Primitive of Root of x squared plus a squared over x squared

Theorem

Logarithm Form

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

Inverse Hyperbolic Sine Form

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

Also see

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