The following is a list of integrals (antiderivative functions) of irrational functions. For a complete list of integral functions, see lists of integrals. Throughout this article the constant of integration is omitted for brevity.

Integrals involving r = √a² + x²

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$\int r\,dx={\frac {1}{2))\left(xr+a^{2}\,\ln \left(x+r\right)\right)$
$\int r^{3}\,dx={\frac {1}{4))xr^{3}+{\frac {3}{8))a^{2}xr+{\frac {3}{8))a^{4}\ln \left(x+r\right)$
$\int r^{5}\,dx={\frac {1}{6))xr^{5}+{\frac {5}{24))a^{2}xr^{3}+{\frac {5}{16))a^{4}xr+{\frac {5}{16))a^{6}\ln \left(x+r\right)$
$\int xr\,dx={\frac {r^{3)){3))$
$\int xr^{3}\,dx={\frac {r^{5)){5))$
$\int xr^{2n+1}\,dx={\frac {r^{2n+3)){2n+3))$
$\int x^{2}r\,dx={\frac {xr^{3)){4))-{\frac {a^{2}xr}{8))-{\frac {a^{4)){8))\ln \left(x+r\right)$
$\int x^{2}r^{3}\,dx={\frac {xr^{5)){6))-{\frac {a^{2}xr^{3)){24))-{\frac {a^{4}xr}{16))-{\frac {a^{6)){16))\ln \left(x+r\right)$
$\int x^{3}r\,dx={\frac {r^{5)){5))-{\frac {a^{2}r^{3)){3))$
$\int x^{3}r^{3}\,dx={\frac {r^{7)){7))-{\frac {a^{2}r^{5)){5))$
$\int x^{3}r^{2n+1}\,dx={\frac {r^{2n+5)){2n+5))-{\frac {a^{2}r^{2n+3)){2n+3))$
$\int x^{4}r\,dx={\frac {x^{3}r^{3)){6))-{\frac {a^{2}xr^{3)){8))+{\frac {a^{4}xr}{16))+{\frac {a^{6)){16))\ln \left(x+r\right)$
$\int x^{4}r^{3}\,dx={\frac {x^{3}r^{5)){8))-{\frac {a^{2}xr^{5)){16))+{\frac {a^{4}xr^{3)){64))+{\frac {3a^{6}xr}{128))+{\frac {3a^{8)){128))\ln \left(x+r\right)$
$\int x^{5}r\,dx={\frac {r^{7)){7))-{\frac {2a^{2}r^{5)){5))+{\frac {a^{4}r^{3)){3))$
$\int x^{5}r^{3}\,dx={\frac {r^{9)){9))-{\frac {2a^{2}r^{7)){7))+{\frac {a^{4}r^{5)){5))$
$\int x^{5}r^{2n+1}\,dx={\frac {r^{2n+7)){2n+7))-{\frac {2a^{2}r^{2n+5)){2n+5))+{\frac {a^{4}r^{2n+3)){2n+3))$
$\int {\frac {r\,dx}{x))=r-a\ln \left|{\frac {a+r}{x))\right|=r-a\,\operatorname {arsinh} {\frac {a}{x))$
$\int {\frac {r^{3}\,dx}{x))={\frac {r^{3)){3))+a^{2}r-a^{3}\ln \left|{\frac {a+r}{x))\right|$
$\int {\frac {r^{5}\,dx}{x))={\frac {r^{5)){5))+{\frac {a^{2}r^{3)){3))+a^{4}r-a^{5}\ln \left|{\frac {a+r}{x))\right|$
$\int {\frac {r^{7}\,dx}{x))={\frac {r^{7)){7))+{\frac {a^{2}r^{5)){5))+{\frac {a^{4}r^{3)){3))+a^{6}r-a^{7}\ln \left|{\frac {a+r}{x))\right|$
$\int {\frac {dx}{r))=\operatorname {arsinh} {\frac {x}{a))=\ln \left({\frac {x+r}{a))\right)$
$\int {\frac {dx}{r^{3))}={\frac {x}{a^{2}r))$
$\int {\frac {x\,dx}{r))=r$
$\int {\frac {x\,dx}{r^{3))}=-{\frac {1}{r))$
$\int {\frac {x^{2}\,dx}{r))={\frac {x}{2))r-{\frac {a^{2)){2))\,\operatorname {arsinh} {\frac {x}{a))={\frac {x}{2))r-{\frac {a^{2)){2))\ln \left({\frac {x+r}{a))\right)$
$\int {\frac {dx}{xr))=-{\frac {1}{a))\,\operatorname {arsinh} {\frac {a}{x))=-{\frac {1}{a))\ln \left|{\frac {a+r}{x))\right|$

