We know how hard these integrals
$$\int_0^1 \frac{x\ln(1+x)}{1+x^2}dx; \int_0^1 \frac{x\ln^2(1+x)}{1+x^2}dx; \int_0^1 \frac{x\ln^3(1+x)}{1+x^2}dx; ...$$
can be. So I decided to come up with a generalization and I succeeded:
With $x=1/(1+y)$, we have
$$\int_0^1 \frac{y\ln^a(1+y)}{1+y^2}dy=(-1)^a\left[\int_{1/2}^1\frac{\ln^a(x)}{x}dx+\int_{1/2}^1\frac{1-2x}{x^2+(1-x)^2}\ln^a(x)dx\right]$$
$$=(-1)^a\left[\frac{(-1)^a}{a+1}\ln^{a+1}(2)+\Re\int_{1/2}^1\frac{1+i}{1-(1+i)x}\ln^a(x)dx\right].$$
For the remaining integral, we need to find $\int_{1/2}^1\frac{z}{1-zx}\ln^a(x)dx:$
$$\int_{1/2}^1\frac{z}{1-zx}\ln^a(x)dx=\int_{0}^1\frac{z}{1-zx}\ln^a(x)dx-\underbrace{\int_0^{1/2}\frac{z}{1-zx}\ln^a(x)dx}_{x=y/2}$$
$$=(-a)^aa!\,\text{Li}_{a+1}(z)-\int_0^{1}\frac{\frac{z}2}{1-\frac{z}2y}\ln^a(y/2)dy$$
$$=(-a)^aa!\,\text{Li}_{a+1}(z)-\sum_{k=0}^a{a\choose k}(-\ln(2))^{a-k}\int_0^1 \frac{\frac{z}2}{1-\frac{z}2y}\ln^k(y)dy$$
$$=(-a)^aa!\,\text{Li}_{a+1}(z)-\sum_{k=0}^a{a\choose k}(-\ln(2))^{a-k}(-1)^kk!\,\text{Li}_{k+1}\left(\frac{z}2\right)$$
$$=(-1)^aa!\,\left[\text{Li}_{a+1}(z)-\sum_{k=0}^a\frac{\ln^{a-k}(2)}{(a-k)!}\text{Li}_{k+1}\left(\frac{z}2\right)\right].\quad\quad (*)$$
Using this integral, we get
$$\int_0^1\frac{x\ln^a(1+x)}{1+x^2}dx=\frac{\ln^{a+1}(2)}{a+1}+a!\,\Re\left[\text{Li}_{a+1}(1+i)-\sum_{k=0}^a\frac{\ln^{a-k}(2)}{(a-k)!}\text{Li}_{k+1}\left(\frac{1+i}2\right)\right].$$
Question: Is this generalization known in the mathematical literature? and if so, may I get the reference?
Thanks,
Bonus: Following the same approach above, using $\frac{1}{x^2+(1-x)^2}=\Im\,\frac{1+i}{1-(1+i)x}$, also gives $$\int_0^1\frac{\ln^a(1+x)}{1+x^2}dx=a!\,\Im\left[\text{Li}_{a+1}(1+i)-\sum_{k=0}^a\frac{\ln^{a-k}(2)}{(a-k)!}\text{Li}_{k+1}\left(\frac{1+i}2\right)\right].$$
Also if we divide the identity in $(*)$ by $z$ then integrate $\int_0^1$ w.r.t $z$, we obtain
$$\int_{1/2}^1\frac{\ln(1-x)\ln^a(x)}{x}dx=(-1)^{a-1}a!\,\left[\zeta(a+2)-\sum_{k=0}^a\frac{\ln^{a-k}(2)}{(a-k)!}\text{Li}_{k+2}\left(\frac{1}2\right)\right].$$