Primitive of Root of x squared minus a squared over x squared

Theorem

Inverse Hyperbolic Cosine Form

$\ds \int \frac {\sqrt {x^2 - a^2} } {x^2} \rd x = \arcosh \dfrac x a - \frac {\sqrt {x^2 - a^2} } x + C$

for $x^2 \ge a^2$.

Logarithm Form

$\ds \int \frac {\sqrt {x^2 - a^2} } {x^2} \rd x = \frac {-\sqrt {x^2 - a^2} } x + \ln \size {x + \sqrt {x^2 - a^2} } + C$

for $x^2 \ge a^2$.

Also see

• Primitive of $\dfrac {\sqrt {x^2 + a^2} } {x^2}$
• Primitive of $\dfrac {\sqrt {a^2 - x^2} } {x^2}$