Table of Indefinite Integrals of Rational Functions

Integrals Involving ax+bax + b

title: Notation
We denote:
$$X = ax + b$$

I. Basic Integrals of XnX^n

  1. ∫Xn dx=1a(n+1)Xn+1(n≠−1)\displaystyle \int X^n \, dx = \frac{1}{a(n+1)} X^{n+1} \qquad (n \neq -1)

(For n=−1n = -1, see formula 2.)

  1. ∫dxX=1aln⁡∣X∣\displaystyle \int \frac{dx}{X} = \frac{1}{a} \ln |X|

II. Integrals of xmXnx^m X^n

  1. ∫xXn dx=1a2(n+2)Xn+2−ba2(n+1)Xn+1(n≠−1,−2)\displaystyle \int x X^n \, dx = \frac{1}{a^2(n+2)} X^{n+2} - \frac{b}{a^2(n+1)} X^{n+1} \qquad (n \neq -1, -2)

(For n=−1,−2n = -1, -2, see formulas 5 and 6.)

  1. ∫xmXn dx=1am+1∫(X−b)mXn dX\displaystyle \int x^m X^n \, dx = \frac{1}{a^{m+1}} \int (X - b)^m X^n \, dX

This substitution is useful when mm is an integer or nn is fractional. Expand (X−b)m(X - b)^m using the binomial theorem.

III. Integrals of xmXn\dfrac{x^m}{X^n}

Case m=1m = 1

  1. ∫x dxX=xa−ba2ln⁡∣X∣\displaystyle \int \frac{x \, dx}{X} = \frac{x}{a} - \frac{b}{a^2} \ln |X|

  2. ∫x dxX2=ba2X+1a2ln⁡∣X∣\displaystyle \int \frac{x \, dx}{X^2} = \frac{b}{a^2 X} + \frac{1}{a^2} \ln |X|

  3. ∫x dxX3=1a2(−1X+b2X2)\displaystyle \int \frac{x \, dx}{X^3} = \frac{1}{a^2} \left( -\frac{1}{X} + \frac{b}{2X^2} \right)

  4. ∫x dxXn=1a2(−1(n−2)Xn−2+b(n−1)Xn−1)(n≠1,2)\displaystyle \int \frac{x \, dx}{X^n} = \frac{1}{a^2} \left( \frac{-1}{(n-2)X^{n-2}} + \frac{b}{(n-1)X^{n-1}} \right) \qquad (n \neq 1, 2)

Case m=2m = 2

  1. ∫x2 dxX=1a3(X22−2bX+b2ln⁡∣X∣)\displaystyle \int \frac{x^2 \, dx}{X} = \frac{1}{a^3} \left( \frac{X^2}{2} - 2bX + b^2 \ln |X| \right)

  2. ∫x2 dxX2=1a3(X−2bln⁡∣X∣−b2X)\displaystyle \int \frac{x^2 \, dx}{X^2} = \frac{1}{a^3} \left( X - 2b \ln |X| - \frac{b^2}{X} \right)

  3. ∫x2 dxX3=1a3(ln⁡∣X∣+2bX−b22X2)\displaystyle \int \frac{x^2 \, dx}{X^3} = \frac{1}{a^3} \left( \ln |X| + \frac{2b}{X} - \frac{b^2}{2X^2} \right)

  4. ∫x2 dxXn=1a3[−1(n−3)Xn−3+2b(n−2)Xn−2−b2(n−1)Xn−1](n≠1,2,3)\displaystyle \int \frac{x^2 \, dx}{X^n} = \frac{1}{a^3} \left[ \frac{-1}{(n-3)X^{n-3}} + \frac{2b}{(n-2)X^{n-2}} - \frac{b^2}{(n-1)X^{n-1}} \right] \qquad (n \neq 1, 2, 3)

Case m=3m = 3

  1. ∫x3 dxX=1a4(X33−3bX22+3b2X−b3ln⁡∣X∣)\displaystyle \int \frac{x^3 \, dx}{X} = \frac{1}{a^4} \left( \frac{X^3}{3} - \frac{3bX^2}{2} + 3b^2 X - b^3 \ln |X| \right)

  2. ∫x3 dxX2=1a4(X22−3bX+3b2ln⁡∣X∣+b3X)\displaystyle \int \frac{x^3 \, dx}{X^2} = \frac{1}{a^4} \left( \frac{X^2}{2} - 3bX + 3b^2 \ln |X| + \frac{b^3}{X} \right)

  3. ∫x3 dxX3=1a4(X−3bln⁡∣X∣−3b2X+b32X2)\displaystyle \int \frac{x^3 \, dx}{X^3} = \frac{1}{a^4} \left( X - 3b \ln |X| - \frac{3b^2}{X} + \frac{b^3}{2X^2} \right)

  4. ∫x3 dxX4=1a4(ln⁡∣X∣+3bX−3b22X2+b33X3)\displaystyle \int \frac{x^3 \, dx}{X^4} = \frac{1}{a^4} \left( \ln |X| + \frac{3b}{X} - \frac{3b^2}{2X^2} + \frac{b^3}{3X^3} \right)

