If you evaluate the square of the function Φ[s, x, α]
you get the following:
FullSimplify[( Sqrt[(s α)/Gamma[1 + 2*s]]
LegendreP[s, s, Tanh[α x]])^2, {α > 0, s > 0}]
(* (s α LegendreP[s, s, Tanh[x α]]^2)/Gamma[1 + 2 s] *)
Thus, you only need to think of how to integrate the LegendreP[s, s, Tanh[α x]])^2
.
When looking at this function it becomes clear that the cases with integer and real values of s are different. Namely, if s is integer, the Legendre polynomials you need are even. Check this:
Plot[{LegendreP[1, 1, Tanh[x ]], LegendreP[2, 2, Tanh[x ]],
LegendreP[3, 3, Tanh[x ]]}, {x, -2, 2}, PlotRange -> All,
PlotStyle -> {Red, Blue, Green}]

Here I plotted the LegendreP[3, 3, Tanh[x ]]
function itself, not its square, just to make all three lines visible together.
It is clear in this case that LegendreP[s, s, Tanh[α x]])^2
is also even, while your integrand, x*LegendreP[s, s, Tanh[x α]]^2
is odd. The integral is, therefore, zero.
The different story would be with the real s values. Check this:
Plot[{LegendreP[0.1, 0.1, Tanh[x ]]^2,
LegendreP[0.4, 0.4, 2, Tanh[x ]]^2,
LegendreP[0.92, 0.92, Tanh[x ]]^2}, {x, -4, 4}, PlotRange -> All,
PlotStyle -> {Red, Blue, Green}]
It is quite clearly visible that the curves are neither even, not odd. Have a look especially at the green one, the most manifested. Check also this:
Manipulate[
Plot[{LegendreP[s, s, Tanh[x ]]^2,
x*LegendreP[s, s, Tanh[x ]]^2}, {x, -4, 4},
PlotStyle -> {Red, Blue}], {{s, 1.026}, 0, 2}]
Here, as you see, the red line shows the behavior of the Legendre squared, while the blue one is that multiplied by x.
In addition they seem to increase with x. Indeed,
Limit[LegendreP[0.92, 0.92, Tanh[x ]]^2, x -> Infinity]
(* ∞ *)
This looks correct, but I would recommend you to check this last statement at some textbook on Legendre polynomials (like, e.g. Abramovitz and Stigun http://people.math.sfu.ca/~cbm/aands/).
The outcome is that for such s values the integral is infinity.
Table[Integrate[x \[Phi][s, x, a]^2, {x, -\[Infinity], \[Infinity]}, Assumptions -> a > 0], {s, 20}]
yields{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}
. $\endgroup$Integrate[Evaluate[(-1)^s Sqrt[(s a)/Gamma[1 + 2*s]] LegendreP[s, s, Tanh[a x]] /. {a -> 2, s -> 5}], {x, -\[Infinity], \[Infinity]}]
..in your method call you are calling function you are passing symbols, Shall you not pass values ? $\endgroup$