Logarithm integral with golden ratio $\int^1_0 \frac{\log(1+\phi x^2)}{1+x}\, dx$ How to evaluate the following integral 
$$ \int^1_0 \frac{\log(1+\phi x^2)}{1+x}\, dx$$
where 
$$\phi =\frac{\sqrt{5}+1}{2}$$
Is there a closed form for the integral ?
My first attempt would involve relating the integral to dilogarithms but that seems complicated !
 A: \begin{align}
\int_{0}^{1}{\ln\left(1 + \phi x^{2}\right) \over 1+x}\,{\rm d}x
&=
\sum_{\sigma = \pm}
\int_{0}^{1}{\ln\left(1 + {\rm i}\,\sigma\,\phi^{1/2}x\right) \over 1 + x}\,{\rm d}x
=
\sum_{\sigma = \pm}
\int_{1}^{2}{\ln\left(1 - {\rm i}\,\sigma\,\phi^{1/2} + {\rm i}\,\sigma\,\phi^{1/2}x\right) \over x}\,{\rm d}x
\\[2mm]&=
\sum_{\sigma = \pm}\left\lbrack\ln\left(1 - {\rm i}\,\sigma\,\phi^{1/2}\right)\ln\left(2\right)
+
\int_{1}^{2}{\ln\left(1 - z_{\sigma}x\right) \over x}\,{\rm d}x\right\rbrack
\end{align}
where $\displaystyle{z_{\sigma}
                    \equiv
                     -\,{{\rm i}\,\sigma\,\phi^{1/2} \over 1 - {\rm i}\,\sigma\,\phi^{1/2}}}$
\begin{align}
\int_{0}^{1}{\ln\left(1 + \phi x^{2}\right) \over 1+x}\,{\rm d}x
&=
\ln\left(1 + \phi\right)\ln\left(2\right)
+
\sum_{\sigma = \pm}\left\lbrack%
\int_{0}^{1}{\ln\left(1 - 2z_{\sigma}x\right) \over x}\,{\rm d}x
-
\int_{0}^{1}{\ln\left(1 - z_{\sigma}x\right) \over x}\,{\rm d}x
\right\rbrack
\\[2mm]&=
\ln\left(1 + \phi\right)\ln\left(2\right)
+
\sum_{\sigma = \pm}\left\lbrack\vphantom{\Large A}%
-{\rm Li_{2}}\left(2z_{\sigma}\right) + {\rm Li_{2}}\left(z_{\sigma}\right)
\right\rbrack
\\[1cm]&
\end{align}
\begin{align}
\int_{0}^{1}{\ln\left(1 + \phi x^{2}\right) \over 1+x}\,{\rm d}x
&=
\ln\left(1 + \phi\right)\ln\left(2\right)
+
2\Re\left\lbrack\vphantom{\Large A}%
{\rm Li_{2}}\left(z\right) - {\rm Li_{2}}\left(2z\right)
\right\rbrack
\\[2mm]&
\mbox{with}\quad z = -\,{{\rm i}\,\phi^{1/2} \over 1 - {\rm i}\,\phi^{1/2}}
\end{align}
See http://en.wikipedia.org/wiki/Polylogarithm#Dilogarithm
A: Let's replace $\phi$ with a positive parameter $a$:
$$
   \mathcal{I}(a) = \int_0^1 \frac{\log(1+a x^2)}{1+x} \mathrm{d}x \stackrel{x^2=t}{=} \int_0^1 \frac{\log(1+ a t)}{1+\sqrt{t}} \frac{\mathrm{d}t}{2 \sqrt{t}}
$$
Differentiating under the integral sign:
$$\begin{eqnarray}
   \mathcal{I}^\prime(a) &=& \int_0^1 \frac{\sqrt{t}}{2\left(1+a t\right)\left(1+ \sqrt{t}\right)} \mathrm{d}t = \left[\frac{\log \left(1+\sqrt{t}\right)}{1+a}+\frac{\log (1+a t)}{2 a
   \left(1+a\right)}-\frac{\arctan\left(\sqrt{a t}\right)}{\sqrt{a}\left(1+a\right)} \right]_0^1 \\
 &=& \frac{\log 2}{1+a} + \frac{\log(1+a)}{2 a \left(1+a\right)} - \frac{\arctan(\sqrt{a})}{\sqrt{a} \left(1+a\right)} \\ 
 &=& \frac{\log(1+a)}{2} \left(\frac{1}{a} - \frac{1}{1+a} \right) + \frac{\mathrm{d}}{\mathrm{d}a}\left( \log 2 \cdot \log\left(1+a\right) - \arctan^2\left(\sqrt{a}\right)\right) \\
 &=& \frac{\log(1+a)}{2 a} + \frac{\mathrm{d}}{\mathrm{d}a}\left( \log 2 \cdot \log\left(1+a\right) + \frac{\log^2\left(1+a\right)}{4} - \arctan^2\left(\sqrt{a}\right)\right)
\end{eqnarray}
$$
Hence, for $\alpha>0$
$$\begin{eqnarray}
  \mathcal{I}\left(\alpha\right) &=& \int_0^\alpha \mathcal{I}^\prime(a) \mathrm{d}a = \left[ \log 2 \cdot \log\left(1+a\right) + \frac{\log^2\left(1+a\right)}{4} - \arctan^2\left(\sqrt{a}\right) \right]_{0}^{\alpha} + \frac{1}{2} \underbrace{\int_0^\alpha \frac{\log(1+a)}{a} \mathrm{d}a }_{-\operatorname{Li}_2\left(-\alpha\right)} \\ 
&=& \frac{\log^2\left(1+\alpha\right)}{4} + \log(2)\log\left(1+\alpha\right) - \arctan^2\left(\sqrt{\alpha}\right) - \frac{1}{2} \operatorname{Li}_2\left(-\alpha\right)
\end{eqnarray}
$$
Quadrature confirmation:
In[45]:= N[
 With[{a = GoldenRatio}, -ArcTan[Sqrt[a]]^2 + Log[2] Log[1 + a] - 
   1/4 Log[1 + a]^2 - 1/2 PolyLog[2, -a]], 30]

Out[45]= 0.226575874320240735184008669784

In[46]:= NIntegrate[Log[1 + GoldenRatio x^2]/(1 + x), {x, 0, 1}, 
  WorkingPrecision -> 70] - %

Out[46]= 0.*10^-31

