If `int(3cot3x-cotx)/(tanx-3tan3x)dx = pf(x) + qg(x)+C`, then which of the following may be correct?

If int (3cot 3 x-cot x)/(tanx-3tan3x)dx=pf(x)+qg(x)+C , where C is a constant of integration, then

int(cotx+cot^3x)/(1+cot^3x)dx

int(cot x+cot^(3)x)/(1+cot^(3)x)dx

(cot x)/(cotx-cot3x)+(tanx)/(tanx-tan3x)=

Let P(x)=cot^(2)x ((1+tanx+tan^(2)x)/(1+cot x+ cot^(2)x))+((cos x-cos 3x+sin3x-sin x)/(2(sin 2x+cos2x)))^(2) . Then, which of the following is (are) correct ?

Let P(x)=cot^(2)x ((1+tanx+tan^(2)x)/(1+cot x+ cot^(2)x))+((cos x-cos 3x+sin3x-sin x)/(2(sin 2x+cos2x)))^(2) . Then, which of the following is (are) correct ?

Let P(x)=cot^(2)x ((1+tanx+tan^(2)x)/(1+cot x+ cot^(2)x))+((cos x-cos 3x+sin3x-sin x)/(2(sin 2x+cos2x)))^(2) . Then, which of the following is (are) correct ?

int(sec^2x)/(sqrt(2tan^2x+3tanx+3))dx=

int cot^3 x dx

int(2tanx)/(2+3tan^(2)x)dx=