Integration problem: $\int \ln\left(\sin(\sqrt{x})+\cos(\sqrt{x})\right)dx$

I need help in solving the following problem:

$$\int \ln\left(\sin(\sqrt{x})+\cos(\sqrt{x})\right)dx$$

I really don't know how to start solving this problem; any tips or solutions will be greatly appreciated.

• $\text{Li}_2$ and $\text{Li}_3$ involved, are you fine with that? – Jack D'Aurizio Jun 28 '15 at 12:35
• Put $\sqrt{x}=t$ then integrate by parts – Pratyush Jun 28 '15 at 12:36
• @JackD'Aurizio Yup.I'm good with it. I'm pretty sure I have almost no ability to solve problems with polylogs, let alone integrals involving them... but it's good to get solutions for problems which one may understand much better in the future. – Kugelblitz Jun 28 '15 at 12:59

By setting $x=u^2$ we have: $$I = \int 2u\log(\sin u+\cos u)\,du = u^2\log(\sin u+\cos u)-\int u^2\frac{1-\tan u}{1+\tan u}\,du$$ and, by putting $v=\frac{\pi}{4}-v$, $$-\int u^2\frac{1-\tan u}{1+\tan u}\,du = \int \left(\frac{\pi}{4}-v\right)^2 \tan v\,dv$$ Now we may exploit $\int\tan v\,dv=-\log\cos v$, so the last integral just depends on: $$\int v \log(\cos v)\,dv$$ that, however, is not an elementary function, but a combination of a logarithm, a dilogarithm and a trilogarithm multiplied by powers of $v$: just write $\cos v$ as $\frac{e^{iv}+e^{-iv}}{2}$, exploit the Taylor series of $\log(1+z)$ and integrate termwise.

• Thank you. If it's not too much trouble, could you add in a step or two to at least begin solving the last integral in your current solution? It'll be a useful reference for me in the future; meanwhile I'm continuing to amass knowledge of special functions and integrals and problems related to the aforementioned. – Kugelblitz Jun 28 '15 at 13:04
• Oh thank you! You did just that with your edit. Thank you sir :D – Kugelblitz Jun 28 '15 at 13:04

I really don't know how to start solving this problem

This is hardly surprising, since the anti-derivative cannot be expressed in terms of elementary functions.

any tips or solutions will be greatly appreciated.

$$\int_0^{\big(\tfrac\pi2\big)^2}\ln\Big(\sin\big(\sqrt{x}\big)+\cos\big(\sqrt{x}\big)\Big)~dx ~=~ \frac\pi2~\bigg(\text{Catalan}-\frac\pi4~\ln2\bigg)$$
$$\int_0^{\big(\tfrac\pi4\big)^2}\ln\Big(\sin\big(\sqrt{x}\big)+\cos\big(\sqrt{x}\big)\Big)~dx ~=~ \frac{21~\zeta(3)-2\pi^2\ln2}{64}$$