\documentclass[a4paper]{article}
\usepackage{amsmath}
\DeclareMathOperator{\arcsec}{arcsec}
\begin{document}
\begin{eqnarray*}
f (x)
 & = & \frac{1}{(x + a)^{2} \sqrt{x^{2} - a^{2}}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
x
 & = & a \sec t \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\sec t
 & = & \frac{x}{a} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
t
 & = & \arcsec \frac{x}{a} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \int \frac{1}{(x + a)^{2} \sqrt{x^{2} - a^{2}}} \, \mbox{d}x \\
& = & \int \frac{1}{(a \sec t + a)^{2} \sqrt{(a \sec t)^{2} - a^{2}}} \, \mbox{d}(a \sec t) \\
& = & \int \frac{1}{(a \sec t + a)^{2} \sqrt{a^{2} \sec^{2} t - a^{2}}} a \sec^{2} t \sin t \, \mbox{d}t \\
& = & \frac{1}{a^{2}} \int \frac{\sec^{2} t \sin t}{(\sec t + 1)^{2} \sqrt{\sec^{2} t - 1}} \, \mbox{d}t \\
& = & \frac{1}{a^{2}} \int \frac{\sec^{2} t \sin t}{(\sec t + 1)^{2} \sqrt{\frac{1}{\cos^{2} t} - \frac{\cos^{2} t}{\cos^{2} t}}} \, \mbox{d}t \\
& = & \frac{1}{a^{2}} \int \frac{\sec^{2} t \sin t}{(\sec t + 1)^{2} \sqrt{\frac{1 - \cos^{2} t}{\cos^{2} t}}} \, \mbox{d}t \\
& = & \frac{1}{a^{2}} \int \frac{\sec^{2} t \sin t}{(\sec t + 1)^{2} \sqrt{\frac{\sin^{2} t}{\cos^{2} t}}} \, \mbox{d}t \\
& = & \frac{1}{a^{2}} \int \frac{\sec^{2} t \sin t}{(\sec t + 1)^{2} \frac{\sin t}{\cos t}} \, \mbox{d}t \\
& = & \frac{1}{a^{2}} \int \frac{\sec t}{(\sec t + 1)^{2}} \, \mbox{d}t \\
& = & \frac{1}{a^{2}} \int \frac{\cos t}{(1 + \cos t)^{2}} \, \mbox{d}t \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \frac{1}{a^{2}} \int \frac{\cos t}{(1 + \cos t)^{2}} \, \mbox{d}t \\
& = & \frac{1}{a^{2}} \left ( \frac{2}{3} \cot^{3} t - \frac{2}{3} \csc^{3} t + \csc t \right ) + c \\
& = & \frac{2}{3 a^{2}} \cot^{3} t - \frac{2}{3 a^{2}} \csc^{3} t + \frac{1}{a^{2}} \csc t + c \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\sin \alpha
 & = & \sqrt{1 - \frac{1}{\sec^{2} \alpha}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\cos \alpha
 & = & \frac{1}{\sec \alpha} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \frac{2}{3 a^{2}} \cot^{3} t - \frac{2}{3 a^{2}} \csc^{3} t + \frac{1}{a^{2}} \csc t + c \\
& = & \frac{2}{3 a^{2}} \frac{\cos^{3} t}{\sin^{3} t} - \frac{2}{3 a^{2}} \frac{1}{\sin^{3} t} + \frac{1}{a^{2}} \frac{1}{\sin t} + c \\
& = & \frac{2}{3 a^{2}} \frac{\frac{1}{\sec^{3} t}}{\left ( \sqrt{1 - \frac{1}{\sec^{2} t}} \, \right )^{3}}
