In September 1943, the Allied forces invaded Southern Italy in combined amphibious operations in Reggio di Calabria (Operation Baytown), Salerno (Operation Avalanche) and Taranto (Operation Slapstick). At first, these operations were relatively successful: despite quite fierce resistance from the German Tenth Army, the Allied progressed quickly northwards on two fronts. However, they were stopped at the so called Winter Line, where the Germans exploited very successfully the natural geographical defenses provided by the Appennine Mountains. In January 1944 started the battle(s) of Montecassino, which ended only five months later after inflicting 55,000 casualties on the Allies (more than double the German ones). Even so, it would take one more month to take Rome. After that, the Allies moved quickly north to the so called Gothic line where they were once again halted in Autumn 1944 for several months (and in fact, the surrender of German forces in Northern Italy only happened in April 1945).
In French common historiography (as presented for instance in History textbooks), the Italian campaign is treated with great importance. I would conjecture that this is due to the heavy involvement of French Free Forces (mostly from Northern Africa) and to the fact that this was the first time in WWII where French forces inflicted significant defeats on German forces, thereby avenging the humiliating defeat of 1940. I would venture that this campaign may perhaps have the same significance in Poland, for the same reasons. The American, British and Canadian point of view seems much more ambiguous; possibly due to the heavy casualties suffered during the Italian campaign and the lack of symbolic military victories to remember.
Whatever its place in current historiography, purely from a strategic point of view, the Italian campaign seems at first sight to be rather a German success. Of course, the Allied eventually came to prevail, but at an extremely high cost. Beside, had the war gone on after April, 1945, it is unclear to me where the forces in Italy would have gone and what their strategic use would have been. On the other hand, the opening of a second ground front in Western Europe was vitally important to relieve the Eastern Front and was actively campaigned for by Stalin.
With the benefits of historical hindsight, is the Italian campaign now considered a strategic mistake (or at least a sub-optimal strategy) by military historians?
For instance, is it now believed (again with hindsight) that canceling the Italian campaign in favor of advancing the combination of Operation Overlord and Operation Dragoon (whose progresses were much more spectacular, at least in part, it seems, because of the lack of natural defenses) would have been a better strategy? Or is it thought that these two operations had to take place in summer (in which case summer 1943 seems obviously too early and thus the Italian campaign was better than nothing)? Let me make it clear that I do not wish to speculate about alternative history and that I believe I have a reasonable understanding of why the Italian campaign was chosen against the alternatives at the time: my question is solely about the current strategic evaluation of this choice now that we know how events unfolded.
I have learnt a lot from the two answers (and have up voted them accordingly) but still I wonder if @RI Swamp Yankee and @Tom Au (or anyone else, obviously) would care to expand their answers in the relative direction. For instance, Tom Au cites the psychological effect of knocking one of the Axis down. Sure, but wouldn't the liberation of France be an even greater psychological victory? Likewise for the economic impact : surely the bulk of industrial production was located in Northern Italy, which remained under German control almost to the end, and surely again the liberation of France would have deprived Germany of an even larger industrial base. And similarly again, or so it seems to me, with the number of German soldiers tied down: considering the excruciatingly slow and murderous progress the Allies made in Italy after the initial landing, perhaps the 240,000 soldiers and 4,000 planes engaged in Monte-Cassino would have been better employed in Normandy or in Provence. I mean, the statistics are rather damning: in just about a month, the 200,000 soldiers of Operation Dragoon inflicted 150,000 casualties on the German forces, provoked the withdrawal of the German army to the Vosges and suffered 20,000 casualties in the process; to be compared with what 240,000 sliders accomplished in Montecassino in 5 months at the price of 55,000 casualties.
Now again, I think I understand why the strategic decision to invade Italy was taken at the time. What I really want to know about more is whether the combined invasions of France (or any massive operation on the Western Front) was deemed possible in 1943/early 1944 or if it was deemed necessary to wait until the summer 1944 (for instance to ensure a favorable climate)?