<s>[INST] $\int_{0}^{\pi/2}\ln\left(1+4\sin^4 x\right)\mathrm{d}x$ and the golden ratio We already know that, for any real number $t$ such that $t\geq-1$,

$$
\int_{0}^{\pi/2} \ln \left(1+t \sin^2 x\right) \mathrm{d}x = \pi \ln \left( \frac{1+\sqrt{1+t}}{2} \right).
$$

Prove that

$$
\int_{0}^{\pi/2} \ln \left(1+4\sin^4 x\right) \mathrm{d}x = \pi \ln \left( \frac{\varphi+\sqrt{\varphi}}{2} \right)
$$ 
  where $\displaystyle \varphi = \frac{1+\sqrt{5}}{2}$ is the golden ratio.

 [/INST] Let's put it with real numbers.
We have

$$
\int_{0}^{\pi/2} \ln \left(1+t\sin^4 x\right) \mathrm{d}x 
= \pi \ln \left( \frac{1}{4} \sqrt{ 1 + \sqrt{1+t} } 
\left( \sqrt{1 + \sqrt{1+t} } + \sqrt{2} \right) \right), \quad t \geq -1.
$$ 

</s>