Rational functions

List of integrals of rational functions

The following is a list of integrals (antiderivative functions) of rational functions. For a more complete list of integrals, see lists of integrals.

$int (ax + b)^n dx$	$= frac{(ax + b)^{n+1}}{a(n + 1)} qquadmbox{(for } nneq -1mbox{)},!$
$intfrac{c}{ax + b} dx$	= frac{c}{a}lnleft>ax + bright\|
$int x(ax + b)^n dx$	$= frac{a(n + 1)x - b}{a^2(n + 1)(n + 2)} (ax + b)^{n+1} qquadmbox{(for }n notin {-1, -2}mbox{)}$

$intfrac{x}{ax + b} dx$	= frac{x}{a} - frac{b}{a^2}lnleft>ax + bright\|
$intfrac{x}{(ax + b)^2} dx$	= frac{b}{a^2(ax + b)} + frac{1}{a^2}lnleft>ax + bright\|
$intfrac{x}{(ax + b)^n} dx$	$= frac{a(1 - n)x - b}{a^2(n - 1)(n - 2)(ax + b)^{n-1}} qquadmbox{(for } nnotin {1, 2}mbox{)}$

$intfrac{x^2}{ax + b} dx$	= frac{1}{a^3}left(frac{(ax + b)^2}{2} - 2b(ax + b) + b^2lnleft>ax + bright\|right)
$intfrac{x^2}{(ax + b)^2} dx$	= frac{1}{a^3}left(ax + b - 2blnleft>ax + bright\| - frac{b^2}{ax + b}right)
$intfrac{x^2}{(ax + b)^3} dx$	= frac{1}{a^3}left(lnleft>ax + bright\| + frac{2b}{ax + b} - frac{b^2}{2(ax + b)^2}right)
$intfrac{x^2}{(ax + b)^n} dx$	$= frac{1}{a^3}left(-frac{(ax + b)^{3-n}}{(n-3)} + frac{2b (a + b)^{2-n}}{(n-2)} - frac{b^2 (ax + b)^{1-n}}{(n - 1)}right) qquadmbox{(for } nnotin {1, 2, 3}mbox{)}$

intfrac{1}{x(ax + b)} dx = -frac{1}{b}lnleft>frac{ax+b}{x}right|

intfrac{1}{x^2(ax+b)} dx = -frac{1}{bx} + frac{a}{b^2}lnleft>frac{ax+b}{x}right|

intfrac{1}{x^2(ax+b)^2} dx = -aleft(frac{1}{b^2(ax+b)} + frac{1}{ab^2x} - frac{2}{b^3}lnleft>frac{ax+b}{x}right|right)

intfrac{1}{x^2+a^2} dx = frac{1}{a}arctanfrac{x}{a},!

intfrac{1}{x^2-a^2} dx = begin{cases} -frac{1}{a},mathrm{arctanh}frac{x}{a} = frac{1}{2a}lnfrac{a-x}{a+x} & mbox{(for }>x| < |a|mbox{)} -frac{1}{a},mathrm{arccoth}frac{x}{a} = frac{1}{2a}lnfrac{x-a}{x+a} & mbox{(for }|x| > |a|mbox{)} end{cases}

for $aneq 0:$

intfrac{1}{ax^2+bx+c} dx = begin{cases}
frac{2}{sqrt{4ac-b^2}}arctanfrac{2ax+b}{sqrt{4ac-b^2}} & mbox{(for }4ac-b^2>0mbox{)}
-frac{2}{sqrt{b^2-4ac}},mathrm{arctanh}frac{2ax+b}{sqrt{b^2-4ac}} = frac{1}{sqrt{b^2-4ac}}lnleft|frac{2ax+b-sqrt{b^2-4ac}}{2ax+b+sqrt{b^2-4ac}}right| & mbox{(for }4ac-b^2<0mbox{)}
-frac{2}{2ax+b} & mbox{(for }4ac-b^2=0mbox{)}
end{cases}

intfrac{x}{ax^2+bx+c} dx

= frac{1}{2a}lnleft>ax^2+bx+cright|-frac{b}{2a}intfrac{dx}{ax^2+bx+c}

0mbox{)} frac{m}{2a}lnleft>ax^2+bx+cright|-frac{2an-bm}{asqrt{b^2-4ac}},mathrm{arctanh}frac{2ax+b}{sqrt{b^2-4ac}} &mbox{(for }4ac-b^2<0mbox{)} frac{m}{2a}lnleft|ax^2+bx+cright|-frac{2an-bm}{a(2ax+b)} &mbox{(for }4ac-b^2=0mbox{)}end{cases}
intfrac{mx+n}{ax^2+bx+c} dx = begin{cases} frac{m}{2a}lnleft>ax^2+bx+cright

intfrac{1}{(ax^2+bx+c)^n} dx= frac{2ax+b}{(n-1)(4ac-b^2)(ax^2+bx+c)^{n-1}}+frac{(2n-3)2a}{(n-1)(4ac-b^2)}intfrac{1}{(ax^2+bx+c)^{n-1}} dx,!

intfrac{x}{(ax^2+bx+c)^n} dx= frac{bx+2c}{(n-1)(4ac-b^2)(ax^2+bx+c)^{n-1}}-frac{b(2n-3)}{(n-1)(4ac-b^2)}intfrac{1}{(ax^2+bx+c)^{n-1}} dx,!

intfrac{1}{x(ax^2+bx+c)} dx= frac{1}{2c}lnleft|frac{x^2}{ax^2+bx+c}right|-frac{b}{2c}intfrac{1}{ax^2+bx+c} dx

int frac{dx}{x^{2^n} + 1} = sum_{k=1}^{2^{n-1}} left { frac{1}{2^{n-1}} left [sin(frac{(2k -1) pi}{2^n}) arctan[left(x - cos(frac{(2k -1) pi}{2^n}) right ) csc(frac{(2k -1) pi}{2^n}) ] right] - frac{1}{2^n} left [cos(frac{(2k -1) pi}{2^n}) ln left | x^2 - 2 x cos(frac{(2k -1) pi}{2^n}) + 1 right | right ] right }

Any rational function can be integrated using above equations and partial fractions in integration, by decomposing the rational function into a sum of functions of the form: