The reason why the subjunctive is used with "dudo que" is because it is a doubtful sentence, while "no dudo que" can refer to an affirmation, for which the verb could be conjugated depending on the tense in which it is used.
Subjunctive in Spanish
The subjunctive form in Spanish is used to mention events whose occurrence is not certain, so when "dudo que," is mentioned, which makes it a hesitant sentence, it usually requires that the following verb be in the subjunctive.
However, when the expression "no dudo que," is used, it refers to a fact which can be verified (affirmative form), or it could be used with the subjunctive form as well.
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