 - \frac{2}{3 a^{2}} \frac{1}{\left ( \sqrt{1 - \frac{1}{\sec^{2} t}} \, \right )^{3}}
 + \frac{1}{a^{2}} \frac{1}{\sqrt{1 - \frac{1}{\sec^{2} t}}} + c \\
& = & \frac{2}{3 a^{2}} \frac{1}{\left ( \sqrt{\sec^{2} t - 1} \, \right )^{3}}
 - \frac{2}{3 a^{2}} \frac{\sec^{3} t}{\left ( \sqrt{\sec^{2} t - 1} \, \right )^{3}}
 + \frac{1}{a^{2}} \frac{\sec t}{\sqrt{\sec^{2} t - 1}} + c \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \frac{2}{3 a^{2}} \frac{1}{\left ( \sqrt{\sec^{2} t - 1} \, \right )^{3}}
 - \frac{2}{3 a^{2}} \frac{\sec^{3} t}{\left ( \sqrt{\sec^{2} t - 1} \, \right )^{3}}
 + \frac{1}{a^{2}} \frac{\sec t}{\sqrt{\sec^{2} t - 1}} + c \\
& = & \frac{2}{3 a^{2}} \frac{1}{\left ( \sqrt{\sec^{2} \arcsec \frac{x}{a} - 1} \, \right )^{3}}
 - \frac{2}{3 a^{2}} \frac{\sec^{3} \arcsec \frac{x}{a}}{\left ( \sqrt{\sec^{2} \arcsec \frac{x}{a} - 1} \, \right )^{3}}
 + \frac{1}{a^{2}} \frac{\sec \arcsec \frac{x}{a}}{\sqrt{\sec^{2} \arcsec \frac{x}{a} - 1}} + c \\
& = & \frac{2}{3 a^{2}} \frac{1}{\left ( \sqrt{\left ( \frac{x}{a} \right )^{2} - 1} \, \right )^{3}}
 - \frac{2}{3 a^{2}} \frac{\left ( \frac{x}{a} \right )^{3}}{\left ( \sqrt{\left ( \frac{x}{a} \right )^{2} - 1} \, \right )^{3}}
 + \frac{1}{a^{2}} \frac{\frac{x}{a}}{\sqrt{\left ( \frac{x}{a} \right )^{2} - 1}} + c \\
& = & \frac{2}{3 a^{2}} \frac{1}{\left ( \sqrt{\frac{x^{2}}{a^{2}} - 1} \, \right )^{3}}
 - \frac{2}{3 a^{2}} \frac{\frac{x^{3}}{a^{3}}}{\left ( \sqrt{\frac{x^{2}}{a^{2}} - 1} \, \right )^{3}}
 + \frac{1}{a^{2}} \frac{\frac{x}{a}}{\sqrt{\frac{x^{2}}{a^{2}} - 1}} + c \\
& = & \frac{2}{3 a^{2}} \frac{a^{3}}{\left ( \sqrt{x^{2} - a^{2}} \, \right )^{3}}
 - \frac{2}{3 a^{2}} \frac{x^{3}}{\left ( \sqrt{x^{2} - a^{2}} \, \right )^{3}}
 + \frac{1}{a^{2}} \frac{x}{\sqrt{x^{2} - a^{2}}} + c \\
& = & \frac{2}{3 a^{2}} \frac{a^{3}}{(x^{2} - a^{2}) \sqrt{x^{2} - a^{2}}}
 - \frac{2}{3 a^{2}} \frac{x^{3}}{(x^{2} - a^{2}) \sqrt{x^{2} - a^{2}}}
 + \frac{1}{a^{2}} \frac{x (x^{2} - a^{2})}{(x^{2} - a^{2}) \sqrt{x^{2} - a^{2}}} + c \\
& = & \frac{\frac{2}{3 a^{2}} a^{3} - \frac{2}{3 a^{2}} x^{3} + \frac{1}{a^{2}} x (x^{2} - a^{2})}{(x^{2} - a^{2}) \sqrt{x^{2} - a^{2}}} + c \\
& = & \frac{\frac{2}{3} a - \frac{2}{3 a^{2}} x^{3} + \frac{1}{a^{2}} x^{3} - x}{(x^{2} - a^{2}) \sqrt{x^{2} - a^{2}}} + c \\
& = & \frac{\frac{2}{3} a + \frac{1}{3 a^{2}} x^{3} - x}{(x^{2} - a^{2}) \sqrt{x^{2} - a^{2}}} + c \\
