\(\int \frac {-40 e^x x^2+e^{2 x} (100 x+140 x^2+40 x^3)+e^{1+x} (40 e^x x^2+e^{2 x} (-40 x-40 x^2))+(40 x^2+e^x (-200 x-180 x^2-40 x^3)+e^{1+x} (-40 x^2+e^x (80 x+40 x^2))) \log (10-4 e^{1+x}+4 x)+(100 x-40 e^{1+x} x+40 x^2) \log ^2(10-4 e^{1+x}+4 x)}{-125-50 x+e^{2 x} (-50 x^2-20 x^3)+e^{1+x} (50+20 e^{2 x} x^2+2 e^{4 x} x^4)+e^{4 x} (-5 x^4-2 x^5)+(e^x (100 x^2+40 x^3)+e^{1+x} (-40 e^x x^2-8 e^{3 x} x^4)+e^{3 x} (20 x^4+8 x^5)) \log (10-4 e^{1+x}+4 x)+(-50 x^2-20 x^3+e^{1+x} (20 x^2+12 e^{2 x} x^4)+e^{2 x} (-30 x^4-12 x^5)) \log ^2(10-4 e^{1+x}+4 x)+(-8 e^{1+2 x} x^4+e^x (20 x^4+8 x^5)) \log ^3(10-4 e^{1+x}+4 x)+(-5 x^4+2 e^{1+x} x^4-2 x^5) \log ^4(10-4 e^{1+x}+4 x)} \, dx\) [956]

Optimal result

Integrand size = 444, antiderivative size = 35 \[ \int \frac {-40 e^x x^2+e^{2 x} \left (100 x+140 x^2+40 x^3\right )+e^{1+x} \left (40 e^x x^2+e^{2 x} \left (-40 x-40 x^2\right )\right )+\left (40 x^2+e^x \left (-200 x-180 x^2-40 x^3\right )+e^{1+x} \left (-40 x^2+e^x \left (80 x+40 x^2\right )\right )\right ) \log \left (10-4 e^{1+x}+4 x\right )+\left (100 x-40 e^{1+x} x+40 x^2\right ) \log ^2\left (10-4 e^{1+x}+4 x\right )}{-125-50 x+e^{2 x} \left (-50 x^2-20 x^3\right )+e^{1+x} \left (50+20 e^{2 x} x^2+2 e^{4 x} x^4\right )+e^{4 x} \left (-5 x^4-2 x^5\right )+\left (e^x \left (100 x^2+40 x^3\right )+e^{1+x} \left (-40 e^x x^2-8 e^{3 x} x^4\right )+e^{3 x} \left (20 x^4+8 x^5\right )\right ) \log \left (10-4 e^{1+x}+4 x\right )+\left (-50 x^2-20 x^3+e^{1+x} \left (20 x^2+12 e^{2 x} x^4\right )+e^{2 x} \left (-30 x^4-12 x^5\right )\right ) \log ^2\left (10-4 e^{1+x}+4 x\right )+\left (-8 e^{1+2 x} x^4+e^x \left (20 x^4+8 x^5\right )\right ) \log ^3\left (10-4 e^{1+x}+4 x\right )+\left (-5 x^4+2 e^{1+x} x^4-2 x^5\right ) \log ^4\left (10-4 e^{1+x}+4 x\right )} \, dx=\frac {2}{1+\frac {1}{5} x^2 \left (-e^x+\log \left (10+4 \left (-e^{1+x}+x\right )\right )\right )^2} \] Output:

2/(1+1/5*(ln(-4*exp(1+x)+4*x+10)-exp(x))^2*x^2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.57 \[ \int \frac {-40 e^x x^2+e^{2 x} \left (100 x+140 x^2+40 x^3\right )+e^{1+x} \left (40 e^x x^2+e^{2 x} \left (-40 x-40 x^2\right )\right )+\left (40 x^2+e^x \left (-200 x-180 x^2-40 x^3\right )+e^{1+x} \left (-40 x^2+e^x \left (80 x+40 x^2\right )\right )\right ) \log \left (10-4 e^{1+x}+4 x\right )+\left (100 x-40 e^{1+x} x+40 x^2\right ) \log ^2\left (10-4 e^{1+x}+4 x\right )}{-125-50 x+e^{2 x} \left (-50 x^2-20 x^3\right )+e^{1+x} \left (50+20 e^{2 x} x^2+2 e^{4 x} x^4\right )+e^{4 x} \left (-5 x^4-2 x^5\right )+\left (e^x \left (100 x^2+40 x^3\right )+e^{1+x} \left (-40 e^x x^2-8 e^{3 x} x^4\right )+e^{3 x} \left (20 x^4+8 x^5\right )\right ) \log \left (10-4 e^{1+x}+4 x\right )+\left (-50 x^2-20 x^3+e^{1+x} \left (20 x^2+12 e^{2 x} x^4\right )+e^{2 x} \left (-30 x^4-12 x^5\right )\right ) \log ^2\left (10-4 e^{1+x}+4 x\right )+\left (-8 e^{1+2 x} x^4+e^x \left (20 x^4+8 x^5\right )\right ) \log ^3\left (10-4 e^{1+x}+4 x\right )+\left (-5 x^4+2 e^{1+x} x^4-2 x^5\right ) \log ^4\left (10-4 e^{1+x}+4 x\right )} \, dx=\frac {10}{5+e^{2 x} x^2-2 e^x x^2 \log \left (10-4 e^{1+x}+4 x\right )+x^2 \log ^2\left (10-4 e^{1+x}+4 x\right )} \] Input:

Integrate[(-40*E^x*x^2 + E^(2*x)*(100*x + 140*x^2 + 40*x^3) + E^(1 + x)*(4 
0*E^x*x^2 + E^(2*x)*(-40*x - 40*x^2)) + (40*x^2 + E^x*(-200*x - 180*x^2 - 
40*x^3) + E^(1 + x)*(-40*x^2 + E^x*(80*x + 40*x^2)))*Log[10 - 4*E^(1 + x) 
+ 4*x] + (100*x - 40*E^(1 + x)*x + 40*x^2)*Log[10 - 4*E^(1 + x) + 4*x]^2)/ 
(-125 - 50*x + E^(2*x)*(-50*x^2 - 20*x^3) + E^(1 + x)*(50 + 20*E^(2*x)*x^2 
 + 2*E^(4*x)*x^4) + E^(4*x)*(-5*x^4 - 2*x^5) + (E^x*(100*x^2 + 40*x^3) + E 
^(1 + x)*(-40*E^x*x^2 - 8*E^(3*x)*x^4) + E^(3*x)*(20*x^4 + 8*x^5))*Log[10 
- 4*E^(1 + x) + 4*x] + (-50*x^2 - 20*x^3 + E^(1 + x)*(20*x^2 + 12*E^(2*x)* 
x^4) + E^(2*x)*(-30*x^4 - 12*x^5))*Log[10 - 4*E^(1 + x) + 4*x]^2 + (-8*E^( 
1 + 2*x)*x^4 + E^x*(20*x^4 + 8*x^5))*Log[10 - 4*E^(1 + x) + 4*x]^3 + (-5*x 
^4 + 2*E^(1 + x)*x^4 - 2*x^5)*Log[10 - 4*E^(1 + x) + 4*x]^4),x]
 

Output:

10/(5 + E^(2*x)*x^2 - 2*E^x*x^2*Log[10 - 4*E^(1 + x) + 4*x] + x^2*Log[10 - 
 4*E^(1 + x) + 4*x]^2)
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {-40 e^x x^2+e^{x+1} \left (40 e^x x^2+e^{2 x} \left (-40 x^2-40 x\right )\right )+\left (40 x^2-40 e^{x+1} x+100 x\right ) \log ^2\left (4 x-4 e^{x+1}+10\right )+e^{2 x} \left (40 x^3+140 x^2+100 x\right )+\left (40 x^2+e^{x+1} \left (e^x \left (40 x^2+80 x\right )-40 x^2\right )+e^x \left (-40 x^3-180 x^2-200 x\right )\right ) \log \left (4 x-4 e^{x+1}+10\right )}{e^{4 x} \left (-2 x^5-5 x^4\right )+\left (-2 x^5+2 e^{x+1} x^4-5 x^4\right ) \log ^4\left (4 x-4 e^{x+1}+10\right )+\left (e^x \left (8 x^5+20 x^4\right )-8 e^{2 x+1} x^4\right ) \log ^3\left (4 x-4 e^{x+1}+10\right )+e^{x+1} \left (2 e^{4 x} x^4+20 e^{2 x} x^2+50\right )+e^{2 x} \left (-20 x^3-50 x^2\right )+\left (-20 x^3-50 x^2+e^{2 x} \left (-12 x^5-30 x^4\right )+e^{x+1} \left (12 e^{2 x} x^4+20 x^2\right )\right ) \log ^2\left (4 x-4 e^{x+1}+10\right )+\left (e^{3 x} \left (8 x^5+20 x^4\right )+e^{x+1} \left (-8 e^{3 x} x^4-40 e^x x^2\right )+e^x \left (40 x^3+100 x^2\right )\right ) \log \left (4 x-4 e^{x+1}+10\right )-50 x-125} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {20 x \left (e^x-\log \left (4 x-4 e^{x+1}+10\right )\right ) \left (-e^x \left (2 x^2+7 x+5\right )-2 e^{x+1} x+2 x+2 e^{2 x+1} (x+1)-\left (-2 x+2 e^{x+1}-5\right ) \log \left (4 x-4 e^{x+1}+10\right )\right )}{\left (2 x-2 e^{x+1}+5\right ) \left (e^{2 x} x^2+x^2 \log ^2\left (4 x-4 e^{x+1}+10\right )-2 e^x x^2 \log \left (4 x-4 e^{x+1}+10\right )+5\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 20 \int \frac {x \left (e^x-\log \left (4 x-4 e^{x+1}+10\right )\right ) \left (-2 e^{x+1} x+2 x+2 e^{2 x+1} (x+1)-e^x \left (2 x^2+7 x+5\right )+\left (2 x-2 e^{x+1}+5\right ) \log \left (4 x-4 e^{x+1}+10\right )\right )}{\left (2 x-2 e^{x+1}+5\right ) \left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 20 \int \left (-\frac {8 e^3 (2 x+3) \left (-2 x+2 e \log \left (4 x-4 e^{x+1}+10\right )-5\right ) x^2}{\left (-2 x+2 e^{x+1}-5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2}+\frac {4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7-4 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^7+4 e^x x^6-8 e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6+8 e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+20 \left (1+\frac {e}{5}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6-20 \left (1+\frac {e}{5}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^6-4 \log \left (4 x-4 e^{x+1}+10\right ) x^6+10 e^x x^5+4 e^2 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5-4 e^{x+2} \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-20 \left (1+\frac {e}{5}\right ) e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5+20 \left (1+\frac {e}{5}\right ) e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+25 \left (1+\frac {14 e}{25}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5-25 \left (1+\frac {14 e}{25}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^5-10 \log \left (4 x-4 e^{x+1}+10\right ) x^5+20 x^5-40 e \log \left (4 x-4 e^{x+1}+10\right ) x^4+120 \left (1+\frac {e}{6}\right ) x^4+20 e^{x+2} x^3+40 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3-20 e^{x+2} \log \left (4 x-4 e^{x+1}+10\right ) x^3-140 \left (1+\frac {e}{7}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^3+225 \left (1+\frac {2 e}{15}\right ) x^3+20 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-100 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+125 x^2+100 e^2 x+100 e^2}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2 \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right ) x}+\frac {-16 x^9+64 e \log \left (4 x-4 e^{x+1}+10\right ) x^8-176 \left (1+\frac {e}{11}\right ) x^8-16 e^{x+2} x^7-96 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7+544 \left (1+\frac {3 e}{34}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^7-760 \left (1+\frac {13 e}{95}\right ) x^7-64 e^{x+2} x^6+64 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6-576 \left (1+\frac {e}{18}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+16 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^6+1680 \left (1+\frac {4 e}{35}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^6-1600 \left (1+\frac {11 e}{80}\right ) x^6-60 e^{x+2} x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5+224 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-1080 \left (1+\frac {2 e}{45}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+24 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^5+2200 \left (1+\frac {9 e}{110}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^5-1625 \left (1+\frac {2}{325} e (15+16 e)\right ) x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^4+160 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^4-600 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^4+1000 e \left (1+\frac {8 e^2}{25}\right ) \log \left (4 x-4 e^{x+1}+10\right ) x^4-625 \left (1+\frac {192 e^2}{125}\right ) x^4-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3+1120 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^3-1800 e^2 x^3-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2+800 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^2-1000 e^2 x^2-400 e^4 x-400 e^4}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2 x}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 20 \int \frac {x \left (e^x-\log \left (4 x-4 e^{x+1}+10\right )\right ) \left (-2 e^{x+1} x+2 x+2 e^{2 x+1} (x+1)-e^x \left (2 x^2+7 x+5\right )-\left (-2 x+2 e^{x+1}-5\right ) \log \left (4 x-4 e^{x+1}+10\right )\right )}{\left (2 x-2 e^{x+1}+5\right ) \left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 20 \int \left (-\frac {8 e^3 (2 x+3) \left (-2 x+2 e \log \left (4 x-4 e^{x+1}+10\right )-5\right ) x^2}{\left (-2 x+2 e^{x+1}-5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2}+\frac {4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7-4 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^7+4 e^x x^6-8 e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6+8 e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+20 \left (1+\frac {e}{5}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6-20 \left (1+\frac {e}{5}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^6-4 \log \left (4 x-4 e^{x+1}+10\right ) x^6+10 e^x x^5+4 e^2 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5-4 e^{x+2} \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-20 \left (1+\frac {e}{5}\right ) e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5+20 \left (1+\frac {e}{5}\right ) e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+25 \left (1+\frac {14 e}{25}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5-25 \left (1+\frac {14 e}{25}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^5-10 \log \left (4 x-4 e^{x+1}+10\right ) x^5+20 x^5-40 e \log \left (4 x-4 