  5. ∫x3 dxXn=1a4[−1(n−4)Xn−4+3b(n−3)Xn−3−3b2(n−2)Xn−2+b3(n−1)Xn−1](n≠1,2,3,4)\displaystyle \int \frac{x^3 \, dx}{X^n} = \frac{1}{a^4} \left[ \frac{-1}{(n-4)X^{n-4}} + \frac{3b}{(n-3)X^{n-3}} - \frac{3b^2}{(n-2)X^{n-2}} + \frac{b^3}{(n-1)X^{n-1}} \right] \qquad (n \neq 1, 2, 3, 4)

IV. Integrals of 1xmXn\dfrac{1}{x^m X^n}

Case m=1m = 1

  1. ∫dxxX=1bln⁡∣xX∣\displaystyle \int \frac{dx}{xX} = \frac{1}{b} \ln \left| \frac{x}{X} \right| 

title: Note:
$\frac{1}{b} \ln|x| - \frac{1}{b} \ln|X| = \frac{1}{b} \ln\left|\frac{x}{X}\right|$, which is equivalent to $-\frac{1}{b} \ln\left|\frac{X}{x}\right|$

  1. ∫dxxX2=1b2(abln⁡∣Xx∣−aX)\displaystyle \int \frac{dx}{xX^2} = \frac{1}{b^2} \left( \frac{a}{b} \ln \left| \frac{X}{x} \right| - \frac{a}{X} \right)

  2. ∫dxxX3=1b3(a22b2ln⁡∣Xx∣−a22X2−abX)\displaystyle \int \frac{dx}{xX^3} = \frac{1}{b^3} \left( \frac{a^2}{2b^2} \ln \left| \frac{X}{x} \right| - \frac{a^2}{2X^2} - \frac{a}{bX} \right)

  3. ∫dxxXn=1bn[1n−1⋅aXn−1−1bln⁡∣xX∣](n≥1)\displaystyle \int \frac{dx}{xX^n} = \frac{1}{b^n} \left[ \frac{1}{n-1} \cdot \frac{a}{X^{n-1}} - \frac{1}{b} \ln \left| \frac{x}{X} \right| \right] \quad (n \geq 1) 

title: Note:
This is a simplified form; the general formula involving a summation is given below.

Case m=2m = 2

  1. ∫dxx2X=−1bx−ab2ln⁡∣Xx∣\displaystyle \int \frac{dx}{x^2 X} = -\frac{1}{bx} - \frac{a}{b^2} \ln \left| \frac{X}{x} \right|

  2. ∫dxx2X2=−1b2x+2ab3ln⁡∣Xx∣−ab2X\displaystyle \int \frac{dx}{x^2 X^2} = -\frac{1}{b^2 x} + \frac{2a}{b^3} \ln \left| \frac{X}{x} \right| - \frac{a}{b^2 X}

  3. ∫dxx2X3=−1b3x+3ab4ln⁡∣Xx∣−3a2b3X2−2ab3X\displaystyle \int \frac{dx}{x^2 X^3} = -\frac{1}{b^3 x} + \frac{3a}{b^4} \ln \left| \frac{X}{x} \right| - \frac{3a}{2b^3 X^2} - \frac{2a}{b^3 X}

Case m=3m = 3

  1. ∫dxx3X=−12bx2+ab2x+a2b3ln⁡∣xX∣\displaystyle \int \frac{dx}{x^3 X} = -\frac{1}{2b x^2} + \frac{a}{b^2 x} + \frac{a^2}{b^3} \ln \left| \frac{x}{X} \right|

  2. ∫dxx3X2=−12b2x2+2ab3x−3a2b4ln⁡∣xX∣+a2b3X\displaystyle \int \frac{dx}{x^3 X^2} = -\frac{1}{2b^2 x^2} + \frac{2a}{b^3 x} - \frac{3a^2}{b^4} \ln \left| \frac{x}{X} \right| + \frac{a^2}{b^3 X}

  3. ∫dxx3X3=−12b3x2+3ab4x−6a2b5ln⁡∣xX∣+4a2b4X−a22b3X2\displaystyle \int \frac{dx}{x^3 X^3} = -\frac{1}{2b^3 x^2} + \frac{3a}{b^4 x} - \frac{6a^2}{b^5} \ln \left| \frac{x}{X} \right| + \frac{4a^2}{b^4 X} - \frac{a^2}{2b^3 X^2}

V. General Formula for ∫dxxmXn\displaystyle \int \frac{dx}{x^m X^n}

The general formula is complex. For particular cases, it is advisable to use the substitution u=xXu = \frac{x}{X} or partial fraction decomposition.

