Integral of (sin x + sin^3 x)/cos 2x

1. A.
   $\frac{1}{2},\frac{3}{4\sqrt{2}}$
2. B.
   $\frac{1}{2},-\frac{3}{4\sqrt{2}}$
3. C.
   $\frac{1}{2},\frac{3}{2\sqrt{2}}$
4. D.
   $\frac{1}{2},-\frac{3}{2\sqrt{2}}$

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1. A.
   $A=\dfrac{1}{2},B=-\dfrac{3}{4\sqrt{2}},f(x)=\dfrac{\sqrt{2}\cos x-1}{\sqrt{2}\cos x+1}$
2. B.
   $A=-\dfrac{1}{2},B=-\dfrac{3}{4\sqrt{2}},f(x)=\dfrac{\sqrt{2}\cos x+1}{\sqrt{2}\cos x-1}$
3. C.
   $A=\dfrac{1}{2},B=-\dfrac{3}{4\sqrt{2}},f(x)=\dfrac{\sqrt{2}\cos x+1}{\sqrt{2}\cos x-1}$
4. D.
   $A=\dfrac{3}{2},B=\dfrac{1}{2},f(x)=\dfrac{\sqrt{2}\cos x-1}{\sqrt{2}\cos x+1}$

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