Answer
$$\int2\sec^3\pi xdx=\frac{\sec\pi x\tan\pi x}{\pi}+\frac{1}{\pi}\ln|\sec \pi x+\tan\pi x|+C$$
Work Step by Step
$$A=\int2\sec^3\pi xdx$$
Use Reduction Formula 99, which states that
$$\int\sec^naxdx=\frac{\sec^{n-2}ax\tan ax}{a(n-1)}+\frac{n-2}{n-1}\int\sec^{n-2}axdx$$
for $n=3$ and $a=\pi$
$$A=2\Big[\frac{\sec\pi x\tan \pi x}{2\pi}+\frac{1}{2}\int\sec\pi xdx\Big]$$ $$A=\frac{\sec\pi x\tan\pi x}{\pi}+\int\sec\pi xdx$$
Next, apply Formula 95, which states that
$$\int\sec axdx=\frac{1}{a}\ln|\sec ax+\tan ax|+C$$
for $a=\pi$
$$A=\frac{\sec\pi x\tan\pi x}{\pi}+\frac{1}{\pi}\ln|\sec \pi x+\tan\pi x|+C$$