A closed-form expression is:

  1. ∫dxxmXn=−1(m−1)bxm−1−(m+n−2)a(m−1)b∫dxxm−1Xn(m>1)\displaystyle \int \frac{dx}{x^m X^n} = -\frac{1}{(m-1)b x^{m-1}} - \frac{(m+n-2)a}{(m-1)b} \int \frac{dx}{x^{m-1} X^n} \quad (m > 1)

For the integral with m=1m=1, we have:

  1. ∫dxxXn=1bn[an−1⋅1Xn−1+ln⁡∣xX∣](n≠1)\displaystyle \int \frac{dx}{x X^n} = \frac{1}{b^n} \left[ \frac{a}{n-1} \cdot \frac{1}{X^{n-1}} + \ln \left| \frac{x}{X} \right| \right] \quad (n \neq 1)

Integrals Involving Linear Expressions (ax+b)(ax+b) and (fx+g)(fx+g)

title: Notation
For formulas **31–34**, define:
$$\Delta = bf - ag$$

I. Integrals of Rational Functions with Linear Numerator and Denominator

  1. ∫ax+bfx+g dx=afx+Δf2ln⁡∣fx+g∣\displaystyle \int \frac{ax + b}{fx + g} \, dx = \frac{a}{f}x + \frac{\Delta}{f^2} \ln |fx + g|

II. Integrals with a Product of Two Linear Terms in the Denominator

  1. ∫dx(ax+b)(fx+g)=1Δln⁡∣fx+gax+b∣(Δ≠0)\displaystyle \int \frac{dx}{(ax + b)(fx + g)} = \frac{1}{\Delta} \ln \left| \frac{fx + g}{ax + b} \right| \qquad (\Delta \neq 0)

  2. ∫x dx(ax+b)(fx+g)=1Δ(baln⁡∣ax+b∣−gfln⁡∣fx+g∣)(Δ≠0)\displaystyle \int \frac{x \, dx}{(ax + b)(fx + g)} = \frac{1}{\Delta} \left( \frac{b}{a} \ln |ax + b| - \frac{g}{f} \ln |fx + g| \right) \qquad (\Delta \neq 0)

  3. ∫dx(ax+b)2(fx+g)=1Δ(1ax+b+fΔln⁡∣fx+gax+b∣)(Δ≠0)\displaystyle \int \frac{dx}{(ax + b)^2 (fx + g)} = \frac{1}{\Delta} \left( \frac{1}{ax + b} + \frac{f}{\Delta} \ln \left| \frac{fx + g}{ax + b} \right| \right) \qquad (\Delta \neq 0)

III. Special Cases with (a+x)(a+x) and (b+x)(b+x)

For formulas 35–39, denominators involve (a+x)(a+x) and (b+x)(b+x), with a≠ba \neq b.

  1. ∫x dx(a+x)(b+x)2=b(a−b)(b+x)+a(a−b)2ln⁡∣b+xa+x∣\displaystyle \int \frac{x \, dx}{(a + x)(b + x)^2} = \frac{b}{(a - b)(b + x)} + \frac{a}{(a - b)^2} \ln \left| \frac{b + x}{a + x} \right|

  2. ∫x2 dx(a+x)(b+x)2=b2(a−b)(b+x)+a2(a−b)2ln⁡∣a+x∣+b(b−2a)(a−b)2ln⁡∣b+x∣\displaystyle \int \frac{x^2 \, dx}{(a + x)(b + x)^2} = \frac{b^2}{(a - b)(b + x)} + \frac{a^2}{(a - b)^2} \ln |a + x| + \frac{b(b - 2a)}{(a - b)^2} \ln |b + x|

  3. ∫dx(a+x)2(b+x)2=−1(a−b)2(1a+x+1b+x)+2(a−b)3ln⁡∣a+xb+x∣\displaystyle \int \frac{dx}{(a + x)^2 (b + x)^2} = -\frac{1}{(a - b)^2} \left( \frac{1}{a + x} + \frac{1}{b + x} \right) + \frac{2}{(a - b)^3} \ln \left| \frac{a + x}{b + x} \right|

  4. ∫x dx(a+x)2(b+x)2=1(a−b)2(aa+x+bb+x)−a+b(a−b)3ln⁡∣a+xb+x∣\displaystyle \int \frac{x \, dx}{(a + x)^2 (b + x)^2} = \frac{1}{(a - b)^2} \left( \frac{a}{a + x} + \frac{b}{b + x} \right) - \frac{a + b}{(a - b)^3} \ln \left| \frac{a + x}{b + x} \right|

  5. ∫x2 dx(a+x)2(b+x)2=−1(a−b)2(a2a+x+b2b+x)+2ab(a−b)3ln⁡∣a+xb+x∣\displaystyle \int \frac{x^2 \, dx}{(a + x)^2 (b + x)^2} = -\frac{1}{(a - b)^2} \left( \frac{a^2}{a + x} + \frac{b^2}{b + x} \right) + \frac{2ab}{(a - b)^3} \ln \left| \frac{a + x}{b + x} \right|

