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

Question

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!

Answer 1

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 $B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$.