Evaluate $\int_{0}^{\pi/2}{\frac{d\theta}{{\left(\cos^3{\theta}+\sin^3{\theta}\right)}^{2/3}}}$

The problem is to evaluate in closed-form:

$$\int_{0}^{\pi/2}{\left(\frac{d\theta}{{\left(\cos^3{\theta}+\sin^3{\theta}\right)}^{2/3}}\right)}$$

An estimate for the integral is $1.76663875028545$.

I have tried all the usual techniques, including by parts and various $u$-subs, but they all seem to make the integral more contrived. The Weierstrass substitution $x=\tan{\left(\theta/2\right)}$ 'simplifies' the integrand to one with only polynomials, with a nice factorization as well:

$$\int_{0}^{\pi/2}{\left(\frac{d\theta}{{\left(\cos^3{\theta}+\sin^3{\theta}\right)}^{2/3}}\right)}=\int_{0}^{1}{\left(\frac{2\left(1+x^2\right)}{{\left(\left(2-{\left(1-x\right)}^2\right)\left(3x^2+{\left(1-x-x^2\right)}^2\right)\right)}^{2/3}}\,dx\right)}$$

However, I do not know how to continue from here, primarily because the exponent $\frac{2}{3}$ in the denominator still remains.

In addition, both Maple and WolframAlpha could not find a closed-form answer (though that does not mean that one does not exist).

Any ideas? Thanks!

• Have you tried using $f(a+b-x)=f(x)$? – Teh Rod Aug 2 '17 at 15:13
• @TehRod : Do you mean that we can make the upper bound $\pi/4$ instead of $\pi/2$? How else can one apply this? – Ant Aug 2 '17 at 15:18
• I don't know, that's just the first thing that came to mind when I saw this. Especially since there does not appear to be a closed form for the antiderivative – Teh Rod Aug 2 '17 at 15:19
• Better numerical estimation: $1.766638750285906$ – Von Neumann Aug 2 '17 at 15:36

First, enforcing the substitution $\theta=\arctan(x)$ reveals

$$\int_0^{\pi/2}\frac{1}{\left(\cos^3(\theta)+\sin^3(\theta)\right)^{2/3}}\,d\theta=\int_0^\infty \frac{1}{\left(1+x^3\right)^{2/3}}\,dx$$

Next, we let $y=x^3$ to obtain

\begin{align} \int_0^\infty \frac{1}{\left(1+x^3\right)^{2/3}}\,dx&=\frac13\int_0^\infty \frac{y^{-2/3}}{(1+y)^{2/3}}\,dy\\\\ &=\frac13 B\left(1/3,1/3\right)\tag 1\\\\ &=\frac13 \frac{\Gamma^2(1/3)}{\Gamma(2/3)}\tag 2 \end{align}

where $B(x,y)$ is the Beta function and $\Gamma(x)$ is the Gamma function. In going from $(1)$ to $(2)$ we used the relationship $B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$.

• I was going to post an answer, but yours is bloody awesome! +1 – Von Neumann Aug 2 '17 at 15:38
• @HenryTuring Wow! Much appreciated. – Mark Viola Aug 2 '17 at 15:38
• (+1) Very nice! I knew about the Beta function beforehand but didn't think to apply it here. Thanks! – Ant Aug 2 '17 at 15:40
• @JohnChessant, you should accept the answer I guess. – Zaid Alyafeai Aug 2 '17 at 20:29