Integrals Involving ax2+bx+cax^2 + bx + c

title: Notation
$$
X = ax^2 + bx + c, \qquad \Delta = 4ac - b^2
$$

I. Integrals of 1Xn\displaystyle \frac{1}{X^n}

∫dxX={2Δarctan⁡(2ax+bΔ),Δ>0,1−Δln⁡∣2ax+b−−Δ2ax+b+−Δ∣,Δ<0.\int \frac{dx}{X} = \begin{cases} \displaystyle \frac{2}{\sqrt{\Delta}} \arctan\left( \frac{2ax + b}{\sqrt{\Delta}} \right), & \Delta > 0, \\[2ex] \displaystyle \frac{1}{\sqrt{-\Delta}} \ln \left| \frac{2ax + b - \sqrt{-\Delta}}{2ax + b + \sqrt{-\Delta}} \right|, & \Delta < 0. \end{cases}

∫dxX2=2ax+bΔ X+4aΔ∫dxX.\int \frac{dx}{X^2} = \frac{2ax + b}{\Delta \, X} + \frac{4a}{\Delta} \int \frac{dx}{X}.

∫dxX3=2ax+bΔ(12X2+3aΔ X)+12a2Δ2∫dxX.\int \frac{dx}{X^3} = \frac{2ax + b}{\Delta} \left( \frac{1}{2X^2} + \frac{3a}{\Delta \, X} \right) + \frac{12a^2}{\Delta^2} \int \frac{dx}{X}.

  1. Reduction formula:

∫dxXn+1=2ax+bnΔ Xn+2(2n−1)anΔ∫dxXn,n≥1.\int \frac{dx}{X^{n+1}} = \frac{2ax + b}{n \Delta \, X^n} + \frac{2(2n-1)a}{n \Delta} \int \frac{dx}{X^n}, \qquad n \ge 1.

II. Integrals of xmXn\displaystyle \frac{x^m}{X^n} (m=1,2m = 1, 2)

∫x dxX=12aln⁡∣X∣−b2a∫dxX.\int \frac{x \, dx}{X} = \frac{1}{2a} \ln|X| - \frac{b}{2a} \int \frac{dx}{X}.

∫x dxX2=−bx+2cΔ X−2bΔ∫dxX.\int \frac{x \, dx}{X^2} = -\frac{bx + 2c}{\Delta \, X} - \frac{2b}{\Delta} \int \frac{dx}{X}.

∫x dxXn+1=−bx+2cnΔ Xn−(2n−1)bnΔ∫dxXn,n≥1.\int \frac{x \, dx}{X^{n+1}} = -\frac{bx + 2c}{n \Delta \, X^n} - \frac{(2n-1)b}{n \Delta} \int \frac{dx}{X^n}, \qquad n \ge 1.

∫x2 dxX=xa−b2a2ln⁡∣X∣+b2−2ac2a2∫dxX.\int \frac{x^2 \, dx}{X} = \frac{x}{a} - \frac{b}{2a^2} \ln|X| + \frac{b^2 - 2ac}{2a^2} \int \frac{dx}{X}.

∫x2 dxX2=(b2−2ac)x+bcaΔ X+4cΔ∫dxX.\int \frac{x^2 \, dx}{X^2} = \frac{(b^2 - 2ac)x + bc}{a \Delta \, X} + \frac{4c}{\Delta} \int \frac{dx}{X}.

III. Reduction Formulas for ∫xm dxXn\displaystyle \int \frac{x^m \, dx}{X^n}

  1. Case m=2m = 2:

∫x2 dxXn+1=−x(2n−1)a Xn+c(2n−1)a∫dxXn+1−(n−1)b(2n−1)a∫x dxXn+1.\int \frac{x^2 \, dx}{X^{n+1}} = -\frac{x}{(2n-1)a \, X^n} + \frac{c}{(2n-1)a} \int \frac{dx}{X^{n+1}} - \frac{(n-1)b}{(2n-1)a} \int \frac{x \, dx}{X^{n+1}}.

  1. General case m<2nm < 2n:

∫xm dxXn+1=−xm−1(2n−m+1)a Xn+(m−1)c(2n−m+1)a∫xm−2 dxXn+1−(n−m+1)b(2n−m+1)a∫xm−1 dxXn+1.\int \frac{x^m \, dx}{X^{n+1}} = -\frac{x^{m-1}}{(2n-m+1)a \, X^n} + \frac{(m-1)c}{(2n-m+1)a} \int \frac{x^{m-2} \, dx}{X^{n+1}} - \frac{(n-m+1)b}{(2n-m+1)a} \int \frac{x^{m-1} \, dx}{X^{n+1}}.

  1. Case m=2nm = 2n:

∫x2n−1 dxXn=1a∫x2n−3 dxXn−1−ca∫x2n−3 dxXn−ba∫x2n−2 dxXn.\int \frac{x^{2n-1} \, dx}{X^n} = \frac{1}{a} \int \frac{x^{2n-3} \, dx}{X^{n-1}} - \frac{c}{a} \int \frac{x^{2n-3} \, dx}{X^n} - \frac{b}{a} \int \frac{x^{2n-2} \, dx}{X^n}.

