antiderivative of rational function

The most notable real functions, which can be integrated in a closed formPlanetmathPlanetmath, are the rational functionsMathworldPlanetmath:

Theorem.  The antiderivative of a rational function is always expressible in a closed form, which only can comprise, except a rational expression summand, summands of logarithms and arcustangents of rational functions.

One can justify the theorem by using the general form of the (unique) partial fraction decomposition

R(x)=H(x)+ i=1m(Ai1x-ai+Ai2(x-ai)2++Aiμi(x-ai)μi)
+ j=1n(Bj1x+Cj1x2+2pjx+qj+Bj2x+Cj2(x2+2pjx+qj)2++Bjνjx+Cjνj(x2+2pjx+qj)νj),

of the rational function R(x) ; here, H(x) is a polynomialMathworldPlanetmathPlanetmathPlanetmath, the first sum expression is determined by the real zeroes ai of the denominator of R(x), the second sum is determined by the real quadratic prime factorsMathworldPlanetmath x2+2pjx+qj of the denominator (which have no real zeroes).

The addends of the form A(x-a)r in the first sum are integrated directly, giving

Ax-adx=Aln|x-a|+constant  (r=1) (1)


A(x-a)rdx=-Ar-11(x-a)r-1+constant  (r>1). (2)

The remaining partial fractionsPlanetmathPlanetmath are of the form Bx+C(x2+2px+q)s where  p2<q  and s is a positive integer.  Now we may write


and make the substitution

x+pq-p2=t, (3)

i.e.  x=tq-p2-p,  getting

Bx+C(x2+2px+q)s𝑑x=Et+F(1+t2)s𝑑t=Etdt(1+t2)s+Fdt(1+t2)s (4)

where E and F are certain constants.  In the case  s=1  we have

tdt1+t2=12ln(1+t2)+constant (5)

and in the case  s>1

tdt(1+t2)s=-12(s-1)1(1+t2)s-1+constant. (6)

The latter addend of the right hand side of (4) is for  s=1  got from

dt1+t2=arctant+constant (7)

and for the cases s>1 on may first write


Using integration by parts in the last integral, this equation can be converted into the reduction formula

dt(1+t2)s=12s-2t(1+t2)s-1+2n-32n-2dt(1+t2)s-1. (8)

The assertion of the theorem follows from (1), …, (8).


Title antiderivative of rational function
Canonical name AntiderivativeOfRationalFunction
Date of creation 2013-03-22 19:21:38
Last modified on 2013-03-22 19:21:38
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 19
Author pahio (2872)
Entry type Theorem
Classification msc 26A36
Synonym integration of rational functions
Related topic IntegrationTechniques