# $\int_{0}^{\frac{\pi}{2}}\cos{x}\sqrt{1+\tan{x}}dx$

Find this integral $$I=\int_{0}^{\frac{\pi}{2}}\cos{x}\sqrt{1+\tan{x}}dx$$

My try:I find this wolf can't find it. **

then I try: let $$\sqrt{1+\tan{x}}=t\Longrightarrow x=\arctan{(t^2-1)}$$ so $$I=\int_{1}^{+\infty}\cos{\left(\arctan{(t^2-1)}\right)}\dfrac{2t^2}{(t^2-1)^2}dt$$

Then I can't

• Mathematica returns a value for the definite integral expressed in terms of the Meijer G function. – Lucian Nov 16 '13 at 16:32
• @Lucian. If you want fun, just compute the antiderivative and post it ! – Claude Leibovici Nov 16 '13 at 16:41
• I'm running FullSimplify[...] on that dreadful formula as we speak... :-) P.S.: I've posted, or at least mentioned, two such monsters on this post. – Lucian Nov 16 '13 at 16:46
• Don't know if this will be helpful or not, but another equivalent form of the integral is $I = \int_0^{\infty}\frac{\sqrt{1+u}}{(1+u^2)^{\frac32}}\space du$. – David H Nov 16 '13 at 17:01
• Another form of this integral is $$I=\int_{0}^{1}\sqrt{1+\frac{y}{\sqrt{1-y^{2}}}}\mathrm{d}y$$ and maybe it can help. – Leonida Nov 16 '13 at 17:44


\begin{align} I&=\int_{0}^{\pi/2}\root{\cos^{2}\pars{x} + \sin\pars{x}\cos\pars{x}}\,\dd x = \int_{0}^{\pi/2}\root{{1 + \cos\pars{2x} \over 2} + {\sin\pars{2x} \over 2}}\,\dd x \\[3mm]&= {1 \over 4}\,\root{2}\int_{0}^{\pi}\root{1 + \cos\pars{x} + \sin\pars{x}}\,\dd x = {1 \over 4}\,\root{2} \int_{0}^{\pi}\root{1 + \root{2}\sin\pars{x + {\pi \over 4}}}\,\dd x \\[3mm]&= {1 \over 4}\,\root{2} \int_{\pi/4}^{5\pi/4}\root{1 + \root{2}\sin\pars{x}}\,\dd x \end{align}

The $\it\underline{last\ integral}$ is evaluated here in terms of a Second Kind Elliptic function.

This is an awful integral ! The formula for the antiderivative write in several pages. The numerical value is : 1.3571445175439115954095406.

• I think this integral can take some constant – math110 Nov 16 '13 at 16:34
• The result is a constant as posted by Lucian. – Claude Leibovici Nov 16 '13 at 16:43

As I wrote earlier, its value is non-elementary, and expressible in terms of the Meijer G function:

The expression of its anti-derivative can be found here. It involves incomplete elliptic integrals of the first and second kind.