TITLE: Evaluate the following integral :$\int\limits_0^{\infty}\frac{\log (1+x^{4})}{\sqrt{x}(1+x)}dx$
QUESTION [6 upvotes]: Evaluate the following integral : 

$$I=\int\limits_0^{\infty}\frac{\log (1+x^{4})}{\sqrt{x}(1+x)}dx$$

I was tried use change variable , 
If I use $x=y^2$ integral becomes : 
$$I=2\int\limits_0^{\infty}\frac{\log (1+x^{8})}{1+x^{2}}dx$$
From here I have one idea  the derivative under sing integral but I got I difficult integration : 
$$I=2\int\limits_0^{\infty}\frac{x^{8}}{(1+ax^{8})(1+x)}dx$$
I already to see you hints or solution!

REPLY [4 votes]: Note
$$1+x^8=(1+e^{i \frac\pi4}x^2)(1+e^{-i \frac\pi4}x^2)(1+e^{i \frac{3\pi}4}x^2)(1+e^{-i \frac{3\pi}4}x^2)$$
$$J(a)=\int_0^\infty \frac{\ln(1+ax^2)}{1+x^2}dx= \pi\ln(1+a^{\frac12})
$$
Then
\begin{align}
I&=\int\limits_0^{\infty}\frac{\ln (1+x^{4})}{\sqrt{x}(1+x)}dx\overset{x\to x^2}
=2\int\limits_0^{\infty}\frac{\ln (1+x^{8})}{1+x^{2}}dx\\
&= 2\left[J(e^{i \frac\pi4})+ J(e^{-i \frac\pi4})+J(e^{i \frac{3\pi}4}) +J(e^{-i \frac{3\pi}4})
\right]\\
&=2\pi \ln\left[(1+e^{i \frac\pi8})(1+e^{-i \frac\pi8})(1+e^{i \frac{3\pi}8})(1+e^{-i \frac{3\pi}8})\right]\\
&=2\pi \ln\left[4(1+\cos\frac\pi8)(1+\cos\frac{3\pi}8) \right]\\
&=4\pi\ln (\sqrt2+\sqrt{2+\sqrt2})
\end{align}