Speeding up Integration over Implicit Region

Similar to my previous question, running

Clear["Global*"];
ineq = (1+y+y^2) r[4]<r[2]+y (y r[1]+r[3]) && y (r[1]-r[2]-(1+y) (r[3]-r[4]))+r[4]<r[2];
reg1=ImplicitRegion[ineq,{{r[1],0,1},{r[2],0,1},{r[3],0,1},{r[4],0,1}}];
(int[y_]=Assuming[1>y>0,Integrate[1,{r[1],r[2],r[3],r[4]}\[Element]reg1]//Simplify])//AbsoluteTiming


Produces $$\left\{165.709,\frac{8 y^4+26 y^3+39 y^2+32 y+12}{24 (y+1)^2 \left(y^2+y+1\right)}\right\}$$

Is there a way to speed this up using Implicit Region? Or is there a faster method to find integration bounds from an inequality? An explanation for why this isn't possible would also be accepted.

• Integration bounds may be obtained from Reduce[1 > y > 0 && First@reg1, {r[1], r[2], r[3], r[4]}] /. _Equal -> False, where the _Equal -> False removes components of measure zero. It will also show some of the complexity. The order of the variable can matter, since you're setting up an iterated integral. – Michael E2 Nov 20 '19 at 23:34
• Doing that shows a hundred lines of complicated integration bounds. Is that the best we can do for this inequality? – Logan Smith Nov 21 '19 at 0:43
• There's an unconventional kluge that gets the result a couple of orders of magnitude faster, happy to post if you want, but it is a kluge that is not very generic... – ciao Nov 23 '19 at 1:08

Using @Michael's suggestion brings the timing down about an order of magnitude. There are 120 components after using LogicalExpand:

components = List @@ LogicalExpand[
Reduce[1 > y > 0 && First @ reg1, {r[1], r[2], r[3], r[4]}] /. _Equal -> False
]; //AbsoluteTiming
Length[components]


{6.53425, Null}

120

Computing integral over each region:

r1 = Assuming[
0<y<1,
Simplify @ Total[
Map[
Integrate[1, z ∈ ImplicitRegion[#, {r[1], r[2], r[3], r[4]}]]&,
components
]
]
]; //AbsoluteTiming


{7.24909, Null}

The result is basically the same as yours, except for a discrete set of values fory:

r1 /. s_Root :> N[s] //TeXForm
`

$$\begin{cases} \frac{8 y^4+26 y^3+39 y^2+32 y+12}{24 (y+1)^2 \left(y^2+y+1\right)} & \frac{1}{2} \left(\sqrt{5}-1\right)0.754878 \\ 0 & \text{True} \end{cases}$$

• This is great for this specific problem, but it fails to generalize. For ineq = r[1] > r[2] , it gives infinity as opposed to 1/2 (half of a square with r[] as the axis). I'm currently trying to understand why it works here, but not in the simpler case. – Logan Smith Dec 6 '19 at 17:38