List of integrals of irrational algebraic 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 integration variable and all parameters are assumed to be real numbers and 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):

• $\int sdx=\frac{1}{2}(xs-a^2\ln |x+s|)$
• $\int xsdx=\frac{1}{3}{s}^3$
• $\int \frac{sdx}{x}=s-|a|\arccos \left|\frac{a}{x}\right|$
• $\int \frac{dx}{s}=\ln \left|\frac{x+s}{a}\right|=\mathrm{sgn}(x)\mathrm{arcosh}\left|\frac{x}{a}\right|=\frac{1}{2}\ln \left(\frac{x+s}{x-s}\right),$ 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.