Primitives involving x squared plus a squared squared

Theorem

This page gathers together the primitives of some expressions involving $\left({x^2 + a^2}\right)^2$.

Primitive of Reciprocal of $\left({x^2 + a^2}\right)^2$

$\ds \int \frac {\d x} {\paren {x^2 + a^2}^2} = \frac x {2 a^2 \paren {x^2 + a^2} } + \frac 1 {2 a^3} \arctan \frac x a + C$

Primitive $x$ over $\left({x^2 + a^2}\right)^2$

$\ds \int \frac {x \rd x} {\paren {x^2 + a^2}^2} = -\frac 1 {2 \paren {x^2 + a^2} } + C$

Primitive $x^2$ over $\left({x^2 + a^2}\right)^2$

$\ds \int \frac {x^2 \rd x} {\paren {x^2 + a^2}^2} = \frac {-x} {2 \paren {x^2 + a^2} } + \frac 1 {2 a} \arctan \frac x a + C$

Primitive $x^3$ over $\left({x^2 + a^2}\right)^2$

$\ds \int \frac {x^3 \rd x} {\paren {x^2 + a^2}^2} = \frac {a^2} {2 \paren {x^2 + a^2} } + \frac 1 2 \map \ln {x^2 + a^2} + C$

Primitive of Reciprocal of $x \left({x^2 + a^2}\right)^2$

$\ds \int \frac {\d x} {x \paren {x^2 + a^2}^2} = \frac 1 {2 a^2 \paren {x^2 + a^2} } + \frac 1 {2 a^4} \map \ln {\frac {x^2} {x^2 + a^2} } + C$

Primitive of Reciprocal of $x^2 \left({x^2 + a^2}\right)^2$

$\ds \int \frac {\d x} {x^2 \paren {x^2 + a^2}^2} = -\frac 1 {a^4 x} - \frac x {2 a^4 \paren {x^2 + a^2} } - \frac 3 {2 a^5} \arctan \frac x a + C$

Primitive of Reciprocal of $x^3 \left({x^2 + a^2}\right)^2$

$\ds \int \frac {\d x} {x^3 \paren {x^2 + a^2}^2} = -\frac 1 {2 a^4 x^2} - \frac 1 {2 a^4 \paren {x^2 + a^2} } - \frac 1 {a^6} \map \ln {\frac {x^2} {x^2 + a^2} } + C$

Also see

• Primitives involving $x^2 + a^2$
• Primitives involving $\left({x^2 + a^2}\right)^n$