Hi, I was wondering if you guys could check my work for indefinite integral of (e^x)(sqrt{1-e^2x)

Let u=e^x

The integral becomes:

integral of sqrt(1-u^2)

Then I do trigonometric substitution where I let u=sinx, and I get:

1/2sin^-1(x) + (x/2)*(1 - x^2)^1/2 + c.

However, the answer in my textbook is -1/3(1-e^(2x))^(3/2).

I was just wondering what I did wrong for the question.

If my complete process is wrong, what should I have done to solve the question?

Thanks in advance