IV. Integrals with xx in the Denominator

∫dxx X=12cln⁡∣x2X∣−b2c∫dxX.\int \frac{dx}{x \, X} = \frac{1}{2c} \ln\left| \frac{x^2}{X} \right| - \frac{b}{2c} \int \frac{dx}{X}.

∫dxx Xn+1=12cn Xn−b2c∫dxXn+1+1c∫dxx Xn,n≥1.\int \frac{dx}{x \, X^{n+1}} = \frac{1}{2c n \, X^n} - \frac{b}{2c} \int \frac{dx}{X^{n+1}} + \frac{1}{c} \int \frac{dx}{x \, X^n}, \qquad n \ge 1.

∫dxx2 X=−1cx+b2c2ln⁡∣x2X∣+(b22c2−ac)∫dxX.\int \frac{dx}{x^2 \, X} = -\frac{1}{c x} + \frac{b}{2c^2} \ln\left| \frac{x^2}{X} \right| + \left( \frac{b^2}{2c^2} - \frac{a}{c} \right) \int \frac{dx}{X}.

  1. Reduction formula:

∫dxxm Xn+1=−1(m−1)c xm−1Xn−(2n+m−1)a(m−1)c∫dxxm−2 Xn+1−(n+m−1)b(m−1)c∫dxxm−1 Xn+1,m>1.\int \frac{dx}{x^{m} \, X^{n+1}} = -\frac{1}{(m-1)c \, x^{m-1} X^n} - \frac{(2n+m-1)a}{(m-1)c} \int \frac{dx}{x^{m-2} \, X^{n+1}} - \frac{(n+m-1)b}{(m-1)c} \int \frac{dx}{x^{m-1} \, X^{n+1}}, \quad m > 1.

V. Integral with a Linear Factor in the Denominator

∫dx(fx+g)X=12(cf2−bfg+ag2)[fln⁡∣(fx+g)2X∣+2ag−bfΔ F(X)],\int \frac{dx}{(fx + g) X} = \frac{1}{2(cf^2 - bfg + ag^2)} \left[ f \ln\left| \frac{(fx + g)^2}{X} \right| + \frac{2ag - bf}{\sqrt{\Delta}} \, F(X) \right],

where
F(X)=arctan⁡(2ax+bΔ)F(X) = \arctan\left( \frac{2ax+b}{\sqrt{\Delta}} \right) if Δ>0\Delta > 0, or
1−Δln⁡∣2ax+b−−Δ2ax+b+−Δ∣\displaystyle \frac{1}{\sqrt{-\Delta}} \ln \left| \frac{2ax+b-\sqrt{-\Delta}}{2ax+b+\sqrt{-\Delta}} \right| if Δ<0\Delta < 0.

Integrals Involving a2±x2a^2 \pm x^2

title: Notation

Let $X = a^2 \pm x^2$, with $a > 0$. Define $Y$ as:

$$
Y =
\begin{cases}
\displaystyle \arctan \left( \frac{x}{a} \right) & \text{for } X = a^2 + x^2, \\[2ex]
\displaystyle \operatorname{arctanh} \left( \frac{x}{a} \right) = \frac{1}{2} \ln \left( \frac{a + x}{a - x} \right) & \text{for } X = a^2 - x^2 \text{ if } |x| < a, \\[2ex]
\displaystyle \operatorname{arccoth} \left( \frac{x}{a} \right) = \frac{1}{2} \ln \left( \frac{x + a}{x - a} \right) & \text{for } X = a^2 - x^2 \text{ if } |x| > a.
\end{cases}
$$

~~~ad-caution
title: Sign Convention:
In all formulas that follow, the **upper sign** (± or ∓) corresponds to the case $X = a^2 + x^2$, and the **lower sign** to the case $X = a^2 - x^2$.
~~~

I. Basic Integrals of 1Xn\frac{1}{X^n}

  1. ∫dxX=1a Y\displaystyle \int \frac{dx}{X} = \frac{1}{a} \, Y

  2. ∫dxX2=x2a2X+12a3 Y\displaystyle \int \frac{dx}{X^2} = \frac{x}{2a^2 X} + \frac{1}{2a^3} \, Y

  3. ∫dxX3=x4a2X2+3x8a4X+38a5 Y\displaystyle \int \frac{dx}{X^3} = \frac{x}{4a^2 X^2} + \frac{3x}{8a^4 X} + \frac{3}{8a^5} \, Y

  4. ∫dxXn+1=x2na2Xn+2n−12na2∫dxXn(n≥1)\displaystyle \int \frac{dx}{X^{n+1}} = \frac{x}{2n a^2 X^n} + \frac{2n - 1}{2n a^2} \int \frac{dx}{X^n} \quad (n \ge 1)