Integrals involving s = √x² − a²

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Assume x² > a² (for x² < a², see next section):

$\int s\,dx={\frac {1}{2))\left(xs-a^{2}\ln \left|x+s\right|\right)$
${\displaystyle \int xs\,dx={\frac {1}{3))s^{3))$
$\int {\frac {s\,dx}{x))=s-|a|\arccos \left|{\frac {a}{x))\right|$
$\int {\frac {dx}{s))=\ln \left|{\frac {x+s}{a))\right|\,.$ Here $\ln \left|{\frac {x+s}{a))\right|=\operatorname {sgn} (x)\,\operatorname {arcosh} \left|{\frac {x}{a))\right|={\frac {1}{2))\ln \left({\frac {x+s}{x-s))\right)\,,$ where the positive value of $\operatorname {arcosh} \left|{\frac {x}{a))\right|$ is to be taken.
$\int {\frac {dx}{xs))={\frac {1}{a))\operatorname {arcsec} \left|{\frac {x}{a))\right|$
$\int {\frac {x\,dx}{s))=s$
$\int {\frac {x\,dx}{s^{3))}=-{\frac {1}{s))$
${\displaystyle \int {\frac {x\,dx}{s^{5))}=-{\frac {1}{3s^{3))))$
${\displaystyle \int {\frac {x\,dx}{s^{7))}=-{\frac {1}{5s^{5))))$
${\displaystyle \int {\frac {x\,dx}{s^{2n+1))}=-{\frac {1}{(2n-1)s^{2n-1))))$
${\displaystyle \int {\frac {x^{2m}\,dx}{s^{2n+1))}=-{\frac {1}{2n-1)){\frac {x^{2m-1)){s^{2n-1))}+{\frac {2m-1}{2n-1))\int {\frac {x^{2m-2}\,dx}{s^{2n-1))))$
$\int {\frac {x^{2}\,dx}{s))={\frac {xs}{2))+{\frac {a^{2)){2))\ln \left|{\frac {x+s}{a))\right|$
$\int {\frac {x^{2}\,dx}{s^{3))}=-{\frac {x}{s))+\ln \left|{\frac {x+s}{a))\right|$
$\int {\frac {x^{4}\,dx}{s))={\frac {x^{3}s}{4))+{\frac {3}{8))a^{2}xs+{\frac {3}{8))a^{4}\ln \left|{\frac {x+s}{a))\right|$
$\int {\frac {x^{4}\,dx}{s^{3))}={\frac {xs}{2))-{\frac {a^{2}x}{s))+{\frac {3}{2))a^{2}\ln \left|{\frac {x+s}{a))\right|$
$\int {\frac {x^{4}\,dx}{s^{5))}=-{\frac {x}{s))-{\frac {1}{3)){\frac {x^{3)){s^{3))}+\ln \left|{\frac {x+s}{a))\right|$
$\int {\frac {x^{2m}\,dx}{s^{2n+1))}=(-1)^{n-m}{\frac {1}{a^{2(n-m)))}\sum _{i=0}^{n-m-1}{\frac {1}{2(m+i)+1)){n-m-1 \choose i}{\frac {x^{2(m+i)+1)){s^{2(m+i)+1))}\qquad {\mbox{())n>m\geq 0{\mbox{)))$
$\int {\frac {dx}{s^{3))}=-{\frac {1}{a^{2))}{\frac {x}{s))$
$\int {\frac {dx}{s^{5))}={\frac {1}{a^{4))}\left[{\frac {x}{s))-{\frac {1}{3)){\frac {x^{3)){s^{3))}\right]$
$\int {\frac {dx}{s^{7))}=-{\frac {1}{a^{6))}\left[{\frac {x}{s))-{\frac {2}{3)){\frac {x^{3)){s^{3))}+{\frac {1}{5)){\frac {x^{5)){s^{5))}\right]$
$\int {\frac {dx}{s^{9))}={\frac {1}{a^{8))}\left[{\frac {x}{s))-{\frac {3}{3)){\frac {x^{3)){s^{3))}+{\frac {3}{5)){\frac {x^{5)){s^{5))}-{\frac {1}{7)){\frac {x^{7)){s^{7))}\right]$
${\displaystyle \int {\frac {x^{2}\,dx}{s^{5))}=-{\frac {1}{a^{2))}{\frac {x^{3)){3s^{3))))$
$\int {\frac {x^{2}\,dx}{s^{7))}={\frac {1}{a^{4))}\left[{\frac {1}{3)){\frac {x^{3)){s^{3))}-{\frac {1}{5)){\frac {x^{5)){s^{5))}\right]$
$\int {\frac {x^{2}\,dx}{s^{9))}=-{\frac {1}{a^{6))}\left[{\frac {1}{3)){\frac {x^{3)){s^{3))}-{\frac {2}{5)){\frac {x^{5)){s^{5))}+{\frac {1}{7)){\frac {x^{7)){s^{7))}\right]$

