Question on integral containing exponential and sine function

How to get estimate on following integral:

$$f(x)=\int\frac{\sin^2(x)}{e^{\sin^2(x)}}dx \,?$$

I tried doing it so by putting $u= \sin^2(x)$, that way we get:

$$\int\frac{\sqrt{u}}{\sqrt{1-u^2}e^u}dx.$$

But, again I could not get a good estimate.

It would also good if I could get a good asymptotic on it.

Sharp Result of sort $f(x)=g(x)+O(h(x))$ would work for my answer.

Solution 1:

We seek to estimate $f(x)=\frac{1}{\exp(\sin^2x)}$ in terms of simple trigonometric functions.

A good estimate of $f(x)$ can be found among the following curves: $$ g(x)=a\cos^4x+b\cos^2x+c $$ where $$g(0)=f(0)\implies a+b+c=1$$and $$ g(\frac{\pi}{2})=f(\frac{\pi}{2})\implies c=e^{-1} $$ hence $$ g(x)=a\cos^4x+(1-e^{-1}-a)\cos^2x+e^{-1} $$ where a good numerical choice for $a$ is $0.3$ and a plot of both functions is as follows

Hence $$ \sin^2x\exp(-\sin^2 x){\approx a\sin^2 x\cos^4 x+b\sin^2 x\cos^2 x+c\sin^2 x \\=0.3\sin^2 x\cos^4 x+0.3321\sin^2 x\cos^2 x+0.3679\sin^2 x. } $$