Once the duty of prayer has been established in its own right, no number of objections can militate decisively against it.  B.M. Palmer writes:-

Truth is not to be weighed in a pair of scales balancing duty against difficulty, as though the one can be cancelled by the other; for no positive duty can be dislodged, until the evidence on which it rests shall be itself swept away.

But the objections to prayer need to be considered, with a view to reclaiming the infidel, protecting against dangerous theories, and guarding against doubts.

The 1st objection is that prayer is an impeachment of the Divine perfections.

God is infinitely wise and good, and his plans are immutable; therefore prayer is an impertinence.

Answer:-

1.  The objection does not lie.  The fact is, that the same God who is declared to be infinitely wise and good (and so he must be, else he must cease to be God), is the same God who commands us to pray.

The simple command to pray, issuing from the Being who is most concerned in giving it, yields beforehand the assurance of its entire consistency with his own integrity.

2.  The necessity of prayer is found imbedded in man’s moral constitution, compelling its exercise.  The instinct to pray is illustrated in the case of psychiatric patients, many of whom ‘unhinged on the subject of religion.’  While mentally stable, man’s reason keeps his religious aspirations under check, ‘when reason is unshipped, and the wrecked intellect drifts at the mercy of the waves, the great secret leaks out because the power of concealment is gone; discovery is made that throughout his whole career the man’s religious nature has always secretly asserted itself.’

But,

it is It is not left to the wreck of reason to bear this compulsory testimony to this startling truth.  There are moments in which the soundest and strongest intellect bows to the force of this inextinguishable conviction; moments of peril and of pain, when the weakness of man pours its pitiful moan into the ear of God; moments of sorrow and bitter grief, when the mask is torn from the face of the earthly life and eternal solemnities break upon the soul.

This instinct to pray is also seen in a curiously perverted way in man’s tendency to swear and curse:-

How universal the vicious habit is! Wherever one goes the ear is smitten with these direful oaths, in which the blessed name which all should revere is tossed from lip to lip with lightness, if not with scorn. Most significantly, too, men swear the most dreadfully when most aroused and thoroughly in earnest, in moments of anger and of deep passion. Nor should it be overlooked that the oaths which are the most common are those which invoke the dreadful God in the exercise of his inflexible justice, and appeal the most directly to the retributions of the judgment day and the pains of eternal perdition. To a thoughtful mind all this is oppressively significant and solemn. By what overruling constraint are men held to the acknowledgment of God, just when they are most forgetful of him ? Why should the wicked, in their moments of wickedness, bear the most emphatic testimony to the penalty that awaits them ? How comes it to pass that bad men should make more frequent and fervent appeals to the Almighty than good men are able to do with all their devotion ? It will not do to deny that these imprecations are prayers; to affirm that they are mere expletives, only designed to fill a blank in one’s speech…Were it a mere expletive, any chance word would fill the chasm; why should the Divine Being be always invoked, and that, not only in the use of his awful name, but in the challenge of his office as the ruler and judge of men?…Unquestionably, what is universal in human practice must have its ground in human nature itself. What can this be but the conviction of God’s being and presence, and of his moral government over men, which even the wicked cannot shake off because laid at the very root of their nature?

3.  If this objection were valid, it would lead to the absurdity of cancelling all religion.  If God is God, then he is worthy of our homage.

A nature endowed with such traits as those we claim for man, yet walled up within itself and with no outlet for all the gratitude and praise with which it is filled, is a confessed lie, and dies of its own exhaustion. It gives endowments with no corresponding object on which to terminate; an understanding with no access to the truth on which it shall feed; a heart with no supreme goodness upon which its affections may fasten; a conscience with no infallible standard to which its decisions may be referred; a will which is only a lawless force flapping loosely about with no commands it shall obey. Shut man out from approach to God in prayer, and his whole religious nature becomes a delusion and a sham.

This objection is absurd, seeking ‘to wall-in the Deity behind the solid masonry of his own attributes.’

4.  This objection proceeds on a total misconception of the office of prayer.  For prayer is not the conference of equals:

In prayer the creature simply expresses his sense of emptiness and want to be supplied from God’s infinite fulness ; he confesses his trangressions that they may be forgiven in the sovereignty of divine mercy; he pours forth his gratitude and praise in acknowledgment of God’s free bounty. In each case alike, the prayer does not contravene a single attribute, but builds upon it as the only ground of expectation and of hope.

Further:

are not all the forms of human labor so many expressions of reliance upon the divine fidelity to his promises; and are they not to be construed as prayers which are acted rather than spoken before God?

B.M Palmer, Theology of Prayer