Using Gamma, eta and zeta function to solve logarithmic integral (ln^2 x ln(1 + x))/x dx from 0 to 1

Integrate (ln^2 x ln(1 + x))/x dx from 0 to 1 (Mis-1891)
Solving logarithmic integral using Gamma function, Dirichlet eta function, Riemann zeta nfunction and taylor seires.
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