Maple is seriously used in my article Approximation of subharmonic functions in the halfplane by the logarithm of the modulus of an analytic function. Math. Notes 78, No 4, 447455 in two places. The purpose of this post is to present these applications. First, I needed to prove the elementary inequality (related to the properties of the minimal harmonic majorant of the function 1/Im z in a certain strip) 2R+sqrt(R)R(R+sqrt(R))y  1/y ≤ 1/4 for y ≥ 1/(R+sqrt(R)) and y ≤ 1/R, the parameter R is greater than or equal to 1. The artless attemt

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fails. The second (and successful) try consists in the use of optimizers:

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To be sure , ;

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Because 0.17 Now we establish this by the use of the derivative.

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The second use of Maple was the calculation of the asymptotics of the following integral (This is the double integral of the Laplacian of 1/Im z over the domain {z: ziR/2 < R/2} \ {z: z ≤ 1}.). That place is the key point of the proof. Its direct calculation in the polar coordinates fails.

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In order to overcome the difficulty, we find the inner integral

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and then we find the outer integral. Because

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is not successful, we find the indefinite integral

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We verify that the domain of the antiderivative includes the range of the integration.
That's all right. By the NewtonLeibnitz formula,

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found by

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It should be noted that a somewhat different expression is written in the article. My inaccuracy, as far as I remember it, consisted in the integration over the whole disk {z: ziR/2 < R/2} instead of {z: ziR/2 < R/2} \ {z: z ≤ 1}. Because only the form of the asymptotics const*R^2 + remainder is used in the article, the exact value of this nonzero constant is of no importance.
It would be nice if somebody else presents similar examples here or elsewhere.
