The efficacy of prayer is not rationally provable because: if supplicatory prayer is not an act of submission it is then a demand. Therefore, determination of efficacy requires knowledge of the heart of the prayer; such knowledge is not within the power of men.
Does it? If a believer prays for something against massive odds and the prayer is answered do I need to know their heart to observe the answered prayer? Now if it repeatedly fails to beat the odds I suppose you could just claim that the problem wasn't the prayer but their heart and who can tell you otherwise? The problem with that position is we are chock full of people who claim their prayers do achieve real world results so whatever criteria you want to claim for the prayer to work, they are claiming they have met that criteria and do it with great enough frequency to keep them praying and recommending prayer.