Remove parentheses.
\[\int 1 \, \frac{d}{(1+{x}^{2})\sqrt{{p}^{2}+{q}^{2}{({tan}^{-1}x)}^{2}}}dx\]
Use Negative Power Rule: \({x}^{-a}=\frac{1}{{x}^{a}}\).
\[\int 1 \, \frac{d}{(1+{x}^{2})\sqrt{{p}^{2}+{q}^{2}{(\frac{1}{tan}x)}^{2}}}dx\]
Simplify \(\frac{1}{tan}x\) to \(\frac{x}{tan}\).
\[\int 1 \, \frac{d}{(1+{x}^{2})\sqrt{{p}^{2}+{q}^{2}{(\frac{x}{tan})}^{2}}}dx\]
Use Division Distributive Property: \({(\frac{x}{y})}^{a}=\frac{{x}^{a}}{{y}^{a}}\).
\[\int 1 \, \frac{d}{(1+{x}^{2})\sqrt{{p}^{2}+{q}^{2}\times \frac{{x}^{2}}{{tan}^{2}}}}dx\]
Use this rule: \(a \times \frac{b}{c}=\frac{ab}{c}\).
\[\int 1 \, \frac{d}{(1+{x}^{2})\sqrt{{p}^{2}+\frac{{q}^{2}{x}^{2}}{{tan}^{2}}}}dx\]
Use this rule: \(\int a \, dx=ax+C\).
\[\frac{dx}{(1+{x}^{2})\sqrt{{p}^{2}+\frac{{q}^{2}{x}^{2}}{{tan}^{2}}}}\]
(d*x)/((1+x^2)*sqrt(p^2+(q^2*x^2)/tan^2))
