
die Betrachtung
observation
The careful and attentive looking at or observing of something or someone.
Die Betrachtung des alten Gemäldes dauerte mehrere Minuten.
The observation of the old painting lasted several minutes.
observation / consideration
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observation
The careful and attentive looking at or observing of something or someone.
Die Betrachtung des alten Gemäldes dauerte mehrere Minuten.
The observation of the old painting lasted several minutes.

reflection / consideration
A mental engagement with a topic or question; an analysis or reflection.
Seine Betrachtungen zur politischen Lage sind sehr tiefgründig.
His reflections on the political situation are very profound.

to consider
To include something as a possibility in one's thoughts; to consider something.
Wir sollten auch alternative Lösungen in Betracht ziehen.
We should also consider alternative solutions.

on closer inspection
Expresses that one examines something more closely or carefully and gains a new insight in the process.
Bei näherer Betrachtung stellte sich heraus, dass der vermeintliche Fehler gar keiner war.
On closer inspection, it turned out that the supposed mistake was not a mistake at all.
The noun 'die Betrachtung' (observation / consideration) is derived directly from the verb 'betrachten' (to look at / consider). If you know the verb, you can infer the meaning of the noun more easily: 'betrachten' means to look at something closely or to think about it.
Er betrachtet das Kunstwerk mit großem Interesse.
He looks at the artwork with great interest.
'Betrachtung' (observation / consideration) is very often used with the genitive to say exactly what is being looked at or thought over. The structure is: 'die Betrachtung' + genitive.
Die Betrachtung der Sterne fasziniert die Menschen seit jeher.
The observation of the stars has always fascinated people.
The context reveals the meaning: If it is about something concrete that can be seen (e.g. a picture, a landscape), visual observation is meant. If it is about an abstract topic (e.g. a question, a problem), mental analysis is meant.
Die stille Betrachtung der Natur beruhigt mich.
The quiet observation of nature calms me.
The expression 'etwas in Betracht ziehen' (to consider something) is very common and means 'etwas erwägen' (to consider something). The thing being considered is in the accusative.
Wir ziehen einen Umzug in eine andere Stadt in Betracht.
We are considering moving to another city.
The expression 'bei näherer Betrachtung' (on closer inspection) often comes at the beginning of a sentence. Then the verb follows directly, and after that the subject (inversion).
Bei näherer Betrachtung erkannte ich die feinen Details des Musters.
On closer inspection, I recognized the fine details of the pattern.
analysis
Both words can refer to thinking about a topic. An Analyse typically requires a systematic examination of details and connections, while a Betrachtung can also be more general and reflective. Use “Analyse” for a structured breakdown and “Betrachtung” for a broader consideration without this strong emphasis.

Die Betrachtung des Problems zeigt, dass auch soziale Folgen wichtig sind.
Considering the problem shows that social consequences are also important.
Die Analyse des Problems vergleicht seine einzelnen Ursachen und Folgen.
The analysis of the problem compares its individual causes and consequences.