Two representation of a particular form of hypergeometric function.

From the basic definition ofhypergeometric function,we know that$${}_{2}F_{1}(1,p;p+1;1) =p \int_{0}^{1} \dfrac{t^{p-1}}{1-t} dt. $$

My professor also told me that\begin{align*}{}_{2}F_{1}(1, p; p+1;1)& = - p \int_{0}^{1} t^{p-1} \ln{(1-t)} dt.\end{align*}

That means we need to show that$$\int_{0}^{1} \dfrac{t^{p-1}}{1-t} dt= -\int_{0}^{1} t^{p-1} \ln{(1-t)} dt.$$

But I am not able to prove this equality at all.I tried using Integration by Parts but did not get the desired result at all.

Plus, the first representation of ${}_{2}F_{1}$ can be derived very easily by using the definition of ${}_{2}F_{1}(1,p;p+1;1)$ from any textbook, like Gradshteyn and Ryzhik: "Table of Integrals, Series, and Products", 8th Edition (2014). But I do not see the logarithmic representation of ${}_{2}F_{1}$ anywhere.

I would appreciate for a proof of equivalency or pointing out me a reference for the second form.