How to solve this integral which involves trigonometry?

Question

I found an integral online that I cannot seem to solve. I tried putting in Wolfram Alpha, however, I did not get any output. The integral is: $$\int\frac{dx}{\sqrt{1-\cos^3(x)}}$$ Does anyone have any hints on how to solve this?

Answer 1

If there is not any typo, then the integral is surely not trivial.

Well, it's better for you. It will introduce you into the world of Elliptic Integrals, which sooner or later you shall study (maybe).

So I will write something for the sake of curiosity!

The result of the integral is

$$\int \frac{1}{\sqrt{1-\cos^3(x)}}\ \text{d}x = \frac{\sin (2 x)-2 \cos ^{\frac{3}{2}}(x) E\left(\left.\frac{x}{2}\right|2\right)}{\sqrt{\cos ^3(x)}}$$

Where the $E$ function is called: Elliptic integral of the second kind.

It's defined as follows (in your case):

$$E\left(\left.\frac{x}{2}\right|2\right) = \int_0^{x/2}\sqrt{1 - 2\sin^2\theta}\ \text{d}\theta$$

Or in general:

$$E\left(\left.\phi\right|m\right) = \int_0^{\phi}\sqrt{1 - m\sin^2\theta}\ \text{d}\theta$$