Integrals that cannot be expressed using elementary functions can be manipulated symbolically using general functions such as the Meijer G-function.
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An even larger, multivolume table is the Integrals and Series by Prudnikov, Brychkov, and Marichev (with volumes 1–3 listing integrals and series of elementary and special functions, volume 4–5 are tables of Laplace transforms).
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Wolfram Alpha can show results, and for some simpler expressions, also the intermediate steps of the integration.
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