Solve by using u substitution as follows:

= t^2

du = 2t * dt

t * dt / (2 + t^4) =>

(1/2) * du / (2 + u^2)

u = sqrt(2) * tan(x)

du = sqrt(2) * sec(x)^2 * dx

(1/2) * sqrt(2) * sec(x)^2 * dx / (2 + 2 * tan(x)^2)) =>

(1/2) * sqrt(2) * sec(x)^2 * dx / (2 * (1 + tan(x)^2)) =>

(1/2) * (1/2) * sqrt(2) * sec(x)^2 * dx / sec(x)^2 =>

(1/4) * sqrt(2) * dx

Integrate

(sqrt(2) / 4) * x + C

u = sqrt(2) * tan(x)

u / sqrt(2) = tan(x)

x = arctan(u / sqrt(2))

u = t^2

x = arctan(t^2 / sqrt(2))

(sqrt(2) / 4) * x + C =>>

(sqrt(2) / 4) * arctan(t^2 / sqrt(2)) + C