Step 1 :Let \(u = \tan(8x)\). This implies that \(du = 8\sec^2(8x)dx\) or \(dx = \frac{1}{8\sec^2(8x)}du\).
Step 2 :Substitute \(u\) and \(dx\) into the integral to get \(\frac{1}{8}\int u^{1/7} du\).
Step 3 :Evaluate the integral to get \(\frac{7}{8}u^{8/7} + C\), where \(C\) is the constant of integration.
Step 4 :Substitute \(u = \tan(8x)\) back into the result to get \(\frac{7}{8}\tan^{8/7}(8x) + C\).
Step 5 :Differentiate \(\frac{7}{8}\tan^{8/7}(8x) + C\) with respect to \(x\) to check the result. The derivative is indeed \(\sqrt[7]{\tan (8 x)} \sec ^{2}(8 x)\), so the result is correct.
Step 6 :\(\boxed{\frac{7}{8}\tan^{8/7}(8x) + C}\) is the final answer.