e^{x+1}+10\right ) x^4+120 \left (1+\frac {e}{6}\right ) x^4+20 e^{x+2} x^3+40 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3-20 e^{x+2} \log \left (4 x-4 e^{x+1}+10\right ) x^3-140 \left (1+\frac {e}{7}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^3+225 \left (1+\frac {2 e}{15}\right ) x^3+20 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-100 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+125 x^2+100 e^2 x+100 e^2}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2 \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right ) x}+\frac {-16 x^9+64 e \log \left (4 x-4 e^{x+1}+10\right ) x^8-176 \left (1+\frac {e}{11}\right ) x^8-16 e^{x+2} x^7-96 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7+544 \left (1+\frac {3 e}{34}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^7-760 \left (1+\frac {13 e}{95}\right ) x^7-64 e^{x+2} x^6+64 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6-576 \left (1+\frac {e}{18}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+16 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^6+1680 \left (1+\frac {4 e}{35}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^6-1600 \left (1+\frac {11 e}{80}\right ) x^6-60 e^{x+2} x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5+224 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-1080 \left (1+\frac {2 e}{45}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+24 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^5+2200 \left (1+\frac {9 e}{110}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^5-1625 \left (1+\frac {2}{325} e (15+16 e)\right ) x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^4+160 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^4-600 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^4+1000 e \left (1+\frac {8 e^2}{25}\right ) \log \left (4 x-4 e^{x+1}+10\right ) x^4-625 \left (1+\frac {192 e^2}{125}\right ) x^4-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3+1120 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^3-1800 e^2 x^3-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2+800 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^2-1000 e^2 x^2-400 e^4 x-400 e^4}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2 x}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 20 \int \frac {x \left (e^x-\log \left (4 x-4 e^{x+1}+10\right )\right ) \left (-2 e^{x+1} x+2 x+2 e^{2 x+1} (x+1)-e^x \left (2 x^2+7 x+5\right )-\left (-2 x+2 e^{x+1}-5\right ) \log \left (4 x-4 e^{x+1}+10\right )\right )}{\left (2 x-2 e^{x+1}+5\right ) \left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 20 \int \left (-\frac {8 e^3 (2 x+3) \left (-2 x+2 e \log \left (4 x-4 e^{x+1}+10\right )-5\right ) x^2}{\left (-2 x+2 e^{x+1}-5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2}+\frac {4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7-4 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^7+4 e^x x^6-8 e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6+8 e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+20 \left (1+\frac {e}{5}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6-20 \left (1+\frac {e}{5}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^6-4 \log \left (4 x-4 e^{x+1}+10\right ) x^6+10 e^x x^5+4 e^2 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5-4 e^{x+2} \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-20 \left (1+\frac {e}{5}\right ) e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5+20 \left (1+\frac {e}{5}\right ) e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+25 \left (1+\frac {14 e}{25}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5-25 \left (1+\frac {14 e}{25}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^5-10 \log \left (4 x-4 e^{x+1}+10\right ) x^5+20 x^5-40 e \log \left (4 x-4 e^{x+1}+10\right ) x^4+120 \left (1+\frac {e}{6}\right ) x^4+20 e^{x+2} x^3+40 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3-20 e^{x+2} \log \left (4 x-4 e^{x+1}+10\right ) x^3-140 \left (1+\frac {e}{7}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^3+225 \left (1+\frac {2 e}{15}\right ) x^3+20 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-100 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+125 x^2+100 e^2 x+100 e^2}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2 \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right ) x}+\frac {-16 x^9+64 e \log \left (4 x-4 e^{x+1}+10\right ) x^8-176 \left (1+\frac {e}{11}\right ) x^8-16 e^{x+2} x^7-96 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7+544 \left (1+\frac {3 e}{34}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^7-760 \left (1+\frac {13 e}{95}\right ) x^7-64 e^{x+2} x^6+64 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6-576 \left (1+\frac {e}{18}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+16 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^6+1680 \left (1+\frac {4 e}{35}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^6-1600 \left (1+\frac {11 e}{80}\right ) x^6-60 e^{x+2} x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5+224 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-1080 \left (1+\frac {2 e}{45}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+24 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^5+2200 \left (1+\frac {9 e}{110}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^5-1625 \left (1+\frac {2}{325} e (15+16 e)\right ) x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^4+160 