& = & \frac{1}{3 a^{2}} \frac{x^{3} - 3 a^{2} x + 2 a^{3}}{(x^{2} - a^{2}) \sqrt{x^{2} - a^{2}}} + c \\
& = & \frac{1}{3 a^{2}} \frac{(x^{2} + a x - 2 a^{2}) (x - a)}{(x + a) (x - a) \sqrt{x^{2} - a^{2}}} + c \\
& = & \frac{1}{3 a^{2}} \frac{x^{2} + a x - 2 a^{2}}{(x + a) \sqrt{x^{2} - a^{2}}} + c \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
f (x)
 & = & \frac{\mbox{d}(F (x))}{\mbox{d}x} \\
& = & \frac{\mbox{d}}{\mbox{d}x} \left ( \frac{1}{3 a^{2}} \frac{x^{2} + a x - 2 a^{2}}{(x + a) \sqrt{x^{2} - a^{2}}} + c \right ) \\
& = & \frac{1}{3 a^{2}}
 \frac{(2x + a) (x + a) \sqrt{x^{2} - a^{2}} - (x^{2} + a x - 2 a^{2}) (1 \cdot \sqrt{x^{2} - a^{2}} + (x + a) \frac{1}{2} \frac{1}{\sqrt{x^{2} - a^{2}}} (2x))}
{(x + a)^{2} (x^{2} - a^{2})} + 0 \\
& = & \frac{1}{3 a^{2}}
 \frac{(2x^{2} + 2ax + ax + a^{2}) (x^{2} - a^{2}) - (x^{2} + a x - 2 a^{2}) (x^{2} - a^{2} + (x + a) x)}{(x + a)^{2} (x^{2} - a^{2}) \sqrt{x^{2} - a^{2}}} \\
& = & \frac{1}{3 a^{2}}
 \frac{(2x^{2} + 3ax + a^{2}) (x^{2} - a^{2}) - (x^{2} + a x - 2 a^{2}) (x^{2} - a^{2} + x^{2} + ax)}{(x + a)^{2} (x^{2} - a^{2}) \sqrt{x^{2} - a^{2}}} \\
& = & \frac{1}{3 a^{2}}
 \frac{(2x^{2} + 3ax + a^{2}) (x^{2} - a^{2}) - (x^{2} + a x - 2 a^{2}) (2x^{2} + ax - a^{2})}{(x + a)^{2} (x^{2} - a^{2}) \sqrt{x^{2} - a^{2}}} \\
& = & \frac{1}{3 a^{2}}
 \frac{(2x^{2} + 3ax + a^{2}) (x^{2} - a^{2}) - (2 x^{4} + a x^{3} - a^{2} x^{2} + 2 a x^{3} + a^{2} x^{2} - a^{3} x - 4 a^{2} x^{2} - 2 a^{3} x + 2 a^{4})}
{(x + a)^{2} (x^{2} - a^{2}) \sqrt{x^{2} - a^{2}}} \\
& = & \frac{1}{3 a^{2}} \frac{(2x^{2} + 3ax + a^{2}) (x^{2} - a^{2}) - (2 x^{4} + 3 a x^{3} - 4 a^{2} x^{2} - 3 a^{3} x + 2 a^{4})}
{(x + a)^{2} (x^{2} - a^{2}) \sqrt{x^{2} - a^{2}}} \\
& = & \frac{1}{3 a^{2}} \frac{(2x^{2} + 3ax + a^{2}) (x^{2} - a^{2}) - (2 x^{2} + 3 a x - 2 a^{2}) (x^{2} - a^{2})}
{(x + a)^{2} (x^{2} - a^{2}) \sqrt{x^{2} - a^{2}}} \\
& = & \frac{1}{3 a^{2}} \frac{(2x^{2} + 3ax + a^{2}) - (2 x^{2} + 3 a x - 2 a^{2})}{(x + a)^{2} \sqrt{x^{2} - a^{2}}} \\
& = & \frac{1}{3 a^{2}} \frac{2x^{2} + 3ax + a^{2} - 2 x^{2} - 3 a x + 2 a^{2}}{(x + a)^{2} \sqrt{x^{2} - a^{2}}} \\
& = & \frac{1}{3 a^{2}} \frac{3 a^{2}}{(x + a)^{2} \sqrt{x^{2} - a^{2}}} \\
& = & \frac{1}{(x + a)^{2} \sqrt{x^{2} - a^{2}}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \frac{1}{3 a^{2}} \frac{x^{2} + a x - 2 a^{2}}{(x + a) \sqrt{x^{2} - a^{2}}} + c \\
\end{eqnarray*}
\end{document}
