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

${\displaystyle \int r\,dx={\frac {1}{2))\left(xr+a^{2}\,\ln \left(x+r\right)\right)}$

${\displaystyle \int r^{3}\,dx={\frac {1}{4))xr^{3}+{\frac {3}{8))a^{2}xr+{\frac {3}{8))a^{4}\ln \left(x+r\right)}$

${\displaystyle \int r^{5}\,dx={\frac {1}{6))xr^{5}+{\frac {5}{24))a^{2}xr^{3}+{\frac {5}{16))a^{4}xr+{\frac {5}{16))a^{6}\ln \left(x+r\right)}$

${\displaystyle \int xr\,dx={\frac {r^{3)){3))}$

${\displaystyle \int xr^{3}\,dx={\frac {r^{5)){5))}$

${\displaystyle \int xr^{2n+1}\,dx={\frac {r^{2n+3)){2n+3))}$

${\displaystyle \int x^{2}r\,dx={\frac {xr^{3)){4))-{\frac {a^{2}xr}{8))-{\frac {a^{4)){8))\ln \left(x+r\right)}$

${\displaystyle \int x^{2}r^{3}\,dx={\frac {xr^{5)){6))-{\frac {a^{2}xr^{3)){24))-{\frac {a^{4}xr}{16))-{\frac {a^{6)){16))\ln \left(x+r\right)}$

${\displaystyle \int x^{3}r\,dx={\frac {r^{5)){5))-{\frac {a^{2}r^{3)){3))}$

${\displaystyle \int x^{3}r^{3}\,dx={\frac {r^{7)){7))-{\frac {a^{2}r^{5)){5))}$

${\displaystyle \int x^{3}r^{2n+1}\,dx={\frac {r^{2n+5)){2n+5))-{\frac {a^{2}r^{2n+3)){2n+3))}$

${\displaystyle \int x^{4}r\,dx={\frac {x^{3}r^{3)){6))-{\frac {a^{2}xr^{3)){8))+{\frac {a^{4}xr}{16))+{\frac {a^{6)){16))\ln \left(x+r\right)}$

${\displaystyle \int x^{4}r^{3}\,dx={\frac {x^{3}r^{5)){8))-{\frac {a^{2}xr^{5)){16))+{\frac {a^{4}xr^{3)){64))+{\frac {3a^{6}xr}{128))+{\frac {3a^{8)){128))\ln \left(x+r\right)}$

${\displaystyle \int x^{5}r\,dx={\frac {r^{7)){7))-{\frac {2a^{2}r^{5)){5))+{\frac {a^{4}r^{3)){3))}$

${\displaystyle \int x^{5}r^{3}\,dx={\frac {r^{9)){9))-{\frac {2a^{2}r^{7)){7))+{\frac {a^{4}r^{5)){5))}$

${\displaystyle \int x^{5}r^{2n+1}\,dx={\frac {r^{2n+7)){2n+7))-{\frac {2a^{2}r^{2n+5)){2n+5))+{\frac {a^{4}r^{2n+3)){2n+3))}$

${\displaystyle \int {\frac {r\,dx}{x))=r-a\ln \left|{\frac {a+r}{x))\right|=r-a\,\operatorname {arsinh} {\frac {a}{x))}$

${\displaystyle \int {\frac {r^{3}\,dx}{x))={\frac {r^{3)){3))+a^{2}r-a^{3}\ln \left|{\frac {a+r}{x))\right|}$

${\displaystyle \int {\frac {r^{5}\,dx}{x))={\frac {r^{5)){5))+{\frac {a^{2}r^{3)){3))+a^{4}r-a^{5}\ln \left|{\frac {a+r}{x))\right|}$

${\displaystyle \int {\frac {r^{7}\,dx}{x))={\frac {r^{7)){7))+{\frac {a^{2}r^{5)){5))+{\frac {a^{4}r^{3)){3))+a^{6}r-a^{7}\ln \left|{\frac {a+r}{x))\right|}$

${\displaystyle \int {\frac {dx}{r))=\operatorname {arsinh} {\frac {x}{a))=\ln \left({\frac {x+r}{a))\right)}$

${\displaystyle \int {\frac {dx}{r^{3))}={\frac {x}{a^{2}r))}$

${\displaystyle \int {\frac {x\,dx}{r))=r}$

${\displaystyle \int {\frac {x\,dx}{r^{3))}=-{\frac {1}{r))}$

${\displaystyle \int {\frac {x^{2}\,dx}{r))={\frac {x}{2))r-{\frac {a^{2)){2))\,\operatorname {arsinh} {\frac {x}{a))={\frac {x}{2))r-{\frac {a^{2)){2))\ln \left({\frac {x+r}{a))\right)}$

