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I ran across the tangent half-angle substitution, $t=tan(\theta)$, to evaluate integrals of rational trigonometric functions. This substitution manages to translate the integrand from a rational trigonometric function to a rational function in terms of t.

Would it make sense to say that rational functions are generally easier to evaluate than rational trigonometric functions? To be more precise, are the algorithms that a language like Mathematica implements to solve integrals more extensive, or more efficient for rational functions as opposed to rational trigonometric functions?

I am wondering if the Reduction Formulas and all the trigonometric substitutions that are taught at an early undergraduate level are actually used in the algorithms of computer languages for integration, or if trigonometric integrals are almost always converted to a rational function through some sort of substitution and then dealt with by the algorithm for rational functions.

Mike
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  • Of interest to you on this topic may be the Risch algorithm: https://en.wikipedia.org/wiki/Risch_algorithm – law-of-fives May 05 '17 at 17:11
  • @law-of-fives Thanks, I find myself returning to that wiki again and again. I went there after reading https://math.stackexchange.com/questions/461527/on-the-origins-of-the-weierstrass-tangent-half-angle-substitution – Mike May 06 '17 at 12:08

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