II. Integrals with xx in the Numerator

  1. ∫x dxX=±12ln⁡∣X∣\displaystyle \int \frac{x \, dx}{X} = \pm \frac{1}{2} \ln |X|

  2. ∫x dxX2=∓12X\displaystyle \int \frac{x \, dx}{X^2} = \mp \frac{1}{2X}

  3. ∫x dxX3=∓14X2\displaystyle \int \frac{x \, dx}{X^3} = \mp \frac{1}{4X^2}

  4. ∫x dxXn+1=∓12nXn(n≥1)\displaystyle \int \frac{x \, dx}{X^{n+1}} = \mp \frac{1}{2n X^n} \quad (n \ge 1)

III. Integrals with x2x^2 in the Numerator

  1. ∫x2 dxX=±x∓a Y\displaystyle \int \frac{x^2 \, dx}{X} = \pm x \mp a \, Y

  2. ∫x2 dxX2=∓x2X±12a Y\displaystyle \int \frac{x^2 \, dx}{X^2} = \mp \frac{x}{2X} \pm \frac{1}{2a} \, Y

  3. ∫x2 dxX3=∓x4X2∓18a2⋅xX±18a3 Y\displaystyle \int \frac{x^2 \, dx}{X^3} = \mp \frac{x}{4X^2} \mp \frac{1}{8a^2} \cdot \frac{x}{X} \pm \frac{1}{8a^3} \, Y

  4. ∫x2 dxXn+1=∓x2nXn±12n∫dxXn(n≥1)\displaystyle \int \frac{x^2 \, dx}{X^{n+1}} = \mp \frac{x}{2n X^n} \pm \frac{1}{2n} \int \frac{dx}{X^n} \quad (n \ge 1)

IV. Integrals with x3x^3 in the Numerator

  1. ∫x3 dxX=±x22−a22ln⁡∣X∣\displaystyle \int \frac{x^3 \, dx}{X} = \pm \frac{x^2}{2} - \frac{a^2}{2} \ln |X|

  2. ∫x3 dxX2=a22X+12ln⁡∣X∣\displaystyle \int \frac{x^3 \, dx}{X^2} = \frac{a^2}{2X} + \frac{1}{2} \ln |X|

  3. ∫x3 dxX3=−12X+a24X2\displaystyle \int \frac{x^3 \, dx}{X^3} = -\frac{1}{2X} + \frac{a^2}{4X^2}

  4. ∫x3 dxXn+1=−12(n−1)Xn−1+a22nXn(n>1)\displaystyle \int \frac{x^3 \, dx}{X^{n+1}} = -\frac{1}{2(n-1) X^{n-1}} + \frac{a^2}{2n X^n} \quad (n > 1)

V. Integrals with xx in the Denominator

  1. ∫dxxX=12a2ln⁡∣x2X∣\displaystyle \int \frac{dx}{x X} = \frac{1}{2a^2} \ln \left| \frac{x^2}{X} \right|

  2. ∫dxxX2=12a2X+12a4ln⁡∣x2X∣\displaystyle \int \frac{dx}{x X^2} = \frac{1}{2a^2 X} + \frac{1}{2a^4} \ln \left| \frac{x^2}{X} \right|

  3. ∫dxxX3=14a2X2+12a4X+12a6ln⁡∣x2X∣\displaystyle \int \frac{dx}{x X^3} = \frac{1}{4a^2 X^2} + \frac{1}{2a^4 X} + \frac{1}{2a^6} \ln \left| \frac{x^2}{X} \right|

  4. ∫dxx2X=−1a4x∓xa4X\displaystyle \int \frac{dx}{x^2 X} = -\frac{1}{a^4 x} \mp \frac{x}{a^4 X} \quad

VI. Integrals with x3x^3 in the Denominator

  1. ∫dxx3X=−12a2x2∓12a4ln⁡∣x2X∣\displaystyle \int \frac{dx}{x^3 X} = -\frac{1}{2a^2 x^2} \mp \frac{1}{2a^4} \ln \left| \frac{x^2}{X} \right|

  2. ∫dxx3X2=−12a4x2∓12a4X∓1a6ln⁡∣x2X∣\displaystyle \int \frac{dx}{x^3 X^2} = -\frac{1}{2a^4 x^2} \mp \frac{1}{2a^4 X} \mp \frac{1}{a^6} \ln \left| \frac{x^2}{X} \right|

  3. ∫dxx3X3=−12a6x2∓1a6X∓14a6X2∓32a8ln⁡∣x2X∣\displaystyle \int \frac{dx}{x^3 X^3} = -\frac{1}{2a^6 x^2} \mp \frac{1}{a^6 X} \mp \frac{1}{4a^6 X^2} \mp \frac{3}{2a^8} \ln \left| \frac{x^2}{X} \right|

VII. Integrals with a Linear Factor (b+cx)(b + cx)

  1. ∫dx(b+cx)X=1a2c2∓b2[ cln⁡∣b+cx∣−c2ln⁡∣X∣∓ba Y ]\displaystyle \int \frac{dx}{(b + cx) X} = \frac{1}{a^2 c^2 \mp b^2} \left[ \, c \ln |b + cx| - \frac{c}{2} \ln |X| \mp \frac{b}{a} \, Y \, \right] 

title: Note:

The sign in the denominator $a^2 c^2 \mp b^2$ and in the term $\mp \frac{b}{a} Y$ is **consistent** with the definition of $X$ (upper for $+$, lower for $-$). It is assumed that $a^2 c^2 \neq b^2$.

