Integral of Log-Sine: Proving \(G\) Dependence

[Shobhit]

Show that

$$\int_0^{\frac{\pi}{4}}\log\left( \sqrt{\sin^3 x}+\sqrt{\cos^3 x}\right) \text{d}x = \dfrac{G}{12}-\dfrac{5\pi}{16}\log{2}+\dfrac{\pi}{8}\log{\left(3-2\sqrt{2}\right)}+\frac{\pi​}{3}\log{\left(2+\sqrt{3} \right)}$$

\(G\) denotes the Catalan's Constant.

 

[Aaron Bex]

Using the substitution \(t=\sin x\), we have
\begin{align*}
\int_0^{\frac{\pi}{4}}\log\left( \sqrt{\sin^3 x}+\sqrt{\cos^3 x}\right) \text{d}x &= \int_0^{1}\log\left( \sqrt{t^3}+\sqrt{(1-t^2)^3}\right) 3t^2\text{d}t \\
&= 3\int_0^{1}\log\left( \sqrt{t}+\sqrt{1-t}\right) t^2\text{d}t \\
&= 3\int_0^{1}\log\left( \dfrac{t+1-t}{\sqrt{t(1-t)}}\right) t^2\text{d}t \\
&= 3\int_0^{1}\log\left( \dfrac{1}{\sqrt{t(1-t)}}\right) t^2\text{d}t + 3\int_0^{1}\log\left( t+1-t\right) t^2\text{d}t \\
&= 3\int_0^{1}\log\left( \dfrac