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^4-600 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^4+1000 e \left (1+\frac {8 e^2}{25}\right ) \log \left (4 x-4 e^{x+1}+10\right ) x^4-625 \left (1+\frac {192 e^2}{125}\right ) x^4-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3+1120 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^3-1800 e^2 x^3-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2+800 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^2-1000 e^2 x^2-400 e^4 x-400 e^4}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2 x}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 20 \int \frac {x \left (e^x-\log \left (4 x-4 e^{x+1}+10\right )\right ) \left (-2 e^{x+1} x+2 x+2 e^{2 x+1} (x+1)-e^x \left (2 x^2+7 x+5\right )-\left (-2 x+2 e^{x+1}-5\right ) \log \left (4 x-4 e^{x+1}+10\right )\right )}{\left (2 x-2 e^{x+1}+5\right ) \left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 20 \int \left (-\frac {8 e^3 (2 x+3) \left (-2 x+2 e \log \left (4 x-4 e^{x+1}+10\right )-5\right ) x^2}{\left (-2 x+2 e^{x+1}-5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2}+\frac {4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7-4 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^7+4 e^x x^6-8 e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6+8 e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+20 \left (1+\frac {e}{5}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6-20 \left (1+\frac {e}{5}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^6-4 \log \left (4 x-4 e^{x+1}+10\right ) x^6+10 e^x x^5+4 e^2 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5-4 e^{x+2} \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-20 \left (1+\frac {e}{5}\right ) e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5+20 \left (1+\frac {e}{5}\right ) e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+25 \left (1+\frac {14 e}{25}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5-25 \left (1+\frac {14 e}{25}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^5-10 \log \left (4 x-4 e^{x+1}+10\right ) x^5+20 x^5-40 e \log \left (4 x-4 e^{x+1}+10\right ) x^4+120 \left (1+\frac {e}{6}\right ) x^4+20 e^{x+2} x^3+40 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3-20 e^{x+2} \log \left (4 x-4 e^{x+1}+10\right ) x^3-140 \left (1+\frac {e}{7}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^3+225 \left (1+\frac {2 e}{15}\right ) x^3+20 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-100 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+125 x^2+100 e^2 x+100 e^2}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2 \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right ) x}+\frac {-16 x^9+64 e \log \left (4 x-4 e^{x+1}+10\right ) x^8-176 \left (1+\frac {e}{11}\right ) x^8-16 e^{x+2} x^7-96 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7+544 \left (1+\frac {3 e}{34}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^7-760 \left (1+\frac {13 e}{95}\right ) x^7-64 e^{x+2} x^6+64 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6-576 \left (1+\frac {e}{18}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+16 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^6+1680 \left (1+\frac {4 e}{35}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^6-1600 \left (1+\frac {11 e}{80}\right ) x^6-60 e^{x+2} x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5+224 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-1080 \left (1+\frac {2 e}{45}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+24 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^5+2200 \left (1+\frac {9 e}{110}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^5-1625 \left (1+\frac {2}{325} e (15+16 e)\right ) x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^4+160 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^4-600 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^4+1000 e \left (1+\frac {8 e^2}{25}\right ) \log \left (4 x-4 e^{x+1}+10\right ) x^4-625 \left (1+\frac {192 e^2}{125}\right ) x^4-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3+1120 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^3-1800 e^2 x^3-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2+800 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^2-1000 e^2 x^2-400 e^4 x-400 e^4}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2 x}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 20 \int \frac {x \left (e^x-\log \left (4 x-4 e^{x+1}+10\right )\right ) \left (-2 e^{x+1} x+2 x+2 e^{2 x+1} (x+1)-e^x \left (2 x^2+7 x+5\right )-\left (-2 x+2 e^{x+1}-5\right ) \log \left (4 x-4 e^{x+1}+10\right )\right )}{\left (2 x-2 e^{x+1}+5\right ) \left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 20 \int \left (-\frac {8 e^3 (2 x+3) \left (-2 x+2 e \log \left (4 x-4 e^{x+1}+10\right )-5\right ) x^2}{\left (-2 x+2 e^{x+1}-5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2}+\frac {4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7-4 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^7+4 e^x x^6-8 e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6+8 e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+20 \left (1+\frac {e}{5}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6-20 \left (1+\frac {e}{5}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^6-4 \log \left (4 x-4 e^{x+1}+10\right ) x^6+10 e^x x^5+4 e^2 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5-4 