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Apart[expr]

rewrites a rational expression as a sum of terms with minimal denominators.

Apart[expr,var]

treats all variables other than var as constants.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Basic Uses  
Advanced Uses  
Options  
Modulus  
Trig  
Applications  
Properties & Relations  
See Also
Tech Notes
Related Guides
History
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Apart

Apart[expr]

rewrites a rational expression as a sum of terms with minimal denominators.

Apart[expr,var]

treats all variables other than var as constants.

Details and Options

  • Apart gives the partial fraction decomposition of a rational expression.
  • Apart[expr,var] writes expr as a polynomial in var together with a sum of ratios of polynomials, where the degree in var of each numerator polynomial is less than that of the corresponding denominator polynomial.
  • Apart takes the following options:
  • Modulus 0modulus to assume for integers
    Trig Falsewhether to do trigonometric as well as algebraic transformations
  • Apart[expr,Trig->True] treats trigonometric functions as rational functions of exponentials, and manipulates them accordingly.
  • Apart automatically threads over lists in expr, as well as equations, inequalities, and logic functions.

Examples

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Basic Examples  (2)

Decompose into partial fractions:

Wolfram Language code: Apart[1 / ((1 + x)(5 + x))]

Compute the partial fraction representation of a rational function:

Wolfram Language code: Apart[(x ^ 5 - 2) / ((1 + x + x ^ 2)(2 + x)(1 - x))]

Scope  (8)

Basic Uses  (6)

Decompose a rational function into partial fractions:

Wolfram Language code: Apart[16x / ((1 + x) ^ 2(5 + x))]

Apart can handle symbolic parameters:

Wolfram Language code: Apart[1 / ((x + a) (x + b) (x + c))]

Treat as the main variable and as a constant:

Wolfram Language code: Apart[(x + y) / ((x + 1)(y + 1)(x - y)), x]

Treat as the main variable and as a constant:

Wolfram Language code: Apart[(x + y) / ((x + 1)(y + 1)(x - y)), y]

Here Apart picks as the main variable and treats as a constant:

Wolfram Language code: Apart[(x + y) / ((x + 1)(y + 1)(x - y))]

Apart can handle non-polynomial expressions:

Wolfram Language code: Apart[2 ^ x / ((4 ^ x + 1) (2 ^ x + 4))]

Apart threads over equations and inequalities:

Wolfram Language code: Apart[1 < (x + 1) / (x - 1) < 2]

Compute the partial fraction representation when radicals are present:

Wolfram Language code: Apart[1 / ((Sqrt[x] + 2)(Sqrt[x] + 3)), x]

Advanced Uses  (2)

Compute the partial fraction representation over the integers modulo :

Wolfram Language code: Apart[(x + 1) / (x - 1), Modulus -> 3]

Compute the partial fraction representation following common trigonometric identities:

Wolfram Language code: Apart[3Sin[x] / ((Sin[x] + 1) (Sin[x] + 4)), Trig -> True]

Options  (3)

Modulus  (2)

Partial fraction decomposition over the rationals:

Wolfram Language code: Apart[(x ^ 2 + 1) / (x + 1)]

Partial fraction decomposition over the integers modulo 2:

Wolfram Language code: Apart[(x ^ 2 + 1) / (x + 1), Modulus -> 2]

Trig  (1)

Partial fraction decompositions of trigonometric expressions:

Wolfram Language code: Apart[Cos[x] / (1 + Sin[x]), Trig -> True]
Wolfram Language code: Apart[1 / Sin[4 x], Trig -> True]

Applications  (1)

Integrals of rational expressions of polynomials are often computed by decomposing into partial fractions:

Wolfram Language code: f = 16x / ((1 + x) ^ 2(5 + x))

To compute the integral, first apply Apart to the rational expression:

Wolfram Language code: Apart[f]

Apply Integrate to each summand individually and sum the results:

Wolfram Language code: ans = Integrate[-(4/(1 + x)^2), x] + Integrate[(5/1 + x), x] + Integrate[-(5/5 + x), x]

Verify the answer:

Wolfram Language code: D[ans, x] - f//Together

Properties & Relations  (2)

Together acts as an inverse of Apart:

Wolfram Language code: Apart[1 / ((1 + x) (5 + x))]
Wolfram Language code: Together[%]

A rational function:

Wolfram Language code: ratfun = (x ^ 7 - x ^ 6 + x ^ 3 - 5x + 3) / (x ^ 6 + 3x ^ 5 + 6x ^ 3 - 39x ^ 2 - 105x + 70);

Rewrite with Apart:

Wolfram Language code: Apart[ratfun, x]

PartialFractions can be made to work like Apart by specifying to not use any algebraic extension for factoring denominators:

Wolfram Language code: PartialFractions[ratfun, x, Extension -> None]
Wolfram Research (1988), Apart, Wolfram Language function, https://reference.wolfram.com/language/ref/Apart.html (updated 2007).

Text

Wolfram Research (1988), Apart, Wolfram Language function, https://reference.wolfram.com/language/ref/Apart.html (updated 2007).

CMS

Wolfram Language. 1988. "Apart." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. https://reference.wolfram.com/language/ref/Apart.html.

APA

Wolfram Language. (1988). Apart. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Apart.html

BibTeX

@misc{reference.wolfram_2026_apart, author="Wolfram Research", title="{Apart}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/Apart.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_apart, organization={Wolfram Research}, title={Apart}, year={2007}, url={https://reference.wolfram.com/language/ref/Apart.html}, note=[Accessed: 12-June-2026]}