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Justice Antonin Scalia, a devout Catholic with a conservative record, wrote the Smith opinion that effectively limited Americans’ ability to cite their religious beliefs in seeking exemption from valid laws. Otherwise, Justice Scalia wrote at the time, it “would open the prospect of constitutionally required religious exemptions from civic obligations of almost every conceivable kind.”

“The criticism he got from the religious community on that was huge,” Mr. Burrus said.

After enactment of the Religious Freedom and Restoration Act, the court used the law in 2006 to uphold the religious freedoms of a New Mexico church whose sacramental tea had been seized by federal agents because it contained an illegal hallucinogen.

Mr. Barnett noted that Chief Justice John G. Roberts Jr., a member of the court’s conservative wing, was aggressive in applying the act in the case Gonzales v. O Centro Espirita Beneficente Uniao do Vegetal, even if members of the conservative wing are not inclined to be lenient about drug use in general.

Free-speech questions

There also is a free-speech component to the case. The 10th Circuit judges who sided with Hobby Lobby cited the landmark Citizens United v. Federal Election Commission case of 2010, which held that corporations hold free-speech rights that entitle them to make independent political expenditures.

“The idea that people don’t lose their rights when they organize in a corporate form is common to both situations,” Mr. Barnett said.

Despite conventional wisdom that breaks the court into liberal and conservative wings and a swing vote — often Justice Anthony M. Kennedy — it is difficult to say which justices might uphold a business’ right to religious freedom, the analysts said.

Mr. Burrus said Justice Stephen G. Breyer, who is typically associated with the court’s liberal wing, is partial to judicial balancing tests, or measuring various interests against each other, or striking a balance between competing interests. The Obama administration has been giving out exemptions to the health care law as it grapples with the high number of people who lost existing health care plans that did not meet Obamacare’s coverage requirements.

The administration also issued an accommodation for nonprofit employers who object to the contraception mandate, so it may be difficult for the government to argue that it is impossible for them to carve out an exception for the for-profit companies.