Answer
$$\frac{{2{{\sec }^{3/2}}x}}{3} + C$$
Work Step by Step
$$\eqalign{
& \int {\tan x} {\sec ^{3/2}}xdx \cr
& {\text{split exponent of }}{\sec ^{3/2}}x \cr
& = \int {\tan x} {\sec ^{1/2}}x\sec xdx \cr
& = \int {{{\sec }^{1/2}}x} \sec x\tan xdx \cr
& {\text{substitute }}u = \sec x,{\text{ }}du = \sec x\tan xdx \cr
& = \int {{u^{1/2}}} du \cr
& {\text{find the antiderivative by the power rule}} \cr
& = \frac{{{u^{3/2}}}}{{3/2}} + C \cr
& = \frac{{2{u^{3/2}}}}{3} + C \cr
& {\text{write in terms of }}x,{\text{ replace }}u = \sec x \cr
& = \frac{{2{{\sec }^{3/2}}x}}{3} + C \cr} $$
