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

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

Integrals involving s = √x² − a²

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

• $\int sdx=\frac{1}{2}(xs-a^2\ln |x+s|)$
• $\int xsdx=\frac{1}{3}{s}^3$
• $\int \frac{sdx}{x}=s-|a|\arccos \left|\frac{a}{x}\right|$
• $\int \frac{dx}{s}=\ln \left|\frac{x+s}{a}\right|.$ Here $\ln \left|\frac{x+s}{a}\right|=\mathrm{sgn}(x)\mathrm{arcosh}\left|\frac{x}{a}\right|=\frac{1}{2}\ln \left(\frac{x+s}{x-s}\right),$ where the positive value of $\mathrm{arcosh}\left|\frac{x}{a}\right|$ is to be taken.
• $\int \frac{dx}{xs}=\frac{1}{a}\mathrm{arcsec}\left|\frac{x}{a}\right|$
• $\int \frac{xdx}{s}=s$
• $\int \frac{xdx}{s^3}=-\frac{1}{s}$
• $\int \frac{xdx}{s^5}=-\frac{1}{3s^3}$
• $\int \frac{xdx}{s^7}=-\frac{1}{5s^5}$
• $\int \frac{xdx}{s^{2n+1}}=-\frac{1}{(2n-1)s^{2n-1}}$
• $\int \frac{x^{2m}dx}{s^{2n+1}}=-\frac{1}{2n-1}\frac{x^{2m-1}}{s^{2n-1}}+\frac{2m-1}{2n-1}\int \frac{x^{2m-2}dx}{s^{2n-1}}$
• $\int \frac{x^{2}dx}{s}=\frac{xs}{2}+\frac{a^2}{2}\ln \left|\frac{x+s}{a}\right|$
• $\int \frac{x^{2}dx}{s^3}=-\frac{x}{s}+\ln \left|\frac{x+s}{a}\right|$
• $\int \frac{x^{4}dx}{s}=\frac{x^{3}s}{4}+\frac{3}{8}{a}^{2}xs+\frac{3}{8}{a}^4\ln \left|\frac{x+s}{a}\right|$
• $\int \frac{x^{4}dx}{s^3}=\frac{xs}{2}-\frac{a^{2}x}{s}+\frac{3}{2}{a}^2\ln \left|\frac{x+s}{a}\right|$
• $\int \frac{x^{4}dx}{s^5}=-\frac{x}{s}-\frac{1}{3}\frac{x^3}{s^3}+\ln \left|\frac{x+s}{a}\right|$
• $\int \frac{x^{2m}dx}{s^{2n+1}}=(-1{)}^{n-m}\frac{1}{a^{2(n-m)}}\sum _{i=0}^{n-m-1}\frac{1}{2(m+i)+1}(\genfrac{}{}{0ex}{}{n-m-1}{i})\frac{x^{2(m+i)+1}}{s^{2(m+i)+1}}\text{(}n>m\ge 0\text{)}$
• $\int \frac{dx}{s^3}=-\frac{1}{a^2}\frac{x}{s}$
• $\int \frac{dx}{s^5}=\frac{1}{a^4}[\frac{x}{s}-\frac{1}{3}\frac{x^3}{s^3}]$
• $\int \frac{dx}{s^7}=-\frac{1}{a^6}[\frac{x}{s}-\frac{2}{3}\frac{x^3}{s^3}+\frac{1}{5}\frac{x^5}{s^5}]$
• $\int \frac{dx}{s^9}=\frac{1}{a^8}[\frac{x}{s}-\frac{3}{3}\frac{x^3}{s^3}+\frac{3}{5}\frac{x^5}{s^5}-\frac{1}{7}\frac{x^7}{s^7}]$
• $\int \frac{x^{2}dx}{s^5}=-\frac{1}{a^2}\frac{x^3}{3s^3}$
• $\int \frac{x^{2}dx}{s^7}=\frac{1}{a^4}[\frac{1}{3}\frac{x^3}{s^3}-\frac{1}{5}\frac{x^5}{s^5}]$
• $\int \frac{x^{2}dx}{s^9}=-\frac{1}{a^6}[\frac{1}{3}\frac{x^3}{s^3}-\frac{2}{5}\frac{x^5}{s^5}+\frac{1}{7}\frac{x^7}{s^7}]$

Integrals involving u = √a² − x²

• $\int udx=\frac{1}{2}(xu+a^2\arcsin \frac{x}{a})\text{(}|x|\le |a|\text{)}$
• $\int xudx=-\frac{1}{3}{u}^3\text{(}|x|\le |a|\text{)}$
• $\int x^{2}udx=-\frac{x}{4}{u}^3+\frac{a^2}{8}(xu+a^2\arcsin \frac{x}{a})\text{(}|x|\le |a|\text{)}$
• $\int \frac{udx}{x}=u-a\ln \left|\frac{a+u}{x}\right|\text{(}|x|\le |a|\text{)}$
• $\int \frac{dx}{u}=\arcsin \frac{x}{a}\text{(}|x|\le |a|\text{)}$
• $\int \frac{x^{2}dx}{u}=\frac{1}{2}(-xu+a^2\arcsin \frac{x}{a})\text{(}|x|\le |a|\text{)}$
• $\int udx=\frac{1}{2}(xu-\mathrm{sgn}x\mathrm{arcosh}\left|\frac{x}{a}\right|)\text{(for }|x|\ge |a|\text{)}$
• $\int \frac{x}{u}dx=-u\text{(}|x|\le |a|\text{)}$

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.