Integrals Involving a3±x3a^3 \pm x^3

title: Notation:
Let $X = a^3 \pm x^3$.

~~~ad-caution
title: Sign Convention:
In formulas containing double signs ($\pm$ or $\mp$), the **upper sign** corresponds to $X = a^3 + x^3$, and the **lower sign** to $X = a^3 - x^3$.
~~~

I. Integrals of 1Xn\displaystyle \frac{1}{X^n}

∫dxX=16a2ln⁡ ⁣∣(a±x)2a2∓ax+x2∣±1a23arctan⁡ ⁣(2x∓aa3).\int \frac{dx}{X} = \frac{1}{6a^2} \ln\!\left| \frac{(a \pm x)^2}{a^2 \mp ax + x^2} \right| \pm \frac{1}{a^2\sqrt{3}} \arctan\!\left( \frac{2x \mp a}{a\sqrt{3}} \right).

∫dxX2=x3a3X+23a3∫dxX.\int \frac{dx}{X^2} = \frac{x}{3a^3 X} + \frac{2}{3a^3} \int \frac{dx}{X}.

II. Integrals with xx in the Numerator

∫x dxX=16aln⁡ ⁣∣a2∓ax+x2(a±x)2∣±1a3arctan⁡ ⁣(2x∓aa3).\int \frac{x \, dx}{X} = \frac{1}{6a} \ln\!\left| \frac{a^2 \mp ax + x^2}{(a \pm x)^2} \right| \pm \frac{1}{a\sqrt{3}} \arctan\!\left( \frac{2x \mp a}{a\sqrt{3}} \right).

∫x dxX2=x23a3X+13a3∫x dxX.\int \frac{x \, dx}{X^2} = \frac{x^2}{3a^3 X} + \frac{1}{3a^3} \int \frac{x \, dx}{X}.

III. Integrals with x2x^2 in the Numerator

∫x2 dxX=±13ln⁡∣X∣.\int \frac{x^2 \, dx}{X} = \pm \frac{1}{3} \ln |X|.

∫x2 dxX2=∓13X.\int \frac{x^2 \, dx}{X^2} = \mp \frac{1}{3X}.

IV. Integrals with x3x^3 in the Numerator

∫x3 dxX=±x∓a3∫dxX.\int \frac{x^3 \, dx}{X} = \pm x \mp a^3 \int \frac{dx}{X}.

∫x3 dxX2=±x3X±13∫dxX.\int \frac{x^3 \, dx}{X^2} = \pm \frac{x}{3X} \pm \frac{1}{3} \int \frac{dx}{X}.

V. Integrals with xx in the Denominator

∫dxxX=13a3ln⁡ ⁣∣x3X∣.\int \frac{dx}{x X} = \frac{1}{3a^3} \ln\!\left| \frac{x^3}{X} \right|.

∫dxxX2=13a3X+13a6ln⁡ ⁣∣x3X∣.\int \frac{dx}{x X^2} = \frac{1}{3a^3 X} + \frac{1}{3a^6} \ln\!\left| \frac{x^3}{X} \right|.

VI. Integrals with x2x^2 in the Denominator

∫dxx2X=−1a3x∓1a3∫x dxX.\int \frac{dx}{x^2 X} = -\frac{1}{a^3 x} \mp \frac{1}{a^3} \int \frac{x \, dx}{X}.

∫dxx2X2=−1a6x∓x23a6X∓43a6∫x dxX.\int \frac{dx}{x^2 X^2} = -\frac{1}{a^6 x} \mp \frac{x^2}{3a^6 X} \mp \frac{4}{3a^6} \int \frac{x \, dx}{X}.

VII. Integrals with x3x^3 in the Denominator

∫dxx3X=−12a3x2∓1a3∫dxX.\int \frac{dx}{x^3 X} = -\frac{1}{2a^3 x^2} \mp \frac{1}{a^3} \int \frac{dx}{X}.

∫dxx3X2=−12a6x2∓x3a6X∓53a6∫dxX.\int \frac{dx}{x^3 X^2} = -\frac{1}{2a^6 x^2} \mp \frac{x}{3a^6 X} \mp \frac{5}{3a^6} \int \frac{dx}{X}.

