2Φ)/(sec^2Φ)^2 now put u=sec^2phi

∫1-x²/(1+x²)² 

add and subtract 1 to numerator ,you get

∫2/(1+x²)²  dx - ∫1/(1+x²) dx  ===>  ∫2/(1+x²)² dx-tanx+c

= let x=tanθ

===> dx=sec²θ dθ

∫2/(1+tan²θ)²  sec²θ dθ -tanx+c

=∫2cos²θ dθ-tanx+c

=∫(cos2θ+1)dθ-tanx+c

=sin2θ/2 +x-tanx+c

=tanθ/1+tan²θ +x-tanx+c

but x=tanθ 

x/(1+x²) +x-tanx+c