A seguinte é unha lista de integrais de funcións irracionais.

Integrais con r=√x2+a2
----------------------

\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 {1}{8}}3a^{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^{3}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\sinh ^{{-1}}{\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}}=\sinh ^{{-1}}{\frac {x}{a}}=\ln \left|x+r\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}}\,\sinh ^{{-1}}{\frac {x}{a}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\ln \left|x+r\right|
\int {\frac {dx}{xr}}=-{\frac {1}{a}}\,\sinh ^{{-1}}{\frac {a}{x}}=-{\frac {1}{a}}\ln \left|{\frac {a+r}{x}}\right|

Integrais con s=√x2-a2
----------------------

\int xs\;dx={\frac {1}{3}}s^{3}
\int {\frac {s\;dx}{x}}=s-a\cos ^{{-1}}\left|{\frac {a}{x}}\right|
\int {\frac {dx}{s}}=\int {\frac {dx}{{\sqrt {x^{2}-a^{2}}}}}=\ln \left|{\frac {x+s}{a}}\right|
\int {\frac {x\;dx}{s}}=s
\int {\frac {x\;dx}{s^{3}}}=-{\frac {1}{s}}
\int {\frac {x\;dx}{s^{5}}}=-{\frac {1}{3s^{3}}}
\int {\frac {x\;dx}{s^{7}}}=-{\frac {1}{5s^{5}}}
\int {\frac {x\;dx}{s^{{2n+1}}}}=-{\frac {1}{(2n-1)s^{{2n-1}}}}...
\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]
\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]

Integrais con t=√a2-x2
----------------------

\int t\;dx={\frac {1}{2}}\left(xt+a^{2}\sin ^{{-1}}{\frac {x}{a}}\right)\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}
\int xt\;dx=-{\frac {1}{3}}t^{3}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}
\int {\frac {t\;dx}{x}}=t-a\ln \left|{\frac {a+t}{x}}\right|\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}
\int {\frac {dx}{t}}=\sin ^{{-1}}{\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}
\int {\frac {x^{2}\;dx}{t}}=-{\frac {x}{2}}t+{\frac {a^{2}}{2}}\sin ^{{-1}}{\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}
\int t\;dx={\frac {1}{2}}\left(xt-\operatorname{sgn} x\,\cosh ^{{-1}}\left|{\frac {x}{a}}\right|\right)\qquad {\mbox{(}}|x|\geq |a|{\mbox{)}}

Integrais con R=√ax2+bx+c
-------------------------

\int {\frac {dx}{{\sqrt {ax^{2}+bx+c}}}}={\frac {1}{{\sqrt {a}}}}\ln \left|2{\sqrt {a}}R+2ax+b\right|\qquad {\mbox{(para }}a>0{\mbox{)}}
\int {\frac {dx}{{\sqrt {ax^{2}+bx+c}}}}={\frac {1}{{\sqrt {a}}}}\,\sinh ^{{-1}}{\frac {2ax+b}{{\sqrt {4ac-b^{2}}}}}\qquad {\mbox{(para }}a>0{\mbox{, }}4ac-b^{2}>0{\mbox{)}}
\int {\frac {dx}{{\sqrt {ax^{2}+bx+c}}}}={\frac {1}{{\sqrt {a}}}}\ln |2ax+b|\quad {\mbox{(para }}a>0{\mbox{, }}4ac-b^{2}=0{\mbox{)}}
\int {\frac {dx}{{\sqrt {ax^{2}+bx+c}}}}=-{\frac {1}{{\sqrt {-a}}}}\arcsin {\frac {2ax+b}{{\sqrt {b^{2}-4ac}}}}\qquad {\mbox{(para }}a<0{\mbox{, }}4ac-b^{2}<0{\mbox{)}}
\int {\frac {dx}{{\sqrt {(ax^{2}+bx+c)^{{3}}}}}}={\frac {4ax+2b}{(4ac-b^{2}){\sqrt {R}}}}
\int {\frac {dx}{{\sqrt {(ax^{2}+bx+c)^{{5}}}}}}={\frac {4ax+2b}{3(4ac-b^{2}){\sqrt {R}}}}\left({\frac {1}{R}}+{\frac {8a}{4ac-b^{2}}}\right)
\int {\frac {dx}{{\sqrt {(ax^{2}+bx+c)^{{2n+1}}}}}}={\frac {4ax+2b}{(2n-1)(4ac-b^{2})R^{{(2n-1)/2}}}}+{\frac {8a(n-1)}{(2n-1)(4ac-b^{2})}}\int {\frac {dx}{R^{{(2n-1)/2}}}}
\int {\frac {x\;dx}{{\sqrt {ax^{2}+bx+c}}}}={\frac {{\sqrt {R}}}{a}}-{\frac {b}{2a}}\int {\frac {dx}{{\sqrt {R}}}}
\int {\frac {x\;dx}{{\sqrt {(ax^{2}+bx+c)^{3}}}}}=-{\frac {2bx+4c}{(4ac-b^{2}){\sqrt {R}}}}
\int {\frac {x\;dx}{{\sqrt {(ax^{2}+bx+c)^{{2n+1}}}}}}=-{\frac {1}{(2n-1)aR^{{(2n-1)/2}}}}-{\frac {b}{2a}}\int {\frac {dx}{R^{{(2n+1)/2}}}}
\int {\frac {dx}{x{\sqrt {ax^{2}+bx+c}}}}=-{\frac {1}{{\sqrt {c}}}}\ln \left({\frac {2{\sqrt {cR}}+bx+2c}{x}}\right)\qquad {\mbox{(para }}c>0{\mbox{)}}
\int {\frac {dx}{x{\sqrt {ax^{2}+bx+c}}}}={\frac {1}{{\sqrt {-c}}}}\sin ^{{-1}}\left({\frac {bx+2c}{|x|{\sqrt {b^{2}-4ac}}}}\right)\qquad {\mbox{(para }}c<0{\mbox{)}}
\int {\frac {dx}{x{\sqrt {ax^{2}+bx+c}}}}=-{\frac {1}{{\sqrt {c}}}}\sinh ^{{-1}}\left({\frac {bx+2c}{|x|{\sqrt {4ac-b^{2}}}}}\right)

Integrais con √ax+b
-------------------

\int {\frac {dx}{x{\sqrt {ax+b}}}}\,=\,{\frac {-2}{{\sqrt {b}}}}\tanh ^{{-1}}{{\sqrt {{\frac {ax+b}{b}}}}}
\int {\frac {{\sqrt {ax+b}}}{x}}\,dx\;=\;2\left({\sqrt {ax+b}}-{\sqrt {b}}\tanh ^{{-1}}{{\sqrt {{\frac {ax+b}{b}}}}}\right)
\int {\frac {x^{n}}{{\sqrt {ax+b}}}}\,dx\;=\;{\frac {2}{a(2n+1)}}\left(x^{{n}}{\sqrt {ax+b}}-bn\int {\frac {x^{{n-1}}}{{\sqrt {ax+b}}}}\,dx\right)
\int x^{n}{\sqrt {ax+b}}\,dx\;=\;{\frac {2}{2n+1}}\left(x^{{n+1}}{\sqrt {ax+b}}+bx^{{n}}{\sqrt {ax+b}}-nb\int x^{{n-1}}{\sqrt {ax+b}}\,dx\right)