How can I calculate $\int \frac{\sec x\tan x}{3x+5}\,\mathrm dx$ [duplicate]

calculus integration recreational-mathematics indefinite-integrals

Integrating elementary functions in elementary terms is completely algorithmic. The algorithm is implemented in all major computer algebra systems, so the fact that Mathematica fails to integrate this in closed form can be viewed as (in effect) a proof that such a closed form does not exist in elementary terms.

To answer the comments:

  1. You may or may not trust Mathematica (I don't always, but do for this). The fact that "its algorithm is not open for inspection" is not relevant -- it would take you much longer to figure out what the code does than to run the algorithm by hand (well, you need some linear algebra).

  2. If you do want to try it in the privacy of your own home, you need to go no further than the late, lamented Manuel Bronstein's excellent book: http://www.amazon.com/Symbolic-Integration-Transcendental-Computation-Mathematics/dp/3540214933

  3. I am quite sure that this particular integral is easy to show non-integrable in elementary terms by hand (if you understand the Risch algorithm).

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