Definitions

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

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

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

 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) $intfrac\left\{1\right\}\left\{x^2+a^2\right\} dx = frac\left\{1\right\}\left\{a\right\}arctanfrac\left\{x\right\}\left\{a\right\},!$ 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 $aneq 0:$

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

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

 intfrac{mx+n}{ax^2+bx+c} dx = begin{cases} frac{m}{2a}lnleft>ax^2+bx+cright

$intfrac\left\{1\right\}\left\{\left(ax^2+bx+c\right)^n\right\} dx= frac\left\{2ax+b\right\}\left\{\left(n-1\right)\left(4ac-b^2\right)\left(ax^2+bx+c\right)^\left\{n-1\right\}\right\}+frac\left\{\left(2n-3\right)2a\right\}\left\{\left(n-1\right)\left(4ac-b^2\right)\right\}intfrac\left\{1\right\}\left\{\left(ax^2+bx+c\right)^\left\{n-1\right\}\right\} dx,!$
$intfrac\left\{x\right\}\left\{\left(ax^2+bx+c\right)^n\right\} dx= frac\left\{bx+2c\right\}\left\{\left(n-1\right)\left(4ac-b^2\right)\left(ax^2+bx+c\right)^\left\{n-1\right\}\right\}-frac\left\{b\left(2n-3\right)\right\}\left\{\left(n-1\right)\left(4ac-b^2\right)\right\}intfrac\left\{1\right\}\left\{\left(ax^2+bx+c\right)^\left\{n-1\right\}\right\} dx,!$
$intfrac\left\{1\right\}\left\{x\left(ax^2+bx+c\right)\right\} dx= frac\left\{1\right\}\left\{2c\right\}lnleft|frac\left\{x^2\right\}\left\{ax^2+bx+c\right\}right|-frac\left\{b\right\}\left\{2c\right\}intfrac\left\{1\right\}\left\{ax^2+bx+c\right\} dx$

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

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:

$frac\left\{ex + f\right\}\left\{left\left(ax^2+bx+cright\right)^n\right\}$.

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