$\int {\sec }^3\left(x\right) \mathit{\ⅆ}x$

$\=\sec \left(x\right)\tan \left(x\right)-\int \tan \left(x\right)\cdot \left(\sec \left(x\right)\tan \left(x\right)\right) \mathit{\ⅆ}x$

$\=\sec \left(x\right)\tan \left(x\right)-\int \sec \left(x\right)\cdot {\tan }^2\left(x\right) \mathit{\ⅆ}x$

$\=\sec \left(x\right)\tan \left(x\right)-\int \sec \left(x\right)\cdot \left({\sec }^2\left(x\right)-1\right) \mathit{\ⅆ}x$

$\=\sec \left(x\right)\tan \left(x\right)-\int {\sec }^3\left(x\right) \ⅆx\+\int \sec \left(x\right) \mathit{\ⅆ}x$

$\=\sec \left(x\right)\tan \left(x\right)-\int {\sec }^3\left(x\right) \mathit{\ⅆ}x\+\ln \(\left|\sec \left(x\right)\+\tan \left(x\right)\right|\)$

$2\int {\sec }^3\left(x\right) \mathit{\ⅆ}x$

$\=\sec \left(x\right)\tan \left(x\right)\+\ln \(\left|\sec \left(x\right)\+\tan \left(x\right)\right|\)$

$\int {\sec }^3\left(x\right) \mathit{\ⅆ}x$

$\=\frac{1}{2}\(\sec \left(x\right)\tan \left(x\right)\+\ln \(\left|\sec \left(x\right)\+\tan \left(x\right)\right|\)\)$

Table 6.2.5(a)   Evaluation of $\int {\sec }^3\left(x\right) \mathit{\ⅆ}x$

Evaluation in Maple

$\int {\sec }^3\left(x\right) \mathit{\ⅆ}x$ = $\frac{1}{2}\frac{\sin \left(x\right)}{{\cos \left(x\right)}^2}\+\frac{1}{2}\ln \left(\sec \left(x\right)\+\tan \left(x\right)\right)$$$

$\begin{array}{c}\int {\sec \left(x\right)}^3\ⅆx\hfill \\ & \text{=}& \tan \left(x\right)\sec \left(x\right)-\int {\tan \left(x\right)}^2\sec \left(x\right)\ⅆx\hfill & \left[\mathrm{parts}\,\sec \left(x\right)\,\tan \left(x\right)\right]\\ & \text{=}& \tan \left(x\right)\sec \left(x\right)-\int \sec \left(x\right)\left({\sec \left(x\right)}^2-1\right)\ⅆx\hfill & \left[\mathrm{rewrite}\,{\tan \left(x\right)}^2={\sec \left(x\right)}^2-1\right]\\ & \text{=}& \tan \left(x\right)\sec \left(x\right)-\int {\sec \left(x\right)}^3\ⅆx+\ln \left(\sec \left(x\right)+\tan \left(x\right)\right)\hfill & \left[\mathrm{rewrite}\,\sec \left(x\right)\left({\sec \left(x\right)}^2-1\right)={\sec \left(x\right)}^3-\sec \left(x\right)\right]\\ & \text{=}& \frac{\tan \left(x\right)\sec \left(x\right)}{2}+\frac{\ln \left(\sec \left(x\right)+\tan \left(x\right)\right)}{2}\hfill & \left[\mathrm{solve}\right]\end{array}$

Table 6.2.5(b)   Annotated stepwise evaluation of $\int {\sec }^3\left(x\right) \mathit{\ⅆ}x$ via the Integration Methods tutor