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Question Number 183976 by cortano1 last updated on 01/Jan/23

$$\int \frac{\sin x-\sqrt{1+\sin x}}{\cos x-\sqrt{1+\cos x}}dx=?$$

Answered by MJS_new last updated on 01/Jan/23

$$t=\frac{1+\sin \frac{x}{2}}{\cos \frac{x}{2}}\iff x=2\arctan \frac{t^2-1}{2t}\to dx=\frac{4dt}{t^2-1}$$$$\Rightarrow$$$$\int \frac{\sin x-\sqrt{1+\sin x}}{\cos x-\sqrt{1+\cos x}}dx=$$$$=4\int \frac{t^4-2t^3+6t-1}{(t^2+1)(t^2+\sqrt{2}(1+\sqrt{5})t+1)(t^2+\sqrt{2}(1-\sqrt{5})t+1)}dt$$$$\mathrm{now}\mathrm{do}\mathrm{the}\mathrm{decomposition}\mathrm{etc}.\mathrm{for}\mathrm{yourself}$$$$\mathrm{because}{\mathrm{I}}^{\prime }\mathrm{m}\mathrm{too}\mathrm{lazy}\mathrm{again}.$$

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