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Using Maxima:

maxima('assume(n>=0)')
w=maxima('integrate(sum(t^(8*n),n,0,N),t,0,1)').sage();w

sum(1/(8*n + 1), n, 0, N)

w.simplify_full()

-1/8euler_gamma + 1/8harmonic_number(N + 1/8) - 1/8*psi(1/8)

Using Maxima:

sage: maxima('assume(n>=0)')
w=maxima('integrate(sum(t^(8*n),n,0,N),t,0,1)').sage();w

sage: w.simplify_full()
-1/8*euler_gamma + 1/8*harmonic_number(N + 1/8) - 1/8*psi(1/8)