Please note that as of very recently, this no longer works:

WolframAlpha["integral (E^(cos x))((cos x)^2)(sin x) dx",
    {{"IndefiniteIntegral", 2}, "Content"},
    PodStates -> {"IndefiniteIntegral__Step-by-step solution"}]

as it pruduces: Missing["NotAvailable"].

After hours of searching, I stumbled on a solution/workaround.

First note that the middle line:

{{"podid",subpodid},"property"}

is ostensibly not necessary (I will return it later), so one can invoke:

WolframAlpha["integral (E^(cos x))((cos x)^2)(sin x) dx",
    PodStates -> {"IndefiniteIntegral__Step-by-step solution"}]

This provides the step-by-step solution, however all of the pods are now there. In addition, one would need to click on "Show all steps" for all of them to appear. Sometimes the button is "Show step-by-step solution," and then another button for showing all steps.

Apparently, due to these extra buttons, WRI added (did I miss the memo?) an undocumented PodState of "IndefiniteIntegral__Show all steps" (or for other podids: "Result__Show all steps", etc.). This solves the "button" problem, however all of the pods are still present.

So we return the {{"podid",subpodid},"property"} line, however we do not want only the subpodid of 2, but rather all of them. Putting this all together, and wrapping it in a Column[] in order to space things out a bit vertically, we have:

Column[
    WolframAlpha[
        "integral (E^(cos x))((cos x)^2)(sin x) dx",
        {{"IndefiniteIntegral", All}, "Content"},
        PodStates -> {
            "IndefiniteIntegral__Step-by-step solution",
            "IndefiniteIntegral__Show all steps"
        }
    ][[All, 2]],
    Left,
    2
]