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.:

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

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

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:: $frac\{ex\; +\; f\}\{left(ax^2+bx+c\; ight)^n\}$.