Is there any faster way to evaluate an integral?

Question

I have this integral

  A=2
  B=1
 FullSimplify[ Integrate[ 
    4 A Exp[-(s2 - s1)] Sin[n1a \[Pi] s1] Sin[n1b \[Pi] s1] Sin[
    n2a \[Pi] s2] Sin[n2b \[Pi] s2], {s1, 0, 1}, {s2, s1, 1}], 
    Assumptions -> {n1a \[Element] Integers, n1b \[Element] Integers, 
    n2a \[Element] Integers, n2b \[Element] Integers}]

But it take forever to evaluate it. Does any one know what is the better way to approach? Am I doing something wrong?

Answer 1

The better way to approach:

sol = Integrate[4 A Exp[-(s2 - s1)] Sin[n1a π s1] Sin[n1b π s1] Sin[
n2a π s2] Sin[n2b π s2], {s2, s1, 1}] // Expand

sol2 = Integrate[#, {s1, 0, 1}] & /@ (sol)

A = 2;
n1a = 2;
n1b = 3;
n2a = 3;
n2b = 5;
sol2// Re // N
(* -0.148968 *)

Numerically:

int[A_, n1a_, n1b_, n2a_, n2b_] := NIntegrate[4 A Exp[-(s2 - s1)] 
Sin[n1a π s1] Sin[n1b π s1] Sin[
n2a π s2] Sin[n2b π s2], {s1, 0, 1}, {s2, s1, 1}, Method -> "LocalAdaptive"]


int[2, 2, 3, 3, 5]
(* -0.148968 *)

EDITED:

Maybe You can use a Limit function to overcome (1/0). I'm converted yours function to Exp because Limit work faster.

sol = Integrate[4 A Exp[-(s2 - s1)] Sin[n1a \[Pi] s1] Sin[n1b \[Pi] s1] Sin[n2a \[Pi] s2] Sin[n2b \[Pi] s2] // TrigToExp // Expand, {s2, s1,1}];
sol2 = Integrate[#, {s1, 0, 1}] & /@ (sol) // Simplify;

Limit[Limit[Limit[Limit[sol2 /. A -> 1, {n1a -> 1}], {n1b -> 1}], {n2a -> 
 4}], {n2b -> 2}] // AbsoluteTiming

(* {2.03222, {(16 I (-1 + E) \[Pi]^2 (4 + 80 \[Pi]^2 + 
 E (-3 - 40 \[Pi]^2 + 144 \[Pi]^4)))/(
E (I + 2 \[Pi]) (1 + 40 \[Pi]^2 + 144 \[Pi]^4)^2)}}*)

In Mathematica 11.2 Limit function was updated a works a 60 time faster,than older version.

  Limit[sol2 /. A -> 1, {n1a -> 1, n1b -> 1, n2a -> 4, n2b -> 2}] // AbsoluteTiming(*Only this code works in Mathematica 11.2 or above*)

   (*{0.0312725, ((1 - E) ((2 I - 2 \[Pi])/(E (I - 2 \[Pi])^2) + (
  2 I + 2 \[Pi])/(E (I + 2 \[Pi])^2) + (
  E^(-I (-I + 6 \[Pi])) (-2 I + 6 \[Pi]))/(-I + 6 \[Pi])^2 - (
  E^(I (I + 6 \[Pi])) (2 I + 6 \[Pi]))/(I + 6 \[Pi])^2 - 
  I (1/(I - 2 \[Pi])^2 + 1/(I + 2 \[Pi])^2 - 1/(-I + 6 \[Pi])^2 - 
   1/(I + 6 \[Pi])^2)))/(4 (I + 2 \[Pi]))}*)