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In court papers, judges have noted that some of the complaining companies have provided contraception coverage in the past, but are changing course during the implementation of Mr. Obama’s signature law. That could play into the legal analysis because injury is difficult to claim when it involves something done “voluntarily for many years,” said Brigitte Amiri, a staff attorney at the ACLU.

Francis J. Manion, senior counsel for the American Center for Law and Justice, said many organizations reviewed their insurance plans amid the high-profile debate and were surprised to find contraceptive coverage.

“It’s more of a [public relations] issue than a legal issue,” said Mr. Manion, whose group is representing some of the plaintiffs.

Matt Bowman, an attorney for the Alliance Defending Freedom, which is handling or is associated with several of the for-profit cases, defended companies’ right to rethink their policies.

“The federal government is not allowed to outlaw repentance,” he said.

Complex question

The catalog of preliminary court rulings to date shows that some plaintiffs are fine with typical forms of contraception, objecting instead to drugs with an “abortifacient effect” such as the Plan B, or “morning-after,” pill that is used after conception.

Some go further. A farm operator in Michigan cited as binding on his conscience Catholic teaching against all forms of contraception, reiterated in the 1968 encyclical Humanae Vitae by Pope Paul VI, which stated that attempts to prevent procreation “before, at the moment of, or after sexual intercourse” amount to a “grave sin.”

Some of the highest-profile cases include Tyndale House Publishers and Mardel — an affiliate of Oklahoma City-based crafts supplier Hobby Lobby — both of which distribute Christian books or supplies.

The Obama administration this week appealed a district court judge’s ruling granting Tyndale a preliminary injunction against the contraceptive insurance mandate.

Their involvement raises the question of whether corporations with clear religious bents have stronger cases than, say, a general contractor. The injunction granted by District Judge Reggie Walton applied only to Tyndale.

Mr. Jost said it is a complex question that the court may have to sort out among the cases from for-profit entities, since opting to work for a corporation whose owners have deeply held beliefs is less apparent than going to work for a hospital or school with a religious affiliation.

“I do that with my eyes open,” he said of the latter case, “and I realize there’s going to be a crucifix in my classroom or something like that.”