$\int\frac{dx}{1+\sinh x}$ is a slightly annoying but still easily solved integral using a weierstrass substitution and PFD. I'm mainly referring to WolframAlpha, but I've seen other computer algebra systems fail to find a closed form for this integral. In my experience, it has no problem with Weierstrass sub or PFD otherwise.
My result is $\frac{1}{\sqrt 2}\ln\left|\frac{\tanh\left(\frac{x}{2}\right)+\sqrt2-1}{\tanh\left(\frac{x}{2}\right)-\sqrt2-1}\right|+C$. At least WolframAlpha differentiates this correctly to get $\frac{1}{1+\sinh x}$.
I think it's a bit of an unwarranted generalization to look at Wolfram Alpha's capabilities and assert that CASes in general have difficulty evaluating this indefinite integral. For instance, Mathematica (version 13.3.1) immediately outputs $$\sqrt{2} \tanh ^{-1}\left(\frac{\tanh \left(\frac{x}{2}\right)-1}{\sqrt{2}}\right).$$
That said, earlier versions of Mathematica, and other CASes in general, have had difficulty returning known elementary closed form antiderivatives for integrands that were tractable with Weierstrass-type substitutions. The development history of Mathematica has, over the decades, included the ability to compute the antiderivatives of more functions, not just in terms of other special functions, but also cases where it previously did not know that an elementary closed form existed.
Regarding the possible reasons why Wolfram Alpha has an issue whereas Wolfram's flagship product Mathematica does not, any answer from a non-developer would be speculative. All we can say is that these two systems are not equivalent. From personal experience, I've seen the difference go both ways--some inputs could be evaluated with Wolfram Alpha that were not in Mathematica, and vice versa. Moreover, such questions are not within the scope of math.se because it is not about mathematics, but about the specific implementation of a computer program for performing mathematical operations.
As a general rule, one should not blindly rely on any computer system for performing symbolic mathematical operations as infallibly correct. It's important to check the reasonableness of the results obtained.