MHB Integral involving polylogarithms up to order 4

[alyafey22]

Prove the following

$$ \int^x_0 \frac{\log(1+t)\log^2(t)}{t}dt = -\log^2(x) \text{Li}_2(-x)+2 \log(x) \text{Li}_3(-x)-2 \text{Li}_4(-x)$$

 

[DreamWeaver]

Prove the following

$$ \int^x_0 \frac{\log(1+t)\log^2(t)}{t}dt = -\log^2(x) \text{Li}_2(-x)+2 \log(x) \text{Li}_3(-x)-2 \text{Li}_4(-x)$$

I'll not post a direct proof, but for those of you less familiar with the Polylogarithm - and I hope you'll excuse my apparent impertinence here, Zaid - here's a little (optional) hint...(Heidy)(Heidy)(Heidy)

Spoiler

For $$ |Re(z)| \le 1 $$ in the cut plane $$\mathbb{C}-[0, \infty)$$, where the real line ('x' axis) is deleted between $$1$$ and $$\infty$$, the Polylogarithm has the integral representation$$\text{Li}_m(z)=\frac{(-1)^{m+1}}{(m-2)!}\int_0^1\frac{(\log x)^{m-2} \log(1-zx) }{x} \,dx$$ Actually, this definition applies more generally, but the conditions given above ensure that the integral is single-valued, rather than a multi-valued complex function...

My shut up now...

 

[polygamma]

Spoiler

$$ \int_{0}^{x} \frac{\log(1+t) \log^2(t)}{t} \ dt = -\text{Li}_{2}(-t) \log^2 (t) \Big|^{x}_{0} + 2 \int_{0}^{x} \frac{\text{Li}_{2} (-t) \log t}{t} dt $$

$$ = - \text{Li}_{2} (-x) \log^{2} (x) + 2 \Big( \text{Li}_{3}(-t) \log t \Big|^{x}_{0} - \int_{0}^{x} \frac{\text{Li}_{3} (-t)}{t} \ dt \Big)$$

$$ = - \text{Li}_{2} (-x) \log^{2} (x) + 2 \text{Li}_{3}(-x) \log x- 2 \text{Li}_{4}(-x) $$