Primitive of Reciprocal of x fourth minus a fourth

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Theorem

$\ds \int \frac {\d x} {x^4 - a^4} = \frac 1 {4 a^3} \ln \size {\frac {x - a} {x + a} } - \frac 1 {2 a^3} \arctan \frac x a$


Proof

\(\ds \int \frac {\d x} {x^4 - a^4}\) \(=\) \(\ds \int \frac {\d x} {\paren {x^2 + a^2} \paren {x^2 - a^2} }\) Difference of Two Squares
\(\ds \) \(=\) \(\ds \int \frac {a^2 \rd x} {a^2 \paren {x^2 + a^2} \paren {x^2 - a^2} }\) multiplying top and bottom by $a^2$
\(\ds \) \(=\) \(\ds \int \frac {\paren {x^2 + a^2 - x^2} \rd x} {a^2 \paren {x^2 + a^2} \paren {x^2 - a^2} }\)
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \int \frac {\paren {x^2 + a^2} \d x} {\paren {x^2 + a^2} \paren {x^2 - a^2} } - \frac 1 {a^2} \int \frac {x^2 \rd x} {\paren {x^2 + a^2} \paren {x^2 - a^2} }\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \int \frac {\d x} {x^2 - a^2} - \frac 1 {a^2} \int \frac {x^2 \rd x} {\paren {x^2 + a^2} \paren {x^2 - a^2} }\) simplifying
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \int \frac {\d x} {x^2 - a^2} - \frac 1 {a^2} \int \frac {\paren {x^2 - a^2 + a^2} \rd x} {\paren {x^2 + a^2} \paren {x^2 - a^2} }\) Difference of Two Squares
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \int \frac {\d x} {x^2 - a^2} - \frac 1 {a^2} \int \frac {\paren {x^2 - a^2} \rd x} {\paren {x^2 + a^2} \paren {x^2 - a^2} }\) Linear Combination of Primitives
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \frac 1 {a^2} \int \frac {a^2 \rd x} {\paren {x^2 + a^2} \paren {x^2 - a^2} }\)
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \int \frac {\d x} {x^2 - a^2} - \frac 1 {a^2} \int \frac {\d x} {x^2 + a^2} - \int \frac {\d x} {\paren {x^2 + a^2} \paren {x^2 - a^2} }\) simplification
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \int \frac {\d x} {x^2 - a^2} - \frac 1 {a^2} \int \frac {\d x} {x^2 + a^2} - \int \frac {\d x} {x^4 - a^4}\) Difference of Two Squares
\(\ds \leadsto \ \ \) \(\ds 2 \int \frac {\d x} {x^4 - a^4}\) \(=\) \(\ds \frac 1 {a^2} \int \frac {\d x} {x^2 - a^2} - \frac 1 {a^2} \int \frac {\d x} {x^2 + a^2}\) gathering terms
\(\ds \leadsto \ \ \) \(\ds \int \frac {\d x} {x^4 - a^4}\) \(=\) \(\ds \frac 1 {2 a^2} \int \frac {\d x} {x^2 - a^2} - \frac 1 {2 a^2} \int \frac {\d x} {x^2 + a^2}\) gathering terms
\(\ds \) \(=\) \(\ds \frac 1 {4 a^3} \ln \size {\frac {x - a} {x + a} } - \frac 1 {2 a^2} \int \frac {\d x} {x^2 + a^2}\) Primitive of $\dfrac 1 {x^2 - a^2}$
\(\ds \) \(=\) \(\ds \frac 1 {4 a^3} \ln \size {\frac {x - a} {x + a} } - \frac 1 {2 a^3} \arctan \frac x a\) Primitive of $\dfrac 1 {x^2 + a^2}$

$\blacksquare$


Sources