Legendre Second Kind: $Q_n(x)$ Functions

[jije1112]

Legendre functions $Q_n(x)$ of the second kind
\begin{equation*}
Q_n(x)=P_n(x) \int \frac{1}{(1-x^2)\cdot P_n^2(x)}\, \mathrm{d}x
\end{equation*}
what to do after this step?
how can I complete ?
I need to reach this formula
\begin{equation*}
Q_n(x)=\frac{1}{2} P_n(x)\ln\left( \frac{ 1+x}{1-x}\right)
\end{equation*}

 

[HallsofIvy]

If that were true then, setting the two forms for $Q_(x)$ equal, it would have to be true that
$$\int \frac{1}{(1- x^2)P^2_n(x)}dx= \frac{1}{2}ln\left(\frac{1+ x}{1- x}\right)$$

Is that true? I suggest you check your formulas.