List of integrals of irrational functions

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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

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

Integrals involving s = √x² − a²

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

• [math]\displaystyle{ \int s\,dx = \frac{1}{2}\left(xs-a^{2}\ln\left|x+s\right|\right) }[/math]
• [math]\displaystyle{ \int xs\,dx = \frac{1}{3}s^3 }[/math]
• [math]\displaystyle{ \int\frac{s\,dx}{x} = s - |a|\arccos\left|\frac{a}{x}\right| }[/math]
• [math]\displaystyle{ \int\frac{dx}{s} = \ln\left|\frac{x+s}{a}\right|\,. }[/math] Here [math]\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)\,, }[/math] where the positive value of [math]\displaystyle{ \operatorname{arcosh}\left|\frac{x}{a}\right| }[/math] is to be taken.
• [math]\displaystyle{ \int\frac{dx}{xs} = \frac{1}{a}\operatorname{arcsec}\left|\frac{x}{a}\right| }[/math]
• [math]\displaystyle{ \int\frac{x\,dx}{s} = s }[/math]
• [math]\displaystyle{ \int\frac{x\,dx}{s^3} = -\frac{1}{s} }[/math]
• [math]\displaystyle{ \int\frac{x\,dx}{s^5} = -\frac{1}{3s^3} }[/math]
• [math]\displaystyle{ \int\frac{x\,dx}{s^7} = -\frac{1}{5s^5} }[/math]
• [math]\displaystyle{ \int\frac{x\,dx}{s^{2n+1}} = -\frac{1}{(2n-1)s^{2n-1}} }[/math]
• [math]\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}} }[/math]
• [math]\displaystyle{ \int\frac{x^2\,dx}{s} = \frac{xs}{2}+\frac{a^2}{2}\ln\left|\frac{x+s}{a}\right| }[/math]
• [math]\displaystyle{ \int\frac{x^2\,dx}{s^3} = -\frac{x}{s}+\ln\left|\frac{x+s}{a}\right| }[/math]
• [math]\displaystyle{ \int\frac{x^4\,dx}{s} = \frac{x^3s}{4}+\frac{3}{8}a^2xs+\frac{3}{8}a^4\ln\left|\frac{x+s}{a}\right| }[/math]
• [math]\displaystyle{ \int\frac{x^4\,dx}{s^3} = \frac{xs}{2}-\frac{a^2x}{s}+\frac{3}{2}a^2\ln\left|\frac{x+s}{a}\right| }[/math]
• [math]\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| }[/math]
• [math]\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\gt m\ge0\mbox{)} }[/math]
• [math]\displaystyle{ \int\frac{dx}{s^3} = -\frac{1}{a^2}\frac{x}{s} }[/math]
• [math]\displaystyle{ \int\frac{dx}{s^5} = \frac{1}{a^4}\left[\frac{x}{s}-\frac{1}{3}\frac{x^3}{s^3}\right] }[/math]
• [math]\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] }[/math]
• [math]\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] }[/math]
• [math]\displaystyle{ \int\frac{x^2\,dx}{s^5} = -\frac{1}{a^2}\frac{x^3}{3s^3} }[/math]
• [math]\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] }[/math]
• [math]\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] }[/math]

Integrals involving u = √a² − x²

• [math]\displaystyle{ \int u\,dx = \frac{1}{2}\left(xu+a^2\arcsin\frac{x}{a}\right) \qquad\mbox{(}|x|\leq|a|\mbox{)} }[/math]
• [math]\displaystyle{ \int xu\,dx = -\frac{1}{3} u^3 \qquad\mbox{(}|x|\leq|a|\mbox{)} }[/math]
• [math]\displaystyle{ \int x^2u\,dx = -\frac{x}{4} u^3+\frac{a^2}{8}(xu+a^2\arcsin\frac{x}{a}) \qquad\mbox{(}|x|\leq|a|\mbox{)} }[/math]
• [math]\displaystyle{ \int\frac{u\,dx}{x} = u-a\ln\left|\frac{a+u}{x}\right| \qquad\mbox{(}|x|\leq|a|\mbox{)} }[/math]
• [math]\displaystyle{ \int\frac{dx}{u} = \arcsin\frac{x}{a} \qquad\mbox{(}|x|\leq|a|\mbox{)} }[/math]
• [math]\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{)} }[/math]
• [math]\displaystyle{ \int u\,dx = \frac{1}{2}\left(xu-\sgn x\,\operatorname{arcosh}\left|\frac{x}{a}\right|\right) \qquad\mbox{(for }|x|\ge|a|\mbox{)} }[/math]
• [math]\displaystyle{ \int \frac{x}{u}\,dx = -u \qquad\mbox{(}|x|\leq|a|\mbox{)} }[/math]

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.

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

Integrals involving S = √ax + b

• [math]\displaystyle{ \int S\,dx = \frac{2 S^{3}}{3 a} }[/math]
• [math]\displaystyle{ \int \frac{dx}{S} = \frac{2S}{a} }[/math]
• [math]\displaystyle{ \int \frac{dx}{x S} = \begin{cases} -\dfrac{2}{\sqrt{b}} \operatorname{arcoth}\left( \dfrac{S}{\sqrt{b}}\right) & \mbox{(for }b \gt 0, \quad a x \gt 0\mbox{)} \\ -\dfrac{2}{\sqrt{b}} \operatorname{artanh}\left( \dfrac{S}{\sqrt{b}}\right) & \mbox{(for }b \gt 0, \quad a x \lt 0\mbox{)} \\ \dfrac{2}{\sqrt{-b}} \arctan\left( \dfrac{S}{\sqrt{-b}}\right) & \mbox{(for }b \lt 0\mbox{)} \\ \end{cases} }[/math]
• [math]\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 \gt 0, \quad a x \gt 0\mbox{)} \\ 2 \left( S - \sqrt{b}\,\operatorname{artanh}\left( \dfrac{S}{\sqrt{b}}\right)\right) & \mbox{(for }b \gt 0, \quad a x \lt 0\mbox{)} \\ 2 \left( S - \sqrt{-b} \arctan\left( \dfrac{S}{\sqrt{-b}}\right)\right) & \mbox{(for }b \lt 0\mbox{)} \\ \end{cases} }[/math]
• [math]\displaystyle{ \int \frac{x^{n}}{S}\,dx = \frac{2}{a (2 n + 1)} \left( x^{n} S - b n \int \frac{x^{n - 1}}{S}\,dx\right) }[/math]
• [math]\displaystyle{ \int x^{n} S\,dx = \frac{2}{a (2 n + 3)} \left(x^{n} S^{3} - n b \int x^{n - 1} S\,dx\right) }[/math]
• [math]\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) }[/math]

References

• Abramowitz, Milton; Stegun, Irene A., eds (1972). "Chapter 3". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. http://www.math.sfu.ca/~cbm/aands/page_13.htm. 
• (in en) Table of Integrals, Series, and Products (8 ed.). Academic Press, Inc.. 2015. ISBN 978-0-12-384933-5.  (Several previous editions as well.)
• Peirce, Benjamin Osgood (1929). "Chapter 3". A Short Table of Integrals (3rd revised ed.). Boston: Ginn and Co. pp. 16–30.