Dont solve integrals individually

If the integral $\int_0^1\frac{\ln^3(1-x)\ln^2(x)}{1-x}dx-3\int_0^1\frac{\ln(1-x)\operatorname{Li}_2^2(1-x)}{1-x}dx=a\zeta^b(c)-d\zeta(a)$ where $a,b,c,d$ are positive integers and $b,c$ are distinct primes. Find $a+b+c+d$ .

Notation: $\operatorname{Li}_2(x)$ is dilogarithm function and $\zeta(z)$ is Riemann zeta function .

Calculating each individual integrals make work lengthy so best way to evaluate one integral that cancels another .

This is an original and proposed issue .