${\displaystyle \int {\frac {dx}{xr))=-{\frac {1}{a))\,\operatorname {arsinh} {\frac {a}{x))=-{\frac {1}{a))\ln \left|{\frac {a+r}{x))\right|}$

Integrals involving s = √x² − a²

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

${\displaystyle \int s\,dx={\frac {1}{2))\left(xs-a^{2}\ln \left|x+s\right|\right)}$

${\displaystyle \int xs\,dx={\frac {1}{3))s^{3))$

${\displaystyle \int {\frac {s\,dx}{x))=s-|a|\arccos \left|{\frac {a}{x))\right|}$

${\displaystyle \int {\frac {dx}{s))=\ln \left|{\frac {x+s}{a))\right|}$

Here ${\displaystyle \ln \left|{\frac {x+s}{a))\right|=\operatorname {sgn} (x)\,\operatorname {arcosh} \left|{\frac {x}{a))\right|={\frac {1}{2))\ln \left({\frac {x+s}{x-s))\right)}$, where the positive value of ${\displaystyle \operatorname {arcosh} \left|{\frac {x}{a))\right|}$ is to be taken.

${\displaystyle \int {\frac {x\,dx}{s))=s}$

${\displaystyle \int {\frac {x\,dx}{s^{3))}=-{\frac {1}{s))}$

${\displaystyle \int {\frac {x\,dx}{s^{5))}=-{\frac {1}{3s^{3))))$

${\displaystyle \int {\frac {x\,dx}{s^{7))}=-{\frac {1}{5s^{5))))$

${\displaystyle \int {\frac {x\,dx}{s^{2n+1))}=-{\frac {1}{(2n-1)s^{2n-1))))$

${\displaystyle \int {\frac {x^{2m}\,dx}{s^{2n+1))}=-{\frac {1}{2n-1)){\frac {x^{2m-1)){s^{2n-1))}+{\frac {2m-1}{2n-1))\int {\frac {x^{2m-2}\,dx}{s^{2n-1))))$

${\displaystyle \int {\frac {x^{2}\,dx}{s))={\frac {xs}{2))+{\frac {a^{2)){2))\ln \left|{\frac {x+s}{a))\right|}$

${\displaystyle \int {\frac {x^{2}\,dx}{s^{3))}=-{\frac {x}{s))+\ln \left|{\frac {x+s}{a))\right|}$

${\displaystyle \int {\frac {x^{4}\,dx}{s))={\frac {x^{3}s}{4))+{\frac {3}{8))a^{2}xs+{\frac {3}{8))a^{4}\ln \left|{\frac {x+s}{a))\right|}$

${\displaystyle \int {\frac {x^{4}\,dx}{s^{3))}={\frac {xs}{2))-{\frac {a^{2}x}{s))+{\frac {3}{2))a^{2}\ln \left|{\frac {x+s}{a))\right|}$

${\displaystyle \int {\frac {x^{4}\,dx}{s^{5))}=-{\frac {x}{s))-{\frac {1}{3)){\frac {x^{3)){s^{3))}+\ln \left|{\frac {x+s}{a))\right|}$

${\displaystyle \int {\frac {x^{2m}\,dx}{s^{2n+1))}=(-1)^{n-m}{\frac {1}{a^{2(n-m)))}\sum _{i=0}^{n-m-1}{\frac {1}{2(m+i)+1)){n-m-1 \choose i}{\frac {x^{2(m+i)+1)){s^{2(m+i)+1))}\qquad {\mbox{())n>m\geq 0{\mbox{)))}$

${\displaystyle \int {\frac {dx}{s^{3))}=-{\frac {1}{a^{2))}{\frac {x}{s))}$

${\displaystyle \int {\frac {dx}{s^{5))}={\frac {1}{a^{4))}\left[{\frac {x}{s))-{\frac {1}{3)){\frac {x^{3)){s^{3))}\right]}$

${\displaystyle \int {\frac {dx}{s^{7))}=-{\frac {1}{a^{6))}\left[{\frac {x}{s))-{\frac {2}{3)){\frac {x^{3)){s^{3))}+{\frac {1}{5)){\frac {x^{5)){s^{5))}\right]}$

${\displaystyle \int {\frac {dx}{s^{9))}={\frac {1}{a^{8))}\left[{\frac {x}{s))-{\frac {3}{3)){\frac {x^{3)){s^{3))}+{\frac {3}{5)){\frac {x^{5)){s^{5))}-{\frac {1}{7)){\frac {x^{7)){s^{7))}\right]}$

${\displaystyle \int {\frac {x^{2}\,dx}{s^{5))}=-{\frac {1}{a^{2))}{\frac {x^{3)){3s^{3))))$