• $\int \frac{dx}{R}=\frac{1}{\sqrt{a}}\ln |2\sqrt{a}R+2ax+b|\text{(for }a>0\text{)}$
• $\int \frac{dx}{R}=\frac{1}{\sqrt{a}}\mathrm{arsinh}\frac{2ax+b}{\sqrt{4ac-b^2}}\text{(for }a>0\text{, }4ac-b^2>0\text{)}$
• $\int \frac{dx}{R}=\frac{1}{\sqrt{a}}\ln |2ax+b|\text{(for }a>0\text{, }4ac-b^2=0\text{)}$
• $\int \frac{dx}{R}=-\frac{1}{\sqrt{-a}}\arcsin \frac{2ax+b}{\sqrt{b^2-4ac}}\text{(for }a<0\text{, }4ac-b^2<0\text{, }|2ax+b|<\sqrt{b^2-4ac}\text{)}$
• $\int \frac{dx}{R^3}=\frac{4ax+2b}{(4ac-b^2)R}$
• $\int \frac{dx}{R^5}=\frac{4ax+2b}{3(4ac-b^2)R}(\frac{1}{R^2}+\frac{8a}{4ac-b^2})$
• $\int \frac{dx}{R^{2n+1}}=\frac{2}{(2n-1)(4ac-b^2)}(\frac{2ax+b}{R^{2n-1}}+4a(n-1)\int \frac{dx}{R^{2n-1}})$
• $\int \frac{x}{R}dx=\frac{R}{a}-\frac{b}{2a}\int \frac{dx}{R}$
• $\int \frac{x}{R^3}dx=-\frac{2bx+4c}{(4ac-b^2)R}$
• $\int \frac{x}{R^{2n+1}}dx=-\frac{1}{(2n-1)a{R}^{2n-1}}-\frac{b}{2a}\int \frac{dx}{R^{2n+1}}$
• $\int \frac{dx}{xR}=-\frac{1}{\sqrt{c}}\ln \left|\frac{2\sqrt{c}R+bx+2c}{x}\right|,c>0$
• $\int \frac{dx}{xR}=-\frac{1}{\sqrt{c}}\mathrm{arsinh}\left(\frac{bx+2c}{|x|\sqrt{4ac-b^2}}\right),c<0$
• $\int \frac{dx}{xR}=\frac{1}{\sqrt{-c}}\arcsin \left(\frac{bx+2c}{|x|\sqrt{b^2-4ac}}\right),c<0,b^2-4ac>0$
• $\int \frac{dx}{xR}=-\frac{2}{bx}\left(\sqrt{a{x}^2+bx}\right),c=0$
• $\int \frac{x^2}{R}dx=\frac{2ax-3b}{4a^2}R+\frac{3b^2-4ac}{8a^2}\int \frac{dx}{R}$
• $\int \frac{dx}{x^{2}R}=-\frac{R}{cx}-\frac{b}{2c}\int \frac{dx}{xR}$
• $\int Rdx=\frac{2ax+b}{4a}R+\frac{4ac-b^2}{8a}\int \frac{dx}{R}$
• $\int xRdx=\frac{R^3}{3a}-\frac{b(2ax+b)}{8a^2}R-\frac{b(4ac-b^2)}{16a^2}\int \frac{dx}{R}$
• $\int x^{2}Rdx=\frac{6ax-5b}{24a^2}{R}^3+\frac{5b^2-4ac}{16a^2}\int Rdx$
• $\int \frac{R}{x}dx=R+\frac{b}{2}\int \frac{dx}{R}+c\int \frac{dx}{xR}$
• $\int \frac{R}{x^2}dx=-\frac{R}{x}+a\int \frac{dx}{R}+\frac{b}{2}\int \frac{dx}{xR}$
• $\int \frac{x^{2}dx}{R^3}=\frac{(2b^2-4ac)x+2bc}{a(4ac-b^2)R}+\frac{1}{a}\int \frac{dx}{R}$

Integrals involving S = √ax + b

• $\int Sdx=\frac{2S^3}{3a}$
• $\int \frac{dx}{S}=\frac{2S}{a}$
• $\int \frac{dx}{xS}=\{\begin{array}{ll}-\frac{2}{\sqrt{b}}\mathrm{arcoth}\left(\frac{S}{\sqrt{b}}\right)& \text{(for }b>0,ax>0\text{)}\\ -\frac{2}{\sqrt{b}}\mathrm{artanh}\left(\frac{S}{\sqrt{b}}\right)& \text{(for }b>0,ax<0\text{)}\\ \frac{2}{\sqrt{-b}}\arctan \left(\frac{S}{\sqrt{-b}}\right)& \text{(for }b<0\text{)}\\ \end{array}$
• $\int \frac{S}{x}dx=\{\begin{array}{ll}2(S-\sqrt{b}\mathrm{arcoth}\left(\frac{S}{\sqrt{b}}\right))& \text{(for }b>0,ax>0\text{)}\\ 2(S-\sqrt{b}\mathrm{artanh}\left(\frac{S}{\sqrt{b}}\right))& \text{(for }b>0,ax<0\text{)}\\ 2(S-\sqrt{-b}\arctan \left(\frac{S}{\sqrt{-b}}\right))& \text{(for }b<0\text{)}\\ \end{array}$
• $\int \frac{x^n}{S}dx=\frac{2}{a(2n+1)}(x^{n}S-bn\int \frac{x^{n-1}}{S}dx)$
• $\int x^{n}Sdx=\frac{2}{a(2n+3)}(x^n{S}^3-nb\int x^{n-1}Sdx)$
• $\int \frac{1}{x^{n}S}dx=-\frac{1}{b(n-1)}(\frac{S}{x^{n-1}}+(n-\frac{3}{2})a\int \frac{dx}{x^{n-1}S})$

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.