Integrals involving u = √a² − x²

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$\int u\,dx={\frac {1}{2))\left(xu+a^{2}\arcsin {\frac {x}{a))\right)\qquad {\mbox{())|x|\leq |a|{\mbox{)))$
$\int xu\,dx=-{\frac {1}{3))u^{3}\qquad {\mbox{())|x|\leq |a|{\mbox{)))$
$\int x^{2}u\,dx=-{\frac {x}{4))u^{3}+{\frac {a^{2)){8))(xu+a^{2}\arcsin {\frac {x}{a)))\qquad {\mbox{())|x|\leq |a|{\mbox{)))$
$\int {\frac {u\,dx}{x))=u-a\ln \left|{\frac {a+u}{x))\right|\qquad {\mbox{())|x|\leq |a|{\mbox{)))$
$\int {\frac {dx}{u))=\arcsin {\frac {x}{a))\qquad {\mbox{())|x|\leq |a|{\mbox{)))$
$\int {\frac {x^{2}\,dx}{u))={\frac {1}{2))\left(-xu+a^{2}\arcsin {\frac {x}{a))\right)\qquad {\mbox{())|x|\leq |a|{\mbox{)))$
$\int u\,dx={\frac {1}{2))\left(xu-\operatorname {sgn} x\,\operatorname {arcosh} \left|{\frac {x}{a))\right|\right)\qquad {\mbox{(for ))|x|\geq |a|{\mbox{)))$
$\int {\frac {x}{u))\,dx=-u\qquad {\mbox{())|x|\leq |a|{\mbox{)))$

Integrals involving R = √ax² + bx + c

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Assume (ax² + bx + c) cannot be reduced to the following expression (px + q)² for some p and q.

$\int {\frac {dx}{R))={\frac {1}{\sqrt {a))}\ln \left|2{\sqrt {a))R+2ax+b\right|\qquad {\mbox{(for ))a>0{\mbox{)))$
$\int {\frac {dx}{R))={\frac {1}{\sqrt {a))}\,\operatorname {arsinh} {\frac {2ax+b}{\sqrt {4ac-b^{2))))\qquad {\mbox{(for ))a>0{\mbox{, ))4ac-b^{2}>0{\mbox{)))$
$\int {\frac {dx}{R))={\frac {1}{\sqrt {a))}\ln |2ax+b|\quad {\mbox{(for ))a>0{\mbox{, ))4ac-b^{2}=0{\mbox{)))$
$\int {\frac {dx}{R))=-{\frac {1}{\sqrt {-a))}\arcsin {\frac {2ax+b}{\sqrt {b^{2}-4ac))}\qquad {\mbox{(for ))a<0{\mbox{, ))4ac-b^{2}<0{\mbox{, ))\left|2ax+b\right|<{\sqrt {b^{2}-4ac)){\mbox{)))$
$\int {\frac {dx}{R^{3))}={\frac {4ax+2b}{(4ac-b^{2})R))$
$\int {\frac {dx}{R^{5))}={\frac {4ax+2b}{3(4ac-b^{2})R))\left({\frac {1}{R^{2))}+{\frac {8a}{4ac-b^{2))}\right)$
$\int {\frac {dx}{R^{2n+1))}={\frac {2}{(2n-1)(4ac-b^{2})))\left({\frac {2ax+b}{R^{2n-1))}+4a(n-1)\int {\frac {dx}{R^{2n-1))}\right)$
$\int {\frac {x}{R))\,dx={\frac {R}{a))-{\frac {b}{2a))\int {\frac {dx}{R))$
$\int {\frac {x}{R^{3))}\,dx=-{\frac {2bx+4c}{(4ac-b^{2})R))$
${\displaystyle \int {\frac {x}{R^{2n+1))}\,dx=-{\frac {1}{(2n-1)aR^{2n-1))}-{\frac {b}{2a))\int {\frac {dx}{R^{2n+1))))$
$\int {\frac {dx}{xR))=-{\frac {1}{\sqrt {c))}\ln \left|{\frac {2{\sqrt {c))R+bx+2c}{x))\right|,~c>0$
$\int {\frac {dx}{xR))=-{\frac {1}{\sqrt {c))}\operatorname {arsinh} \left({\frac {bx+2c}{|x|{\sqrt {4ac-b^{2))))}\right),~c<0$
$\int {\frac {dx}{xR))={\frac {1}{\sqrt {-c))}\operatorname {arcsin} \left({\frac {bx+2c}{|x|{\sqrt {b^{2}-4ac))))\right),~c<0,b^{2}-4ac>0$
$\int {\frac {dx}{xR))=-{\frac {2}{bx))\left({\sqrt {ax^{2}+bx))\right),~c=0$
$\int {\frac {x^{2)){R))\,dx={\frac {2ax-3b}{4a^{2))}R+{\frac {3b^{2}-4ac}{8a^{2))}\int {\frac {dx}{R))$
$\int {\frac {dx}{x^{2}R))=-{\frac {R}{cx))-{\frac {b}{2c))\int {\frac {dx}{xR))$
$\int R\,dx={\frac {2ax+b}{4a))R+{\frac {4ac-b^{2)){8a))\int {\frac {dx}{R))$
$\int xR\,dx={\frac {R^{3)){3a))-{\frac {b(2ax+b)}{8a^{2))}R-{\frac {b(4ac-b^{2})}{16a^{2))}\int {\frac {dx}{R))$
$\int x^{2}R\,dx={\frac {6ax-5b}{24a^{2))}R^{3}+{\frac {5b^{2}-4ac}{16a^{2))}\int R\,dx$
$\int {\frac {R}{x))\,dx=R+{\frac {b}{2))\int {\frac {dx}{R))+c\int {\frac {dx}{xR))$
$\int {\frac {R}{x^{2))}\,dx=-{\frac {R}{x))+a\int {\frac {dx}{R))+{\frac {b}{2))\int {\frac {dx}{xR))$
$\int {\frac {x^{2}\,dx}{R^{3))}={\frac {(2b^{2}-4ac)x+2bc}{a(4ac-b^{2})R))+{\frac {1}{a))\int {\frac {dx}{R))$

