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.

## Contents

* 1 Integrals involving r = √a2 \+ x2
* 2 Integrals involving s = √x2 − a2
* 3 Integrals involving u = √a2 − x2
* 4 Integrals involving R = √ax2 \+ bx \+ c
* 5 Integrals involving S = √ax \+ b
* 6 References

## Integrals involving r = √a2 \+ x2

* ∫rdx=12(xr+a2ln⁡(x+r))
* ∫r3dx=14xr3+38a2xr+38a4ln⁡(x+r)
* ∫r5dx=16xr5+524a2xr3+516a4xr+516a6ln⁡(x+r)
* ∫xrdx=r33
* ∫xr3dx=r55
* ∫xr2n+1dx=r2n+32n+3
* ∫x2rdx=xr34−a2xr8−a48ln⁡(x+r)
* ∫x2r3dx=xr56−a2xr324−a4xr16−a616ln⁡(x+r)
* ∫x3rdx=r55−a2r33
* ∫x3r3dx=r77−a2r55
* ∫x3r2n+1dx=r2n+52n+5−a2r2n+32n+3
* ∫x4rdx=x3r36−a2xr38+a4xr16+a616ln⁡(x+r)
* ∫x4r3dx=x3r58−a2xr516+a4xr364+3a6xr128+3a8128ln⁡(x+r)
* ∫x5rdx=r77−2a2r55+a4r33
* ∫x5r3dx=r99−2a2r77+a4r55
* ∫x5r2n+1dx=r2n+72n+7−2a2r2n+52n+5+a4r2n+32n+3
* ∫rdxx=r−aln⁡|a+rx|=r−aarsinh⁡ax
* ∫r3dxx=r33+a2r−a3ln⁡|a+rx|
* ∫r5dxx=r55+a2r33+a4r−a5ln⁡|a+rx|
* ∫r7dxx=r77+a2r55+a4r33+a6r−a7ln⁡|a+rx|
* ∫dxr=arsinh⁡xa=ln⁡(x+ra)
* ∫dxr3=xa2r
* ∫xdxr=r
* ∫xdxr3=−1r
* ∫x2dxr=x2r−a22arsinh⁡xa=x2r−a22ln⁡(x+ra)
* ∫dxxr=−1aarsinh⁡ax=−1aln⁡|a+rx|

## Integrals involving s = √x2 − a2

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

* ∫sdx=12(xs−a2ln⁡|x+s|)
* ∫xsdx=13s3
* ∫sdxx=s−|a|arccos⁡|ax|
* ∫dxs=ln⁡|x+sa|. Here ln⁡|x+sa|=sgn⁡(x)arcosh⁡|xa|=12ln⁡(x+sx−s), where the positive value of arcosh⁡|xa| is to be taken.
* ∫dxxs=1aarcsec⁡|xa|
* ∫xdxs=s
* ∫xdxs3=−1s
* ∫xdxs5=−13s3
* ∫xdxs7=−15s5
* ∫xdxs2n+1=−1(2n−1)s2n−1
* ∫x2mdxs2n+1=−12n−1x2m−1s2n−1+2m−12n−1∫x2m−2dxs2n−1
* ∫x2dxs=xs2+a22ln⁡|x+sa|
* ∫x2dxs3=−xs+ln⁡|x+sa|
* ∫x4dxs=x3s4+38a2xs+38a4ln⁡|x+sa|
* ∫x4dxs3=xs2−a2xs+32a2ln⁡|x+sa|
* ∫x4dxs5=−xs−13x3s3+ln⁡|x+sa|
* ∫x2mdxs2n+1=(−1)n−m1a2(n−m)∑i=0n−m−112(m+i)+1(n−m−1i)x2(m+i)+1s2(m+i)+1(n>m≥0)
* ∫dxs3=−1a2xs
* ∫dxs5=1a4[xs−13x3s3]
* ∫dxs7=−1a6[xs−23x3s3+15x5s5]
* ∫dxs9=1a8[xs−33x3s3+35x5s5−17x7s7]
* ∫x2dxs5=−1a2x33s3
* ∫x2dxs7=1a4[13x3s3−15x5s5]
* ∫x2dxs9=−1a6[13x3s3−25x5s5+17x7s7]

