The value of the integral $\int e^{\sin ^2 x}\left(\cos x+\cos ^3 x\right) \sin x d x$ is

$e^{\sin ^2 x}\left(1+\frac{1}{2} \cos ^2 x\right)+c$

Put $t=\sin ^2 x$

The integral reduces to

$I=\frac{1}{2} \int e^t(2-t) d t=\frac{3}{2} e^t-\frac{t e^t}{2}+c$

$=\frac{1}{2} e^{\sin ^2 x}\left(3-\sin ^2 x\right)+c$

$=e^{\sin ^2 x}\left(1+\frac{1}{2} \cos ^2 x\right)+c$

Hence (2) are the correct answer.