Find the indefinite integral $\int\frac{(x+1)e^x}{x(1+xe^x)}dx$

Question

Find the indefinite integral

$$\int\frac{(x+1)e^x}{x(1+xe^x)}dx$$

I feel like this function does not have an anti-derivative in the form of elementary functions.

Answer 1

You're right: Maple confirms that it does not have an elementary antiderivative.

Answer 2

As written, you're correct. However, I suspect that it is supposed to be $$\int\frac{(x+1)e^x}{1+xe^x}\,dx,$$ which does have a nice antiderivative family, as the numerator is the derivative of the denominator.

Answer 3

You're entirely correct that the posted problem has no elementary antiderivative; you can easily confirm this using Wolfram Alpha.

However, if perchance your problem has a typo in it, and you meant to post:

$$\int\frac{(x+1)e^x}{(1+xe^x)}dx$$

Then note that with

• $\;u = 1 + xe^x,\;$

• $\;du = xe^x + e^x = (x+1)e^x$,

  thus giving us an integral of the form $$\int \dfrac {du}{u} = \ln |u| + C = \ln|1 + xe^x| + C$$