Integrals Involving a4±x4a^4 \pm x^4

I. Integrals with a4+x4a^4 + x^4

∫dxa4+x4=14a32ln⁡ ⁣(x2+a2 x+a2x2−a2 x+a2)+12a32arctan⁡ ⁣(a2 xa2−x2).\int \frac{dx}{a^4 + x^4} = \frac{1}{4a^3\sqrt{2}} \ln\!\left( \frac{x^2 + a\sqrt{2}\,x + a^2}{x^2 - a\sqrt{2}\,x + a^2} \right) + \frac{1}{2a^3\sqrt{2}} \arctan\!\left( \frac{a\sqrt{2}\,x}{a^2 - x^2} \right).

∫x dxa4+x4=12a2arctan⁡ ⁣(x2a2).\int \frac{x \, dx}{a^4 + x^4} = \frac{1}{2a^2} \arctan\!\left( \frac{x^2}{a^2} \right).

∫x2 dxa4+x4=−14a2ln⁡ ⁣(x2+a2 x+a2x2−a2 x+a2)+12a2arctan⁡ ⁣(a2 xa2−x2).\int \frac{x^2 \, dx}{a^4 + x^4} = -\frac{1}{4a\sqrt{2}} \ln\!\left( \frac{x^2 + a\sqrt{2}\,x + a^2}{x^2 - a\sqrt{2}\,x + a^2} \right) + \frac{1}{2a\sqrt{2}} \arctan\!\left( \frac{a\sqrt{2}\,x}{a^2 - x^2} \right).

∫x3 dxa4+x4=14ln⁡ ⁣(a4+x4).\int \frac{x^3 \, dx}{a^4 + x^4} = \frac{1}{4} \ln\!\left( a^4 + x^4 \right).

II. Integrals with a4−x4a^4 - x^4

∫dxa4−x4=14a3ln⁡ ⁣∣a+xa−x∣+12a3arctan⁡ ⁣(xa).\int \frac{dx}{a^4 - x^4} = \frac{1}{4a^3} \ln\!\left| \frac{a + x}{a - x} \right| + \frac{1}{2a^3} \arctan\!\left( \frac{x}{a} \right).

∫x dxa4−x4=14a2ln⁡ ⁣∣a2+x2a2−x2∣.\int \frac{x \, dx}{a^4 - x^4} = \frac{1}{4a^2} \ln\!\left| \frac{a^2 + x^2}{a^2 - x^2} \right|.

∫x2 dxa4−x4=14aln⁡ ⁣∣a+xa−x∣−12aarctan⁡ ⁣(xa).\int \frac{x^2 \, dx}{a^4 - x^4} = \frac{1}{4a} \ln\!\left| \frac{a + x}{a - x} \right| - \frac{1}{2a} \arctan\!\left( \frac{x}{a} \right).

∫x3 dxa4−x4=−14ln⁡ ⁣∣a4−x4∣.\int \frac{x^3 \, dx}{a^4 - x^4} = -\frac{1}{4} \ln\!\left| a^4 - x^4 \right|.

III. Partial Fraction Decomposition (Special Cases)

  1. Distinct linear factors:

1(a+bx)(f+gx)=1bf−ag(ba+bx−gf+gx),\frac{1}{(a + bx)(f + gx)} = \frac{1}{bf - ag} \left( \frac{b}{a + bx} - \frac{g}{f + gx} \right),

assuming bf−ag≠0bf - ag \neq 0.
106) Three distinct linear factors:

1(x+a)(x+b)(x+c)=Ax+a+Bx+b+Cx+c,\frac{1}{(x + a)(x + b)(x + c)} = \frac{A}{x + a} + \frac{B}{x + b} + \frac{C}{x + c},

with

A=1(b−a)(c−a),B=1(a−b)(c−b),C=1(a−c)(b−c).A = \frac{1}{(b - a)(c - a)}, \quad B = \frac{1}{(a - b)(c - b)}, \quad C = \frac{1}{(a - c)(b - c)}.

  1. Four distinct linear factors:

1(x+a)(x+b)(x+c)(x+d)=Ax+a+Bx+b+Cx+c+Dx+d,\frac{1}{(x + a)(x + b)(x + c)(x + d)} = \frac{A}{x + a} + \frac{B}{x + b} + \frac{C}{x + c} + \frac{D}{x + d},

with

A=1(b−a)(c−a)(d−a),B=1(a−b)(c−b)(d−b),A = \frac{1}{(b - a)(c - a)(d - a)}, \quad B = \frac{1}{(a - b)(c - b)(d - b)},

C=1(a−c)(b−c)(d−c),D=1(a−d)(b−d)(c−d).C = \frac{1}{(a - c)(b - c)(d - c)}, \quad D = \frac{1}{(a - d)(b - d)(c - d)}.

  1. Irreducible quadratic factors:

1(a+bx2)(f+gx2)=1bf−ag(ba+bx2−gf+gx2),\frac{1}{(a + bx^2)(f + gx^2)} = \frac{1}{bf - ag} \left( \frac{b}{a + bx^2} - \frac{g}{f + gx^2} \right),

assuming bf−ag≠0bf - ag \neq 0.

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