Answer
\[ = - \ln \left| {\cos x} \right| - \frac{{{{\sin }^2}x}}{2} + C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {{{\sec }^{ - 2}}x{{\tan }^3}xdx} \hfill \\
\hfill \\
rewrite\,\,the\,\,{\text{integrand}} \hfill \\
\hfill \\
\,\,\int_{}^{} {{{\sec }^{ - 2}}x\tan x{{\tan }^2}xdx} \hfill \\
\hfill \\
use\,\,\,{\tan ^2}x = {\sec ^2}x - 1 \hfill \\
\hfill \\
= \int_{}^{} {{{\sec }^{ - 2}}x\tan x\,\left( {{{\sec }^2}x - 1} \right)dx} \hfill \\
\hfill \\
multiply \hfill \\
\hfill \\
= \int_{}^{} {\,\left( {\tan x - {{\sec }^{ - 2}}x\tan x} \right)dx} \hfill \\
\hfill \\
= \int_{}^{} {\,\left( {\tan x - \sin x\cos x} \right)dx} \hfill \\
\hfill \\
integrate \hfill \\
\hfill \\
= - \ln \left| {\cos x} \right| - \frac{{{{\sin }^2}x}}{2} + C \hfill \\
\end{gathered} \]