${\displaystyle \int {\frac {x^{2}\,dx}{s^{7))}={\frac {1}{a^{4))}\left[{\frac {1}{3)){\frac {x^{3)){s^{3))}-{\frac {1}{5)){\frac {x^{5)){s^{5))}\right]}$

${\displaystyle \int {\frac {x^{2}\,dx}{s^{9))}=-{\frac {1}{a^{6))}\left[{\frac {1}{3)){\frac {x^{3)){s^{3))}-{\frac {2}{5)){\frac {x^{5)){s^{5))}+{\frac {1}{7)){\frac {x^{7)){s^{7))}\right]}$

Integrals involving u = √a² − x²

${\displaystyle \int u\,dx={\frac {1}{2))\left(xu+a^{2}\arcsin {\frac {x}{a))\right)\qquad {\mbox{())|x|\leq |a|{\mbox{)))}$

${\displaystyle \int xu\,dx=-{\frac {1}{3))u^{3}\qquad {\mbox{())|x|\leq |a|{\mbox{)))}$

${\displaystyle \int x^{2}u\,dx=-{\frac {x}{4))u^{3}+{\frac {a^{2)){8))(xu+a^{2}\arcsin {\frac {x}{a)))\qquad {\mbox{())|x|\leq |a|{\mbox{)))}$

${\displaystyle \int {\frac {u\,dx}{x))=u-a\ln \left|{\frac {a+u}{x))\right|\qquad {\mbox{())|x|\leq |a|{\mbox{)))}$

${\displaystyle \int {\frac {dx}{u))=\arcsin {\frac {x}{a))\qquad {\mbox{())|x|\leq |a|{\mbox{)))}$

${\displaystyle \int {\frac {x^{2}\,dx}{u))={\frac {1}{2))\left(-xu+a^{2}\arcsin {\frac {x}{a))\right)\qquad {\mbox{())|x|\leq |a|{\mbox{)))}$

${\displaystyle \int u\,dx={\frac {1}{2))\left(xu-\operatorname {sgn} x\,\operatorname {arcosh} \left|{\frac {x}{a))\right|\right)\qquad {\mbox{(for ))|x|\geq |a|{\mbox{)))}$

${\displaystyle \int {\frac {x}{u))\,dx=-u\qquad {\mbox{())|x|\leq |a|{\mbox{)))}$

Integrals involving R = √ax² + bx + c

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

${\displaystyle \int {\frac {dx}{R))={\frac {1}{\sqrt {a))}\ln \left|2{\sqrt {a))R+2ax+b\right|\qquad {\mbox{(for ))a>0{\mbox{)))}$

${\displaystyle \int {\frac {dx}{R))={\frac {1}{\sqrt {a))}\,\operatorname {arsinh} {\frac {2ax+b}{\sqrt {4ac-b^{2))))\qquad {\mbox{(for ))a>0{\mbox{, ))4ac-b^{2}>0{\mbox{)))}$

${\displaystyle \int {\frac {dx}{R))={\frac {1}{\sqrt {a))}\ln |2ax+b|\quad {\mbox{(for ))a>0{\mbox{, ))4ac-b^{2}=0{\mbox{)))}$

${\displaystyle \int {\frac {dx}{R))=-{\frac {1}{\sqrt {-a))}\arcsin {\frac {2ax+b}{\sqrt {b^{2}-4ac))}\qquad {\mbox{(for ))a<0{\mbox{, ))4ac-b^{2}<0{\mbox{, ))\left|2ax+b\right|<{\sqrt {b^{2}-4ac)){\mbox{)))}$

${\displaystyle \int {\frac {dx}{R^{3))}={\frac {4ax+2b}{(4ac-b^{2})R))}$

${\displaystyle \int {\frac {dx}{R^{5))}={\frac {4ax+2b}{3(4ac-b^{2})R))\left({\frac {1}{R^{2))}+{\frac {8a}{4ac-b^{2))}\right)}$

${\displaystyle \int {\frac {dx}{R^{2n+1))}={\frac {2}{(2n-1)(4ac-b^{2})))\left({\frac {2ax+b}{R^{2n-1))}+4a(n-1)\int {\frac {dx}{R^{2n-1))}\right)}$

${\displaystyle \int {\frac {x}{R))\,dx={\frac {R}{a))-{\frac {b}{2a))\int {\frac {dx}{R))}$

${\displaystyle \int {\frac {x}{R^{3))}\,dx=-{\frac {2bx+4c}{(4ac-b^{2})R))}$

${\displaystyle \int {\frac {x}{R^{2n+1))}\,dx=-{\frac {1}{(2n-1)aR^{2n-1))}-{\frac {b}{2a))\int {\frac {dx}{R^{2n+1))))$

