# NIntegral and Integral compare: which one is more accurate when integrand is not integrable?

I am calculating the following integral $$I=\int_0^1 e^{-c_1x^2+c_2x}\mathrm{erf}(c_3x+c_4)dx,$$ with Integrate and NIntegrate respectively.

There is no analytical solution for this integral and I got two different numerical results, with quite large difference: $$-0.117035 + 0.5i$$.

My question is, which one is more accurate in this case.

Here is my code:

c1 = Pi;
c2 = SetPrecision[10.1 + 10.1 I, 30];
c3 = Sqrt[Pi];
c4 = SetPrecision[1.1 + 6.1 I, 30];
I1 = NIntegrate[Exp[-c1 x^2 + c2 x] Erf[c3 x + c4], {x, 0, 1},
PrecisionGoal -> 20, WorkingPrecision -> 30];
I2 = Integrate[Exp[-c1 x^2 + c2 x] Erf[c3 x + c4], {x, 0, 1}];
error = N[I1 - I2]
(*error=-0.117035 + 0.5i*)


Better define the parameters with infinite precision. Then you can use ultrahigh WorkingPrecision.

c1 = Pi;
c2 = (101 + 101 I)/10;
c3 = Sqrt[Pi];
c4 = (11 + 61 I)/10;

I3 = NIntegrate[Exp[-c1 x^2 + c2 x] Erf[c3 x + c4], {x, 0, 1},
WorkingPrecision -> 100]

(*   5.98422153733426875937314115532253471792710605523929063484400170963328\
0746439954199937370126209185221*10^10 -
4.7006274392441169485465943452912663047232799365447530746425809872967\
02374014674651258602665889195814*10^13 I   *)

• @Akkr14: thanks for the answer^_^! After Rationalize the parameters, I recalculated $I_1$, $I_2$ and $I_3$. It turned out that $I_1$ and $I_3$ via NIntegrate are almost the same, but $I_2$ is still much different than them. Does that mean the result obtained by Integrate is not trustable in this case? – Ligang Sun Feb 28 at 9:52
• Remove the semicolon after Integrate... to see, that it gives back the input. It does not find an analytical solution, most likely the is no one. But when you apply N[I2] the integral is automaticlly calculated numericaly with NIntegrate, but with standard machine precision, which is 16. This is of course less accurate than with WorkingPrecission->30 . – Akku14 Feb 28 at 14:33
• @Akkr14: thanks a lot! it is very clear now. – Ligang Sun Feb 29 at 17:11