I'm having problems with this primitive in the real number domain:
Where $x \ne 0$ and $a x^2 + b x + c > 0$:
$$\int \frac {\mathrm d x} {x \sqrt {a x^2 + b x + c} }$$ for $b^2 - 4 a c < 0$
According to item $14.283$ in Spiegel's "Mathematical Handbook of Formulas and Tables" (Schaum, 1968) it should work out as:
$$\dfrac {-1} {\sqrt c} {\sinh^{-1} } {\dfrac {b x + 2 c} {|x| \sqrt {4 a c - b^2} } }$$
or in an alternative form:
$$\dfrac {-1} {\sqrt c} {\arcsin^{-1} } {\dfrac {b x + 2 c} {|x| \sqrt {b^2 - 4 a c} } }$$
(ignoring the arbitrary constant of integration, to minimise clutter).
Now I can't work out where the absolute value operator comes in on the $x$.
I have got this far:
$x = \dfrac 1 u$
$\leadsto \dfrac {\mathrm d x} {\mathrm d u} = \dfrac {-1} {u^2}$
$\leadsto \displaystyle \int \frac {\mathrm d x} {x \sqrt {a x^2 + b x + c} } = \int {\dfrac 1 {\dfrac 1 u \sqrt {a \left({\frac 1 u}\right)^2 + b \left({\frac 1 u}\right) + c} } } \dfrac {-\, \mathrm d u} {u^2}$
(by integration by substitution)
$= -\displaystyle \int \dfrac {\mathrm d u} {\frac {u^2} u \sqrt {\frac 1 {u^2} (a + b u + c u^2) } }$
$= -\displaystyle \int \dfrac {\mathrm d u} {\sqrt {a + b u + c u^2} }$
as those $u$ factors all cancel.
Then there is a standard result:
$\displaystyle \int \dfrac {\mathrm d u} {\sqrt {a + b u + c u^2} } = \begin {cases} \dfrac {-1} {\sqrt c} \ln \left\vert {2 \sqrt c \sqrt {a + b u + c u^2} + 2 c u + b}\right\vert + C & : b^2 - 4 a c > 0 \\ \dfrac {-1} {\sqrt c} \sinh^{-1} \left({\dfrac {2 c u + b} {\sqrt{4 a c - b^2} } }\right) + C & : b^2 - 4 a c < 0 \\ \dfrac {-1} {\sqrt c} \ln |2 c u + b| + C & : b^2 - 4 a c = 0 \end {cases}$
I'm only interested in the case where $b^2 - 4 a c < 0$.
Anyway, I substitute back the $u = \dfrac 1 x$ and after algebra I work it out to be:
$= \sinh^{-1} \left({\dfrac {b x + 2 c} {x \sqrt{4 a c - b^2} } }\right)$
Taking a step back I can't see where the $|x|$ would come from in Spiegel's result. Because $\sinh^{-1}$ is a bijection over the reals, an odd function continuous at $0$, there should be no discontinuity in the derivative, so the $|x|$ puzzles me.
The problem is in this place: You have $$\frac {u^2} u \sqrt {\frac 1 {u^2} (a + b u + c u^2) } = \sqrt {a + b u + c u^2} $$ It is not true. It should be $$\frac {u^2} u \sqrt {\frac 1 {u^2} (a + b u + c u^2) } = \pm\sqrt {a + b u + c u^2}. $$