How can I force Mathematica to calculate this integral?

Question

I have the following expression. I realize Mathematica has a little bit of trouble with Abs[] as it assumes variables to be complex and etc., but I am trying to compute an integral and it just keeps running without any output...

The expression is

Abs[-((20 Log[2])/Log[10]) + (
  20 Log[2/Abs[
     1 + (0. + 0.262217 I) f - 
      0.0343788 f^2 - (0. + 0.00278569 I) f^3 + 
      0.000139504 f^4 + (0. + 3.49311*10^-6 I) f^5]])/Log[10]]

And the integral to evaluate is

Integrate[ErrorButt, {f, 0, 1}, Assumptions -> f \[Element] Reals]

Any guess?

Answer 1

Clear["Global`*"]

Use ComplexExpand

expr = Assuming[Element[f, Reals],
  Abs[-((20 Log[2])/
          Log[10]) + (20 Log[
           2/Abs[1 + (0. + 0.262217 I) f - 
              0.0343788 f^2 - (0. + 0.00278569 I) f^3 + 
              0.000139504 f^4 + (0. + 3.49311*10^-6 I) f^5]])/Log[10]] // 
     Rationalize[#, 0] & // ComplexExpand // Simplify]

(1/Log[10])10 Log[(1 - (85947 f^2)/2500000 + (8719 f^4)/62500000)^2 + ((
     262217 f)/1000000 - (278569 f^3)/100000000 + (171923 f^5)/49217745791)^2]

The exact integral is

int = Integrate[expr, {f, 0, 1}]

(* (1/Log[10])10 (-10 + 
   Log[605596718724483025606799576062383049649/
     605596625186874553920250000000000000000] - 
   98435491582 RootSum[
     605596625186874553920250000000000000000 + 
       93921375003607187692937652250000 #1^2 - 
       399705884555840943078443405000 #1^4 + 
       8596710556978237869461280425 #1^6 - 46737404025411666287075776 #1^8 + 
       7389379482250000000000000000 #1^10 &, (-30761091119375000000000000000 \
Log[1 - #1] - 3816565488490199500000 Log[1 - #1] #1^2 + 
         12181761216365932500 Log[1 - #1] #1^4 - 
         174666889326537175 Log[1 - #1] #1^6 + 
         474802363195168 Log[
           1 - #1] #1^8)/(93921375003607187692937652250000 #1 - 
         799411769111681886156886810000 #1^3 + 
         25790131670934713608383841275 #1^5 - 
         186949616101646665148303104 #1^7 + 
         36946897411250000000000000000 #1^9) &] + 
   98435491582 RootSum[
     605596625186874553920250000000000000000 + 
       93921375003607187692937652250000 #1^2 - 
       399705884555840943078443405000 #1^4 + 
       8596710556978237869461280425 #1^6 - 46737404025411666287075776 #1^8 + 
       7389379482250000000000000000 #1^10 &, (-30761091119375000000000000000 \
Log[-#1] - 3816565488490199500000 Log[-#1] #1^2 + 
         12181761216365932500 Log[-#1] #1^4 - 
         174666889326537175 Log[-#1] #1^6 + 
         474802363195168 Log[-#1] #1^8)/(93921375003607187692937652250000 #1 \
- 799411769111681886156886810000 #1^3 + 25790131670934713608383841275 #1^5 - 
         186949616101646665148303104 #1^7 + 
         36946897411250000000000000000 #1^9) &]) *)

Its approximate numeric value is

int // N[#, 20] & // Chop[#, 10^-20] & // N

(* 2.23955*10^-7 *)

Comparing with numeric integration

int == NIntegrate[expr, {f, 0, 1}, WorkingPrecision -> 20]

(* True *)