American English speaker: The answers that "can't be bothered" is and is not a common American phase are both sorta correct.

Americans would use the logically equivalent "I won't bother" or "I didn't bother".

"I didn't bother doing X" usually means something along the lines of "I treated X as optional or unnecessary" (justified or not).

"I didn't bother doing my homework last night." "I didn't bother rinsing my dishes before I put them in the dishwasher." "I didn't bother sorting my clothes before washing them." "He got fired because he didn't bother telling his boss he would be out of town." "I am not going to bother adding outside sources to this answer."

American English speaker: The answers that "can't be bothered" is and is not a common American phase are both sorta correct.

Americans would use the logically equivalent "I won't bother" or "I didn't bother".

"I didn't bother doing X" usually means something along the lines of "I treated X as optional or unnecessary" (justified or not).

• "I didn't bother doing my homework last night."
• "I didn't bother rinsing my dishes before I put them in the dishwasher."
• "I didn't bother sorting my clothes before washing them."
• "He got fired because he didn't bother telling his boss he would be out of town."
• "I am not going to bother adding outside sources to this answer."