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[tex]x^2-4x-13=(x-2)^2-17[/tex]

[tex]x-2=\sqrt{17}\sec y[/tex]
[tex]\mathrm dx=\sqrt{17}\sec y\tan y\,\mathrm dy[/tex]

[tex]\displaystyle\int\frac{\mathrm dx}{x^2-4x-13}=\int\frac{\sqrt{17}\sec y\tan y}{(\sqrt{17}\sec y)^2-17}\,\mathrm dy[/tex]
[tex]=\displaystyle\frac1{\sqrt{17}}\int\frac{\sec y\tan y}{\sec^2y-1}\,\mathrm dy[/tex]
[tex]=\displaystyle\frac1{\sqrt{17}}\int\frac{\sec y\tan y}{\tan^2y}\,\mathrm dy[/tex]
[tex]=\displaystyle\frac1{\sqrt{17}}\int\frac{\sec y}{\tan y}\,\mathrm dy[/tex]
[tex]=\displaystyle\frac1{\sqrt{17}}\int\frac{\frac1{\cos y}}{\frac{\sin y}{\cos y}}\,\mathrm dy[/tex]
[tex]=\displaystyle\frac1{\sqrt{17}}\int\csc y\,\mathrm dy[/tex]
[tex]=-\dfrac1{\sqrt{17}}\ln|\csc y+\cot y|+C[/tex]

[tex]\sec y=\dfrac{x-2}{\sqrt{17}}\iff y=\sec^{-1}\dfrac{x-2}{\sqrt{17}}[/tex]
[tex]\implies\csc y=\dfrac{x-2}{\sqrt{(x-2)^2-17}}=\dfrac{x-2}{\sqrt{x^2-4x-13}}[/tex]
[tex]\implies\cot y=\dfrac{\sqrt{17}}{\sqrt{(x-2)^2-17}}=\dfrac{\sqrt{17}}{\sqrt{x^2-4x-13}}[/tex]

[tex]\displaystyle\int\frac{\mathrm dx}{x^2-4x-13}=-\dfrac1{\sqrt{17}}\ln\left|\frac{x-2+\sqrt{17}}{\sqrt{x^2-4x-13}}\right|+C[/tex]

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