<s>[INST] Integral of the form $\int_a^b \frac{\ln(c+dx)}{P(x)}dx$ I found here a "great theorem" which states that: $$\int_a^b \frac{\ln(c+dx)}{P(x)}dx =\frac{\ln((ad+c)(bd+c))}{2}\int_a^b\frac{dx}{P(x)}$$
I don't know how to prove this, but I am pretty sure that we should work by symmetry with a substitution of the form $\frac{mx+n}{sx+p}$, then add the result with the initial integral.
An easier case which shows this idea is the well-known integral $\int_0^1 \frac{\ln(1+x)}{1+x^2}dx$ which can be dealt with the substitution $\frac{1-x}{1+x}$ which produces $\int_0^1 \frac{\ln 2 -\ln(1+x)}{1+x^2}dx$ and adding this with the initial integral simplifies the logarithm. 
In our case, after finding the magic substitution we will have: $$\int_a^b \frac{\ln(c+dx)}{P(x)}dx=\int_a^b \frac{\ln((ad+c)(bd+c)) - \ln(c+dx)}{P(x)}dx$$ 
Unfortunately I dont know what $P(x)$ is, but it's hard to believe that it can be any polynomial of the form $x^2+sx+p$. I would appreciate some help to prove this "great theorem".
 [/INST] \begin{align}J=\int_a^b \frac{\ln(c+dx)}{P(x)}dx\end{align}
Formally,
1)"Clean up" the logarithm. Perform the change of variable $u=c+dx$,
\begin{align}J=\frac{1}{d}\int_{c+da}^{c+db} \frac{\ln u}{P\left(\frac{u-c}{d}\right)}du\end{align}
2) Change of the bounds of the integral to new ones, m,M such that $m\times M=1$.
Perform the change of variable $v=\frac{1}{\sqrt{(c+db)(c+da)}}u$
\begin{align}J&=\frac{1}{d}\int_{\sqrt{\frac{c+da}{c+db}}}^{\sqrt{\frac{c+db}{c+da}}} \frac{\ln\left( v\sqrt{(c+db)(c+da)}\right)}{P\left(\frac{v\sqrt{(c+db)(c+da)}-c}{d}\right)}dv\\
&=\frac{1}{d}\int_{\sqrt{\frac{c+da}{c+db}}}^{\sqrt{\frac{c+db}{c+da}}} \frac{\ln  v}{P\left(\frac{v\sqrt{(c+db)(c+da)}-c}{d}\right)}dv+\\
&\frac{\ln\left((c+db)(c+da)\right)}{2d}\int_{\sqrt{\frac{c+da}{c+db}}}^{\sqrt{\frac{c+db}{c+da}}} \frac{1}{P\left(\frac{v\sqrt{(c+db)(c+da)}-c}{d}\right)}dv
\end{align}
In the latter integral perform the change of variable $z=\dfrac{v\sqrt{(c+db)(c+da)}-c}{d}$,
\begin{align}J&=\frac{1}{d}\int_{\sqrt{\frac{c+da}{c+db}}}^{\sqrt{\frac{c+db}{c+da}}} \frac{\ln  v}{P\left(\frac{v\sqrt{(c+db)(c+da)}-c}{d}\right)}dv+\frac{\ln\left((c+db)(c+da)\right)}{2}\int_a^b\frac{1}{P(z)}\,dz\end{align}
If for all $v$ real, $v^2 P\left(\frac{\frac{1}{v}\sqrt{(c+db)(c+da)}-c}{d}\right)=P\left(\frac{v\sqrt{(c+db)(c+da)}-c}{d}\right)$ then ,
\begin{align}\int_{\sqrt{\frac{c+da}{c+db}}}^{\sqrt{\frac{c+db}{c+da}}} \frac{\ln  v}{P\left(\frac{v\sqrt{(c+db)(c+da)}-c}{d}\right)}dv=0\end{align}
(Perform the change of variable $w=\dfrac{1}{v}$)
Thus,
\begin{align}J=\frac{\ln\left((c+db)(c+da)\right)}{2}\int_a^b\frac{1}{P(z)}\,dz\end{align}
PS:
Don't expect to use this formula with P a polynomial of degree>2 or even equal to 1.
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