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How can you integrate

$$\int \exp{\left( -x-e^{-x}\right)}dx$$

I have tried some substitutions but they make things harder than before. Is there some trick that I can use to solve this?

Finishing the integration based on the anwers:

Let $u=-e^{-x} \implies du=e^{-x}dx \iff dx= e^x du$

Substituting:

$$\int e^u du=e^u +c $$

Plugging back in:

$$\int \exp{\left( -x-e^{-x}\right)}dx=\boxed{e^{-e^{-x}}+c}$$

Nullspace
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$$\int \exp(-x-e^{-x})dx= \int e^{-x} e^{-e^{-x}}dx$$ Taking $-e^{-x}=t$, you'll get that $e^{-x}dx=dt$ so... can you finish?

Tito Eliatron
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using the substitution $y=e^{-x}$ the integral becomes trivial

$$-\int e^{-y}dy$$

tommik
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    I think it's $-\int e^{-y}dy$ – Tito Eliatron May 13 '21 at 10:44
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    @TitoEliatron: of course....typo amended :( That is the reason why I wrote "the integral becomes trivial". Thanks for noticing that – tommik May 13 '21 at 10:45
  • @tommik Please do not answer low quality question. See here. – vitamin d May 13 '21 at 10:48
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    @tommik Thanks! It helped a lot. I finished the integration and edited my post to include the answer. – Nullspace May 13 '21 at 10:52
  • @vitamind I don't understand why this is low quality? I asked for a hint or a suggestion in order to be able to continue and not for a completed answer. I finished the integral myself and edited my opening post to include the answer. – Nullspace May 13 '21 at 10:54
  • @Nullspace Your question before the edit was: "I have tried some substitutions but they make things harder than before. Is there some trick that I can use to solve this?" No context, no visible attempt. – vitamin d May 13 '21 at 10:56
  • @Nullspace I appreciate that you took the time to attempt the question after the hint was given in the comments. You said you tried some substitutions : you could have mentioned these. The source of the question can be mentioned. Your level, or any similar questions you've seen can be mentioned. All these count as context. I'm not always asking for an attempt, just something to convince us that you have put effort into framing and asking the question, beyond merely stating just the question. That's all. – Sarvesh Ravichandran Iyer May 13 '21 at 13:12
  • @TeresaLisbon That makes sense. I'll try to include some context and my attempts next time. Thanks! – Nullspace May 13 '21 at 14:56
  • @Nullspace Very appreciated, thanks for the reply. – Sarvesh Ravichandran Iyer May 13 '21 at 16:03