#### Background

I am studying the area of the implicit function curve

$$y^2=\sin \left(e^x\right).$$

I found that I need to calculate this integral:

$$I=\int\_{-\infty }^{\ln \pi} \sqrt{\sin e^t} \, \rm{d}t.$$

#### Try 1

let $t = \sin e^x$, then

$$\int \frac{\sqrt{t}}{\sqrt{1-t^2} }\frac{1}{\arcsin t}\,\rm{d}t$$

I can't figure out how to calculate this.

#### Try 2

let $t = e^x$, then

$$\int\_{0}^{\pi} \frac{\sqrt{\sin (t)}}{t}\,\rm{d}t$$

This form looks simple, but I still don't know how to calculate.

#### Question

Does $I$ have closed form?