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The integral of sec(x) is ln|secx+tanx| + C
Since the derivative is taken to the third power, we have to consider the chain rule; the original equation must be to the fourth power, and in order for that to be canceled out, the equation must also have had a coefficient of 1/4. 2x is also subject to the chain rule. I would suggest u substitution.
integral(sec(2x))^3 dx
u=2x
du=2dx
dx=1/2du
integral (sec(u))^3 *1/2 du
1/8 secxtanx + 1/8(ln|secx+tanx|^4) + C
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