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Question Number 153109 by peter frank last updated on 04/Sep/21

$$\int \frac{\mathrm{dx}}{{\mathrm{x}}^2+2\mathrm{x}+2\sqrt{{\mathrm{x}}^2+2\mathrm{x}-4}}$$

Commented by MJS_new last updated on 05/Sep/21

$$\mathrm{the}\mathrm{path}\mathrm{is}\mathrm{clear}\mathrm{but}\mathrm{the}\mathrm{constants}\mathrm{are}\mathrm{weird}$$$$t=\frac{x+1+\sqrt{x^2+2x-4}}{\sqrt{5}}\iff x=\frac{\sqrt{5}{t}^2-2t+\sqrt{5}}{2t}$$$$\to dx=\frac{\sqrt{x^2+2x-4}}{t}dt$$$$\mathrm{now}\mathrm{we}\mathrm{have}$$$$\frac{2\sqrt{5}}{5}\int \frac{t^2-1}{t^4+\frac{4\sqrt{5}}{5}{t}^3+\frac{6}{5}{t}^2+\frac{4\sqrt{5}}{5}t+1}dt$$$$u=t+\frac{\sqrt{5}}{5}\iff t=u-\frac{\sqrt{5}}{5}\to dt=du$$$$\frac{2\sqrt{5}}{5}\int \frac{u^2-\frac{2\sqrt{5}}{5}u-\frac{4}{5}}{u^4-\frac{24\sqrt{5}}{25}u+\frac{48}{25}}du$$$$u^4-\frac{24\sqrt{5}}{25}u+\frac{48}{25}=$$$$=(u^2-\frac{\sqrt{10(3+\sqrt{21})}}{5}u+\frac{3+\sqrt{21}-\sqrt{6(-3+\sqrt{21})}}{5})\times$$$$\times (u^2+\frac{\sqrt{10(3+\sqrt{21})}}{5}u+\frac{3+\sqrt{21}+\sqrt{6(+3+\sqrt{21})}}{5})$$$$\mathrm{and}\mathrm{now}\mathrm{we}\mathrm{need}\mathrm{to}\mathrm{decompose}...$$

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