e^{x+2} \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-20 \left (1+\frac {e}{5}\right ) e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5+20 \left (1+\frac {e}{5}\right ) e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+25 \left (1+\frac {14 e}{25}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5-25 \left (1+\frac {14 e}{25}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^5-10 \log \left (4 x-4 e^{x+1}+10\right ) x^5+20 x^5-40 e \log \left (4 x-4 e^{x+1}+10\right ) x^4+120 \left (1+\frac {e}{6}\right ) x^4+20 e^{x+2} x^3+40 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3-20 e^{x+2} \log \left (4 x-4 e^{x+1}+10\right ) x^3-140 \left (1+\frac {e}{7}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^3+225 \left (1+\frac {2 e}{15}\right ) x^3+20 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-100 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+125 x^2+100 e^2 x+100 e^2}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2 \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right ) x}+\frac {-16 x^9+64 e \log \left (4 x-4 e^{x+1}+10\right ) x^8-176 \left (1+\frac {e}{11}\right ) x^8-16 e^{x+2} x^7-96 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7+544 \left (1+\frac {3 e}{34}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^7-760 \left (1+\frac {13 e}{95}\right ) x^7-64 e^{x+2} x^6+64 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6-576 \left (1+\frac {e}{18}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+16 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^6+1680 \left (1+\frac {4 e}{35}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^6-1600 \left (1+\frac {11 e}{80}\right ) x^6-60 e^{x+2} x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5+224 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-1080 \left (1+\frac {2 e}{45}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+24 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^5+2200 \left (1+\frac {9 e}{110}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^5-1625 \left (1+\frac {2}{325} e (15+16 e)\right ) x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^4+160 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^4-600 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^4+1000 e \left (1+\frac {8 e^2}{25}\right ) \log \left (4 x-4 e^{x+1}+10\right ) x^4-625 \left (1+\frac {192 e^2}{125}\right ) x^4-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3+1120 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^3-1800 e^2 x^3-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2+800 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^2-1000 e^2 x^2-400 e^4 x-400 e^4}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2 x}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 20 \int \frac {x \left (e^x-\log \left (4 x-4 e^{x+1}+10\right )\right ) \left (-2 e^{x+1} x+2 x+2 e^{2 x+1} (x+1)-e^x \left (2 x^2+7 x+5\right )-\left (-2 x+2 e^{x+1}-5\right ) \log \left (4 x-4 e^{x+1}+10\right )\right )}{\left (2 x-2 e^{x+1}+5\right ) \left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 20 \int \left (-\frac {8 e^3 (2 x+3) \left (-2 x+2 e \log \left (4 x-4 e^{x+1}+10\right )-5\right ) x^2}{\left (-2 x+2 e^{x+1}-5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2}+\frac {4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7-4 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^7+4 e^x x^6-8 e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6+8 e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+20 \left (1+\frac {e}{5}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6-20 \left (1+\frac {e}{5}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^6-4 \log \left (4 x-4 e^{x+1}+10\right ) x^6+10 e^x x^5+4 e^2 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5-4 e^{x+2} \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-20 \left (1+\frac {e}{5}\right ) e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5+20 \left (1+\frac {e}{5}\right ) e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+25 \left (1+\frac {14 e}{25}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5-25 \left (1+\frac {14 e}{25}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^5-10 \log \left (4 x-4 e^{x+1}+10\right ) x^5+20 x^5-40 e \log \left (4 x-4 e^{x+1}+10\right ) x^4+120 \left (1+\frac {e}{6}\right ) x^4+20 e^{x+2} x^3+40 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3-20 e^{x+2} \log \left (4 x-4 e^{x+1}+10\right ) x^3-140 \left (1+\frac {e}{7}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^3+225 \left (1+\frac {2 e}{15}\right ) x^3+20 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-100 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+125 x^2+100 e^2 x+100 e^2}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2 \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right ) x}+\frac {-16 x^9+64 e \log \left (4 x-4 e^{x+1}+10\right ) x^8-176 \left (1+\frac {e}{11}\right ) x^8-16 e^{x+2} x^7-96 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7+544 \left (1+\frac {3 e}{34}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^7-760 \left (1+\frac {13 e}{95}\right ) x^7-64 e^{x+2} x^6+64 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6-576 \left (1+\frac {e}{18}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+16 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^6+1680 \left (1+\frac {4 e}{35}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^6-1600 \left (1+\frac {11 e}{80}\right ) x^6-60 e^{x+2} x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5+224 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-1080 \left (1+\frac {2 e}{45}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+24 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^5+2200 \left (1+\frac {9 e}{110}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^5-1625 \left (1+\frac {2}{325} e (15+16 e)\right ) x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^4+160 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^4-600 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^4+1000 e \left (1+\frac {8 e^2}{25}\right ) \log \left (4 x-4 e^{x+1}+10\right ) x^4-625 \left (1+\frac {192 e^2}{125}\right ) x^4-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3+1120 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^3-1800 e^2 x^3-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2+800 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^2-1000 e^2 x^2-400 e^4 x-400 e^4}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2 x}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 20 \int \frac {x \left (e^x-\log \left (4 x-4 e^{x+1}+10\right )\right ) \left (-2 e^{x+1} x+2 x+2 e^{2 x+1} (x+1)-e^x \left (2 x^2+7 x+5\right )-\left (-2 x+2 e^{x+1}-5\right ) \log \left (4 x-4 e^{x+1}+10\right )\right )}{\left (2 x-2 e^{x+1}+5\right ) \left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 20 \int \left (-\frac {8 e^3 (2 x+3) \left (-2 x+2 e \log \left (4 x-4 e^{x+1}+10\right )-5\right ) x^2}{\left (-2 x+2 e^{x+1}-5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2}+\frac {4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7-4 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^7+4 e^x x^6-8 e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6+8 e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+20 \left (1+\frac {e}{5}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6-20 \left (1+\frac {e}{5}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^6-4 \log \left (4 x-4 e^{x+1}+10\right ) x^6+10 e^x x^5+4 e^2 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5-4 e^{x+2} \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-20 \left (1+\frac {e}{5}\right ) e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5+20 \left (1+\frac {e}{5}\right ) e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+25 \left (1+\frac {14 e}{25}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5-25 \left (1+\frac {14 e}{25}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^5-10 \log \left (4 x-4 e^{x+1}+10\right ) x^5+20 x^5-40 e \log \left (4 x-4 e^{x+1}+10\right ) x^4+120 \left (1+\frac {e}{6}\right ) x^4+20 e^{x+2} x^3+40 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3-20 e^{x+2} \log \left (4 x-4 e^{x+1}+10\right ) x^3-140 \left (1+\frac {e}{7}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^3+225 \left (1+\frac {2 e}{15}\right ) x^3+20 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-100 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+125 x^2+100 e^2 x+100 e^2}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2 \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right ) x}+\frac {-16 x^9+64 e \log \left (4 x-4 e^{x+1}+10\right ) x^8-176 \left (1+\frac {e}{11}\right ) x^8-16 e^{x+2} x^7-96 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7+544 \left (1+\frac {3 e}{34}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^7-760 \left (1+\frac {13 e}{95}\right ) x^7-64 e^{x+2} x^6+64 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6-576 \left (1+\frac {e}{18}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+16 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^6+1680 \left (1+\frac {4 e}{35}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^6-1600 \left (1+\frac {11 e}{80}\right ) x^6-60 e^{x+2} x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5+224 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-1080 \left (1+\frac {2 e}{45}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+24 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^5+2200 \left (1+\frac {9 e}{110}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^5-1625 \left (1+\frac {2}{325} e (15+16 e)\right ) x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^4+160 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^4-600 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^4+1000 e \left (1+\frac {8 e^2}{25}\right ) \log \left (4 x-4 e^{x+1}+10\right ) x^4-625 \left (1+\frac {192 e^2}{125}\right ) x^4-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3+1120 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^3-1800 e^2 x^3-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2+800 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^2-1000 e^2 x^2-400 e^4 x-400 e^4}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2 x}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 20 \int \frac {x \left (e^x-\log \left (4 x-4 e^{x+1}+10\right )\right ) \left (-2 e^{x+1} x+2 x+2 e^{2 x+1} (x+1)-e^x \left (2 x^2+7 x+5\right )-\left (-2 x+2 e^{x+1}-5\right ) \log \left (4 x-4 e^{x+1}+10\right )\right )}{\left (2 x-2 e^{x+1}+5\right ) \left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 20 \int \left (-\frac {8 e^3 (2 x+3) \left (-2 x+2 e \log \left (4 x-4 e^{x+1}+10\right )-5\right ) x^2}{\left (-2 x+2 e^{x+1}-5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2}+\frac {4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7-4 