Answer
$$ - 2\ln \left| {\cos {x^2}} \right| + C$$
Work Step by Step
$$\eqalign{
& \int {4x} \tan \left( {{x^2}} \right)dx \cr
& {\text{substitute }}u = {x^2},{\text{ }}du = 2xdx \cr
& = 4\int x \tan \left( {{x^2}} \right)dx \cr
& = 4\int {\tan } u\left( {\frac{1}{2}du} \right) \cr
& = 2\int {\tan } udu \cr
& {\text{find the antiderivative}}{\text{, }}\int {\tan \theta } d\theta = - \ln \left| {\cos \theta } \right| + C \cr
& = 2\left( { - \ln \left| {\cos \theta } \right|} \right) + C \cr
& = - 2\ln \left| {\cos \theta } \right| + C \cr
& {\text{write in terms of }}x \cr
& = - 2\ln \left| {\cos {x^2}} \right| + C \cr} $$
