Evaluate the indefinite integral. (Use symbolic notation and fractions where needed. Use $\mathrm{C}$ for the arbitrary constant. Absorb into $\mathrm{C}$ as much as possible.) \[ \int 52 x^{-1 / 5} \tan \left(x^{4 / 5}\right) d x= \]

Step 1 ：Define the function \(f = 52 x^{-1 / 5} \tan \left(x^{4 / 5}\right)\)

Step 2 ：Calculate the indefinite integral of the function \(f\)

Step 3 ：The output from the calculation is the integral of the function, which is \(-65.0 \log(\cos(x^{0.8}))\)

Step 4 ：However, we need to add the constant of integration, \(C\), to the result

Step 5 ：The indefinite integral of \(52 x^{-1 / 5} \tan \left(x^{4 / 5}\right) d x\) is \(\boxed{-65.0 \log(\cos(x^{0.8})) + C}\)