${\displaystyle \int {\frac {dx}{xR))=-{\frac {1}{\sqrt {c))}\ln \left|{\frac {2{\sqrt {c))R+bx+2c}{x))\right|,~c>0}$

${\displaystyle \int {\frac {dx}{xR))=-{\frac {1}{\sqrt {c))}\operatorname {arsinh} \left({\frac {bx+2c}{|x|{\sqrt {4ac-b^{2))))}\right),~c<0}$

${\displaystyle \int {\frac {dx}{xR))={\frac {1}{\sqrt {-c))}\operatorname {arcsin} \left({\frac {bx+2c}{|x|{\sqrt {b^{2}-4ac))))\right),~c<0,b^{2}-4ac>0}$

${\displaystyle \int {\frac {dx}{xR))=-{\frac {2}{bx))\left({\sqrt {ax^{2}+bx))\right),~c=0}$

${\displaystyle \int {\frac {x^{2)){R))\,dx={\frac {2ax-3b}{4a^{2))}R+{\frac {3b^{2}-4ac}{8a^{2))}\int {\frac {dx}{R))}$

${\displaystyle \int {\frac {dx}{x^{2}R))=-{\frac {R}{cx))-{\frac {b}{2c))\int {\frac {dx}{xR))}$

${\displaystyle \int R\,dx={\frac {2ax+b}{4a))R+{\frac {4ac-b^{2)){8a))\int {\frac {dx}{R))}$

${\displaystyle \int xR\,dx={\frac {R^{3)){3a))-{\frac {b(2ax+b)}{8a^{2))}R-{\frac {b(4ac-b^{2})}{16a^{2))}\int {\frac {dx}{R))}$

${\displaystyle \int x^{2}R\,dx={\frac {6ax-5b}{24a^{2))}R^{3}+{\frac {5b^{2}-4ac}{16a^{2))}\int R\,dx}$

${\displaystyle \int {\frac {R}{x))\,dx=R+{\frac {b}{2))\int {\frac {dx}{R))+c\int {\frac {dx}{xR))}$

${\displaystyle \int {\frac {R}{x^{2))}\,dx=-{\frac {R}{x))+a\int {\frac {dx}{R))+{\frac {b}{2))\int {\frac {dx}{xR))}$

${\displaystyle \int {\frac {x^{2}\,dx}{R^{3))}={\frac {(2b^{2}-4ac)x+2bc}{a(4ac-b^{2})R))+{\frac {1}{a))\int {\frac {dx}{R))}$

Integrals involving S = √ax + b

${\displaystyle \int S\,dx={\frac {2S^{3)){3a))}$

${\displaystyle \int {\frac {dx}{S))={\frac {2S}{a))}$

${\displaystyle \int {\frac {dx}{xS))={\begin{cases}-{\dfrac {2}{\sqrt {b))}\operatorname {arcoth} \left({\dfrac {S}{\sqrt {b))}\right)&{\mbox{(for ))b>0,\quad ax>0{\mbox{)))\\-{\dfrac {2}{\sqrt {b))}\operatorname {artanh} \left({\dfrac {S}{\sqrt {b))}\right)&{\mbox{(for ))b>0,\quad ax<0{\mbox{)))\\{\dfrac {2}{\sqrt {-b))}\arctan \left({\dfrac {S}{\sqrt {-b))}\right)&{\mbox{(for ))b<0{\mbox{)))\\\end{cases))}$

${\displaystyle \int {\frac {S}{x))\,dx={\begin{cases}2\left(S-{\sqrt {b))\,\operatorname {arcoth} \left({\dfrac {S}{\sqrt {b))}\right)\right)&{\mbox{(for ))b>0,\quad ax>0{\mbox{)))\\2\left(S-{\sqrt {b))\,\operatorname {artanh} \left({\dfrac {S}{\sqrt {b))}\right)\right)&{\mbox{(for ))b>0,\quad ax<0{\mbox{)))\\2\left(S-{\sqrt {-b))\arctan \left({\dfrac {S}{\sqrt {-b))}\right)\right)&{\mbox{(for ))b<0{\mbox{)))\\\end{cases))}$

${\displaystyle \int {\frac {x^{n)){S))\,dx={\frac {2}{a(2n+1)))\left(x^{n}S-bn\int {\frac {x^{n-1)){S))\,dx\right)}$

${\displaystyle \int x^{n}S\,dx={\frac {2}{a(2n+3)))\left(x^{n}S^{3}-nb\int x^{n-1}S\,dx\right)}$

${\displaystyle \int {\frac {1}{x^{n}S))\,dx=-{\frac {1}{b(n-1)))\left({\frac {S}{x^{n-1))}+\left(n-{\frac {3}{2))\right)a\int {\frac {dx}{x^{n-1}S))\right)}$