Primitive of Reciprocal of x fourth plus a fourth

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Theorem

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


Proof

\(\ds \int \frac {\d x} {x^4 + a^4}\) \(=\) \(\ds \int \paren {\frac {x + a \sqrt 2} {2 a^3 \sqrt 2 \paren {x^2 + a x \sqrt 2 + a^2} } - \frac {x - a \sqrt 2} {2 a^3 \sqrt 2 \paren {x^2 - a x \sqrt 2 + a^2} } } \rd x\) Partial Fraction Expansion
\(\text {(1)}: \quad\) \(\ds \) \(=\) \(\ds \frac 1 {4 a^3 \sqrt 2} \int \frac {\paren {2 x + 2 a \sqrt 2} \rd x} {x^2 + a x \sqrt 2 + a^2} - \frac 1 {4 a^3 \sqrt 2} \int \frac {\paren {2 x - 2 a \sqrt 2} \rd x} {x^2 - a x \sqrt 2 + a^2}\) Linear Combination of Primitives


Then:

\(\ds \int \frac {\paren {2 x + 2 a \sqrt 2} \rd x} {x^2 + a x \sqrt 2 + a^2}\) \(=\) \(\ds \int \frac {\paren {2 x + a \sqrt 2} \rd x} {x^2 + a x \sqrt 2 + a^2} + a \sqrt 2 \int \frac {\d x} {x^2 + a x \sqrt 2 + a^2}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \ln \size {x^2 + a x \sqrt 2 + a^2} + a \sqrt 2 \int \frac {\d x} {x^2 + a x \sqrt 2 + a^2}\) Primitive of Function under its Derivative
\(\ds \) \(=\) \(\ds \ln \size {x^2 + a x \sqrt 2 + a^2} + a \sqrt 2 \paren {\frac {\sqrt 2} a \, \map \arctan {1 + \frac {x \sqrt 2} a} }\) Primitive of $\dfrac 1 {x^2 + a x \sqrt 2 + a^2}$
\(\text {(2)}: \quad\) \(\ds \) \(=\) \(\ds \ln \size {x^2 + a x \sqrt 2 + a^2} + 2 \map \arctan {1 + \frac {x \sqrt 2} a}\) simplifying


Similarly:

\(\ds \int \frac {\paren {2 x - 2 a \sqrt 2} \rd x} {x^2 - a x \sqrt 2 + a^2}\) \(=\) \(\ds \int \frac {\paren {2 x - a \sqrt 2} \rd x} {x^2 - a x \sqrt 2 + a^2} - a \sqrt 2 \int \frac {\d x} {x^2 - a x \sqrt 2 + a^2}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \ln \size {x^2 - a x \sqrt 2 + a^2} - a \sqrt 2 \int \frac {\d x} {x^2 - a x \sqrt 2 + a^2}\) Primitive of Function under its Derivative
\(\ds \) \(=\) \(\ds \ln \size {x^2 - a x \sqrt 2 + a^2} - a \sqrt 2 \paren {\frac {-\sqrt 2} a \, \map \arctan {1 - \frac {x \sqrt 2} a} }\) Primitive of $\dfrac 1 {x^2 - a x \sqrt 2 + a^2}$
\(\text {(3)}: \quad\) \(\ds \) \(=\) \(\ds \ln \size {x^2 - a x \sqrt 2 + a^2} + 2 \, \map \arctan {1 - \frac {x \sqrt 2} a}\) simplifying


Thus:

\(\ds \int \frac {\d x} {x^4 + a^4}\) \(=\) \(\ds \frac 1 {4 a^3 \sqrt 2} \int \frac {\paren {2 x + 2 a \sqrt 2} \rd x} {x^2 + a x \sqrt 2 + a^2} - \frac 1 {4 a^3 \sqrt 2} \int \frac {\paren {2 x - 2 a \sqrt 2} \rd x} {x^2 - a x \sqrt 2 + a^2}\) from $(1)$
\(\ds \) \(=\) \(\ds \frac 1 {4 a^3 \sqrt 2} \paren {\ln \size {x^2 + a x \sqrt 2 + a^2} + 2 \, \map \arctan {1 + \frac {x \sqrt 2} a} }\) from $(2)$
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \frac 1 {4 a^3 \sqrt 2} \paren {\ln \size {x^2 - a x \sqrt 2 + a^2} + 2 \, \map \arctan {1 - \frac {x \sqrt 2} a} }\) from $(3)$
\(\ds \) \(=\) \(\ds \frac 1 {4 a^3 \sqrt 2} \map \ln {\frac {x^2 + a x \sqrt 2 + a^2} {x^2 - a x \sqrt 2 + a^2} } - \frac 1 {2 a^3 \sqrt 2} \paren {\map \arctan {1 - \frac {x \sqrt 2} a} - \map \arctan {1 + \frac {x \sqrt 2} a} }\) simplifying

$\blacksquare$


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