![\int \frac{cos x}{1 + cos x} dx\\=\int \frac{cos x(1-cosx)}{(1 + cos x)(1-cosx)} dx\\=\int \frac{cos x-cos^2 x}{(1-cos^2 x)} dx\\=\int \frac{cos x-cos^2 x}{sin^2 x} dx\\=\int \bigg (\frac{cos x}{sin^2 x}- \frac{cos^2 x}{sin^2 x}\bigg) dx\\=\int (cotx \: cosecx- cot^2 x) dx\\=\int [cotx \: cosecx- (cosec^2 x-1)] dx\\=\int (cotx \: cosecx- cosec^2 x+1) dx\\=\int (cotx \: cosec) dx- \int cosec^2 x dx+\int 1 dx\\= - cosec x + cot x + x + c](/tpl/images/0655/6512/768d0.png)
Please integrate it do it fast
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