List of integrals of rational functions


The following is a list of integrals of rational functions.
Any rational function can be integrated by partial fraction decomposition of the function into a sum of functions of the form:
which can then be integrated term by term.
For other types of functions, see lists of integrals.

Miscellaneous integrands

  • Integrands of the form ''x''''m''(''a x'' + ''b'')''n''

Many of the following antiderivatives have a term of the form ln |ax + b|. Because this is undefined when x = −b / a, the most general form of the antiderivative replaces the constant of integration with a locally constant function. However, it is conventional to omit this from the notation. For example,
is usually abbreviated as
where C is to be understood as notation for a locally constant function of x. This convention will be adhered to in the following.

  • Integrands of the form ''x''''m'' / (''a x''2 + ''b x'' + ''c'')''n''

For

  • Integrands of the form ''x''''m'' (''a'' + ''b x''''n'')''p''

  • The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.
  • These reduction formulas can be used for integrands having integer and/or fractional exponents.

  • Integrands of the form (''A'' + ''B x'') (''a'' + ''b x'')''m'' (''c'' + ''d x'')''n'' (''e'' + ''f x'')''p''

  • The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m, n and p toward 0.
  • These reduction formulas can be used for integrands having integer and/or fractional exponents.
  • Special cases of these reductions formulas can be used for integrands of the form by setting B to 0.

  • Integrands of the form ''x''''m'' (''A'' + ''B x''''n'') (''a'' + ''b x''''n'')''p'' (''c'' + ''d x''''n'')''q''

  • The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m, p and q toward 0.
  • These reduction formulas can be used for integrands having integer and/or fractional exponents.
  • Special cases of these reductions formulas can be used for integrands of the form and by setting m and/or B to 0.

  • Integrands of the form (''d'' + ''e x'')''m'' (''a'' + ''b x'' + ''c x''2)''p'' when ''b''2 − 4 ''a c'' = 0

  • The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.
  • These reduction formulas can be used for integrands having integer and/or fractional exponents.
  • Special cases of these reductions formulas can be used for integrands of the form when by setting m to 0.

  • Integrands of the form (''d'' + ''e x'')''m'' (''A'' + ''B x'') (''a'' + ''b x'' + ''c x''2)''p''

  • The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.
  • These reduction formulas can be used for integrands having integer and/or fractional exponents.
  • Special cases of these reductions formulas can be used for integrands of the form and by setting m and/or B to 0.

  • Integrands of the form ''x''''m'' (''a'' + ''b x''''n'' + ''c x''2''n'')''p'' when ''b''2 − 4 ''a c'' = 0

  • The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.
  • These reduction formulas can be used for integrands having integer and/or fractional exponents.
  • Special cases of these reductions formulas can be used for integrands of the form when by setting m to 0.

  • Integrands of the form ''x''''m'' (''A'' + ''B x''''n'') (''a'' + ''b x''''n'' + ''c x''2''n'')''p''

  • The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.
  • These reduction formulas can be used for integrands having integer and/or fractional exponents.
  • Special cases of these reductions formulas can be used for integrands of the form and by setting m and/or B to 0.
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