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I know that this is an unusual function to talk about. Is there any chance that the integral exists in terms of standard functions?

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    Short answer: sorry, no. – 2017-02-20
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    As claimed by @EthanBolker, the integral doesn't seem to exist in terms of [standardized functions](http://www.wolframalpha.com/input/?i=integrate+sin+(e%5E(x%5E2))). – 2017-02-20
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    @Ethan Bolker: If I use $tan$ instead of $sin$, then? – 2017-02-20
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    Still no. Not cosine either. – 2017-02-20
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    See https://en.wikipedia.org/wiki/Nonelementary_integral – 2017-02-20
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    @Ethan Bolker: What if I remove the square, then? – 2017-02-20
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    Still no. See the wikipedia entry in my previous comment. And please don't change questions on the fly - it confuses potential answerers. You can edit the question to clarify. In this case no simplification along the lines you've tried will help. – 2017-02-20
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    @Ethan Bolker: I looked up the wikipedia article. It said you can't even rely on taylor series for integrating some functions. Can I integrate $sin{e^x}$ using taylor series? – 2017-02-20
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    Taylor series will work for this one and for many like it. See the wikipedia article sentence beginning "This is because Taylor series can always be integrated as one would an ordinary polynomial ..." – 2017-02-20

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$$\int \sin(e^x)\,dx \stackrel{x\mapsto\log t}{=}\int\frac{\sin t}{t}\,dt = \text{Si}(t) = \color{red}{\text{Si}(e^x)}$$ hence the answer depends on you considering the sine integral an elementary function or not.