When does WolframAlpha refuse to give the precise value?

Question

I was computing the length of the curve $y=e^x$ between $x=1$ and $x=2$. When I input "arc length of e^x between 1 and 2" into WolframAlpha, it gives me only an approximation as follows:

However, using calculus, one obtains the precise value of the length:

$$\sqrt{1+e^4}+2^{-1}\log\frac{\sqrt{1+e^4}-1}{\sqrt{1+e^4}+1}-\sqrt{1+e^2}-2^{-1}\log\frac{\sqrt{1+e^2}-1}{\sqrt{1+e^2}+1}.$$

Is this result too complicated for WolframAlpha so that WolframAlpha can only give an approximation? More generally, when does WolframAlpha refuse to give the precise value?

Answer 1

What is "amazing" is that, typing in Wolfram Alpha

integrate sqrt(1+e^(2x))

we properly obtain $$\sqrt{e^{2 x}+1}-\tanh ^{-1}\left(\sqrt{e^{2 x}+1}\right)$$

and typing

integrate sqrt(1+e^(2x)) from x=1 to x=2

we properly obtain $$-\sqrt{1+e^2}+\sqrt{1+e^4}+\tanh ^{-1}\left(\sqrt{1+e^2}\right)-\tanh ^{-1}\left(\sqrt{1+e^4}\right)\approx 4.7852 +0. \times 10^{-6} \,i$$

Answer 2

The input

arc length of e^x between 1 and 2

invokes an additional step where Wolfram Alpha needs to parse the meaning of "arc length," and it formulates an integral of the function $e^x$ over a suitable interval. When you open the code that Wolfram Alpha generates--in essence, its interpretation of your input--you will find that it invokes NIntegrate[] rather than Integrate[], which explains why your output does not contain the symbolic computation in closed form, only the numeric output.

This also explains why if you use the word "Integrate" in your input, Wolfram Alpha will parse that as you wanting the output of some Integrate[] expression.

Generally speaking, if I were to use Wolfram Alpha, I would attempt to phrase my input in the Wolfram language as much as possible. The natural language processing capabilities are impressive, but as you can see, they do not--indeed, cannot--lend themselves to a precise level of control, nor an unambiguous input.