## Integrals involving u = √a2 − x2

* ∫udx=12(xu+a2arcsin⁡xa)(|x|≤|a|)
* ∫xudx=−13u3(|x|≤|a|)
* ∫x2udx=−x4u3+a28(xu+a2arcsin⁡xa)(|x|≤|a|)
* ∫udxx=u−aln⁡|a+ux|(|x|≤|a|)
* ∫dxu=arcsin⁡xa(|x|≤|a|)
* ∫x2dxu=12(−xu+a2arcsin⁡xa)(|x|≤|a|)
* ∫udx=12(xu−sgn⁡xarcosh⁡|xa|)(for |x|≥|a|)
* ∫xudx=−u(|x|≤|a|)

## 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.

* ∫dxR=1aln⁡|2aR+2ax+b|(for a>0)
* ∫dxR=1aarsinh⁡2ax+b4ac−b2(for a>0, 4ac−b2>0)
* ∫dxR=1aln⁡|2ax+b|(for a>0, 4ac−b2=0)
* ∫dxR=−1−aarcsin⁡2ax+bb2−4ac(for a<0, 4ac−b2<0, |2ax+b|<b2−4ac)
* ∫dxR3=4ax+2b(4ac−b2)R
* ∫dxR5=4ax+2b3(4ac−b2)R(1R2+8a4ac−b2)
* ∫dxR2n+1=2(2n−1)(4ac−b2)(2ax+bR2n−1+4a(n−1)∫dxR2n−1)
* ∫xRdx=Ra−b2a∫dxR
* ∫xR3dx=−2bx+4c(4ac−b2)R
* ∫xR2n+1dx=−1(2n−1)aR2n−1−b2a∫dxR2n+1
* ∫dxxR=−1cln⁡|2cR+bx+2cx|,c>0
* ∫dxxR=−1carsinh⁡(bx+2c|x|4ac−b2),c<0
* ∫dxxR=1−carcsin⁡(bx+2c|x|b2−4ac),c<0,b2−4ac>0
* ∫dxxR=−2bx(ax2+bx),c=0
* ∫x2Rdx=2ax−3b4a2R+3b2−4ac8a2∫dxR
* ∫dxx2R=−Rcx−b2c∫dxxR
* ∫Rdx=2ax+b4aR+4ac−b28a∫dxR
* ∫xRdx=R33a−b(2ax+b)8a2R−b(4ac−b2)16a2∫dxR
* ∫x2Rdx=6ax−5b24a2R3+5b2−4ac16a2∫Rdx
* ∫Rxdx=R+b2∫dxR+c∫dxxR
* ∫Rx2dx=−Rx+a∫dxR+b2∫dxxR
* ∫x2dxR3=(2b2−4ac)x+2bca(4ac−b2)R+1a∫dxR

## Integrals involving S = √ax \+ b

* ∫Sdx=2S33a
* ∫dxS=2Sa
* ∫dxxS={−2barcoth⁡(Sb)(for b>0,ax>0)−2bartanh⁡(Sb)(for b>0,ax<0)2−barctan⁡(S−b)(for b<0)
* ∫Sxdx={2(S−barcoth⁡(Sb))(for b>0,ax>0)2(S−bartanh⁡(Sb))(for b>0,ax<0)2(S−−barctan⁡(S−b))(for b<0)
* ∫xnSdx=2a(2n+1)(xnS−bn∫xn−1Sdx)
* ∫xnSdx=2a(2n+3)(xnS3−nb∫xn−1Sdx)
* ∫1xnSdx=−1b(n−1)(Sxn−1+(n−32)a∫dxxn−1S)

## 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.

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Lists of integrals

* Rational functions
* Irrational functions
* Trigonometric functions
* Inverse trigonometric functions
* Hyperbolic functions
* Inverse hyperbolic functions
* Exponential functions
* Logarithmic functions
* Gaussian functions

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