Primitive of Reciprocal of x squared by x fourth plus a fourth

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Theorem

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


Proof

\(\ds \) \(\) \(\ds \int \frac {\d x} {x^2 \paren {x^4 + a^4} }\)
\(\ds \) \(=\) \(\ds \int \frac {a^4 \rd x} {a^4 x^2 \paren {x^4 + a^4} }\) multiplying top and bottom by $a^4$
\(\ds \) \(=\) \(\ds \int \frac {\paren {x^4 + a^4 - x^4} \rd x} {a^4 x^2 \paren {x^4 + a^4} }\)
\(\ds \) \(=\) \(\ds \frac 1 {a^4} \int \frac {\paren {x^4 + a^4} \rd x} {x^2 \paren {x^4 + a^4} } - \frac 1 {a^4} \int \frac {x^4 \rd x} {x^2 \paren {x^4 + a^4} }\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac 1 {a^4} \int \frac {\d x} {x^2} - \frac 1 {a^4} \int \frac {x^2 \rd x} {x^4 + a^4}\) simplifying
\(\ds \) \(=\) \(\ds \frac 1 {a^4} \paren {\frac {-1} x} - \frac 1 {a^4} \int \frac {x^2 \rd x} {x^4 + a^4}\) Primitive of Power
\(\ds \) \(=\) \(\ds \frac {-1} {a^4 x} - \frac 1 {a^4} \paren {\frac 1 {4 a \sqrt 2} \map \ln {\frac {x^2 - a x \sqrt 2 + a^2} {x^2 + a x \sqrt 2 + a^2} } - \frac 1 {2 a \sqrt 2} \paren {\map \arctan {1 - \frac {x \sqrt 2} a} - \map \arctan {1 + \frac {x \sqrt 2} a} } }\) Primitive of $\dfrac {x^2} {x^4 + a^4}$
\(\ds \) \(=\) \(\ds \frac {-1} {a^4 x} - \frac {-1} {4 a^5 \sqrt 2} \map \ln {\frac {x^2 - a x \sqrt 2 + a^2} {x^2 + a x \sqrt 2 + a^2} } + \frac 1 {2 a^5 \sqrt 2} \paren {\map \arctan {1 - \frac {x \sqrt 2} a} - \map \arctan {1 + \frac {x \sqrt 2} a} }\) simplifying

$\blacksquare$


Sources

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