integration

let tanx = z² , sec²x dx = 2z dz

therefore I = ∫(z. 2z dz)/sec²x
I= ∫2z²1+z⁴ dz

= ∫{z²+11+z⁴ } + ∫{z²-11+z⁴ }

= ∫{z^-2+1z²+z^-2 } +∫{-z^-2+1z²+z^-2 }

now,
z² +z^-2 = (z-1/z)² +2 (for first fraction)
z² +z^-2 = (z+1/z)² -2 (for 2nd frac.)

replace in previous step then
take z-1/z =t for first frac. and
z+1/z = p for 2nd frac.

then just standard integrals

1st frac. -

dtt² +2 = √2.tan^-1 (t/√2)

2nd frac. -

dpp²-2 = (1/2√2 ).ln(|p-2p+2 |)

substituing for x we get the final ans.