Integrals involving S = √ax + b

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$\int S\,dx={\frac {2S^{3)){3a))$
$\int {\frac {dx}{S))={\frac {2S}{a))$
$\int {\frac {dx}{xS))={\begin{cases}-{\dfrac {2}{\sqrt {b))}\operatorname {arcoth} \left({\dfrac {S}{\sqrt {b))}\right)&{\mbox{(for ))b>0,\quad ax>0{\mbox{)))\\-{\dfrac {2}{\sqrt {b))}\operatorname {artanh} \left({\dfrac {S}{\sqrt {b))}\right)&{\mbox{(for ))b>0,\quad ax<0{\mbox{)))\\{\dfrac {2}{\sqrt {-b))}\arctan \left({\dfrac {S}{\sqrt {-b))}\right)&{\mbox{(for ))b<0{\mbox{)))\\\end{cases))$
$\int {\frac {S}{x))\,dx={\begin{cases}2\left(S-{\sqrt {b))\,\operatorname {arcoth} \left({\dfrac {S}{\sqrt {b))}\right)\right)&{\mbox{(for ))b>0,\quad ax>0{\mbox{)))\\2\left(S-{\sqrt {b))\,\operatorname {artanh} \left({\dfrac {S}{\sqrt {b))}\right)\right)&{\mbox{(for ))b>0,\quad ax<0{\mbox{)))\\2\left(S-{\sqrt {-b))\arctan \left({\dfrac {S}{\sqrt {-b))}\right)\right)&{\mbox{(for ))b<0{\mbox{)))\\\end{cases))$
$\int {\frac {x^{n)){S))\,dx={\frac {2}{a(2n+1)))\left(x^{n}S-bn\int {\frac {x^{n-1)){S))\,dx\right)$
$\int x^{n}S\,dx={\frac {2}{a(2n+3)))\left(x^{n}S^{3}-nb\int x^{n-1}S\,dx\right)$
$\int {\frac {1}{x^{n}S))\,dx=-{\frac {1}{b(n-1)))\left({\frac {S}{x^{n-1))}+\left(n-{\frac {3}{2))\right)a\int {\frac {dx}{x^{n-1}S))\right)$

References

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Abramowitz, Milton; Stegun, Irene A., eds. (1972). "Chapter 3". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover.
Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. ISBN 978-0-12-384933-5. LCCN 2014010276. (Several previous editions as well.)
Peirce, Benjamin Osgood (1929) [1899]. "Chapter 3". A Short Table of Integrals (3rd revised ed.). Boston: Ginn and Co. pp. 16–30.

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