Finding $\int\frac{\sqrt{1+\sqrt{1+\sqrt{1+\cos(2\sqrt{x+5})}}}}{\sqrt{x}} dx$

Question

The following integral is well posed? we must correct?

$$\int\frac{\sqrt{1+\sqrt{1+\sqrt{1+\cos(2\sqrt{x+5})}}}}{\sqrt{x}}dx$$

Any hint would be appreciated.

Answer 1

I'm not sure what the OP means by 'well posed'. It's an indefinite integral, so there is no question about convergence.

As for the antiderivative, it exists, but most likely can't be found in closed form, as it was said in the comments.

However, we can simplify the integral a little.

First, make a change of variable:

$$y=\sqrt{x+5}$$

$$x=y^2-5$$

$$dx=2y~dy$$

Then we obtain

$$2 \int \sqrt{1+\sqrt{1+\sqrt{1+\cos(2y)}}} \frac{y ~dy}{\sqrt{y^2-5}}$$

Now use the double angle formula:

$$|\cos y| = \sqrt{\frac{1+\cos 2y}{2}}$$

We get:

$$2 \int \sqrt{1+\sqrt{1 + \sqrt{2} |\cos y|}} \frac{y ~dy}{\sqrt{y^2-5}}$$

Here $||$ denotes absolute value. We got rid of one of the nested roots. I don't think we can simplify it any further.