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^7+4 e^x x^6-8 e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6+8 e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+20 \left (1+\frac {e}{5}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6-20 \left (1+\frac {e}{5}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^6-4 \log \left (4 x-4 e^{x+1}+10\right ) x^6+10 e^x x^5+4 e^2 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5-4 e^{x+2} \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-20 \left (1+\frac {e}{5}\right ) e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5+20 \left (1+\frac {e}{5}\right ) e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+25 \left (1+\frac {14 e}{25}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5-25 \left (1+\frac {14 e}{25}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^5-10 \log \left (4 x-4 e^{x+1}+10\right ) x^5+20 x^5-40 e \log \left (4 x-4 e^{x+1}+10\right ) x^4+120 \left (1+\frac {e}{6}\right ) x^4+20 e^{x+2} x^3+40 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3-20 e^{x+2} \log \left (4 x-4 e^{x+1}+10\right ) x^3-140 \left (1+\frac {e}{7}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^3+225 \left (1+\frac {2 e}{15}\right ) x^3+20 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-100 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+125 x^2+100 e^2 x+100 e^2}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2 \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right ) x}+\frac {-16 x^9+64 e \log \left (4 x-4 e^{x+1}+10\right ) x^8-176 \left (1+\frac {e}{11}\right ) x^8-16 e^{x+2} x^7-96 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7+544 \left (1+\frac {3 e}{34}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^7-760 \left (1+\frac {13 e}{95}\right ) x^7-64 e^{x+2} x^6+64 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6-576 \left (1+\frac {e}{18}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+16 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^6+1680 \left (1+\frac {4 e}{35}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^6-1600 \left (1+\frac {11 e}{80}\right ) x^6-60 e^{x+2} x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5+224 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-1080 \left (1+\frac {2 e}{45}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+24 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^5+2200 \left (1+\frac {9 e}{110}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^5-1625 \left (1+\frac {2}{325} e (15+16 e)\right ) x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^4+160 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^4-600 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^4+1000 e \left (1+\frac {8 e^2}{25}\right ) \log \left (4 x-4 e^{x+1}+10\right ) x^4-625 \left (1+\frac {192 e^2}{125}\right ) x^4-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3+1120 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^3-1800 e^2 x^3-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2+800 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^2-1000 e^2 x^2-400 e^4 x-400 e^4}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2 x}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 20 \int \frac {x \left (e^x-\log \left (4 x-4 e^{x+1}+10\right )\right ) \left (-2 e^{x+1} x+2 x+2 e^{2 x+1} (x+1)-e^x \left (2 x^2+7 x+5\right )-\left (-2 x+2 e^{x+1}-5\right ) \log \left (4 x-4 e^{x+1}+10\right )\right )}{\left (2 x-2 e^{x+1}+5\right ) \left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 20 \int \left (-\frac {8 e^3 (2 x+3) \left (-2 x+2 e \log \left (4 x-4 e^{x+1}+10\right )-5\right ) x^2}{\left (-2 x+2 e^{x+1}-5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2}+\frac {4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7-4 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^7+4 e^x x^6-8 e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6+8 e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+20 \left (1+\frac {e}{5}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6-20 \left (1+\frac {e}{5}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^6-4 \log \left (4 x-4 e^{x+1}+10\right ) x^6+10 e^x x^5+4 e^2 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5-4 e^{x+2} \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-20 \left (1+\frac {e}{5}\right ) e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5+20 \left (1+\frac {e}{5}\right ) e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+25 \left (1+\frac {14 e}{25}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5-25 \left (1+\frac {14 e}{25}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^5-10 \log \left (4 x-4 e^{x+1}+10\right ) x^5+20 x^5-40 e \log \left (4 x-4 e^{x+1}+10\right ) x^4+120 \left (1+\frac {e}{6}\right ) x^4+20 e^{x+2} x^3+40 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3-20 e^{x+2} \log \left (4 x-4 e^{x+1}+10\right ) x^3-140 \left (1+\frac {e}{7}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^3+225 \left (1+\frac {2 e}{15}\right ) x^3+20 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-100 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+125 x^2+100 e^2 x+100 e^2}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2 \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right ) x}+\frac {-16 x^9+64 e \log \left (4 x-4 e^{x+1}+10\right ) x^8-176 \left (1+\frac {e}{11}\right ) x^8-16 e^{x+2} x^7-96 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7+544 \left (1+\frac {3 e}{34}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^7-760 \left (1+\frac {13 e}{95}\right ) x^7-64 e^{x+2} x^6+64 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6-576 \left (1+\frac {e}{18}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+16 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^6+1680 \left (1+\frac {4 e}{35}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^6-1600 \left (1+\frac {11 e}{80}\right ) x^6-60 e^{x+2} x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5+224 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-1080 \left (1+\frac {2 e}{45}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+24 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^5+2200 \left (1+\frac {9 e}{110}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^5-1625 \left (1+\frac {2}{325} e (15+16 e)\right ) x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^4+160 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^4-600 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^4+1000 e \left (1+\frac {8 e^2}{25}\right ) \log \left (4 x-4 e^{x+1}+10\right ) x^4-625 \left (1+\frac {192 e^2}{125}\right ) x^4-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3+1120 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^3-1800 e^2 x^3-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2+800 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^2-1000 e^2 x^2-400 e^4 x-400 e^4}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2 x}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 20 \int \frac {x \left (e^x-\log \left (4 x-4 e^{x+1}+10\right )\right ) \left (-2 e^{x+1} x+2 x+2 e^{2 x+1} (x+1)-e^x \left (2 x^2+7 x+5\right )-\left (-2 x+2 e^{x+1}-5\right ) \log \left (4 x-4 e^{x+1}+10\right )\right )}{\left (2 x-2 e^{x+1}+5\right ) \left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 20 \int \left (-\frac {8 e^3 (2 x+3) \left (-2 x+2 e \log \left (4 x-4 e^{x+1}+10\right )-5\right ) x^2}{\left (-2 x+2 e^{x+1}-5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2}+\frac {4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7-4 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^7+4 e^x x^6-8 e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6+8 e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+20 \left (1+\frac {e}{5}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6-20 \left (1+\frac {e}{5}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^6-4 \log \left (4 x-4 e^{x+1}+10\right ) x^6+10 e^x x^5+4 e^2 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5-4 e^{x+2} \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-20 \left (1+\frac {e}{5}\right ) e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5+20 \left (1+\frac {e}{5}\right ) e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+25 \left (1+\frac {14 e}{25}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5-25 \left (1+\frac {14 e}{25}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^5-10 \log \left (4 x-4 e^{x+1}+10\right ) x^5+20 x^5-40 e \log \left (4 x-4 e^{x+1}+10\right ) x^4+120 \left (1+\frac {e}{6}\right ) x^4+20 e^{x+2} x^3+40 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3-20 e^{x+2} \log \left (4 x-4 e^{x+1}+10\right ) x^3-140 \left (1+\frac {e}{7}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^3+225 \left (1+\frac {2 e}{15}\right ) x^3+20 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-100 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+125 x^2+100 e^2 x+100 e^2}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2 \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right ) x}+\frac {-16 x^9+64 e \log \left (4 x-4 e^{x+1}+10\right ) x^8-176 \left (1+\frac {e}{11}\right ) x^8-16 e^{x+2} x^7-96 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7+544 \left (1+\frac {3 e}{34}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^7-760 \left (1+\frac {13 e}{95}\right ) x^7-64 e^{x+2} x^6+64 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6-576 \left (1+\frac {e}{18}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+16 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^6+1680 \left (1+\frac {4 e}{35}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^6-1600 \left (1+\frac {11 e}{80}\right ) x^6-60 e^{x+2} x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5+224 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-1080 \left (1+\frac {2 e}{45}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+24 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^5+2200 \left (1+\frac {9 e}{110}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^5-1625 \left (1+\frac {2}{325} e (15+16 e)\right ) x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^4+160 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^4-600 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^4+1000 e \left (1+\frac {8 e^2}{25}\right ) \log \left (4 x-4 e^{x+1}+10\right ) x^4-625 \left (1+\frac {192 e^2}{125}\right ) x^4-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3+1120 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^3-1800 e^2 x^3-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2+800 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^2-1000 e^2 x^2-400 e^4 x-400 e^4}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2 x}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 20 \int \frac {x \left (e^x-\log \left (4 x-4 e^{x+1}+10\right )\right ) \left (-2 e^{x+1} x+2 x+2 e^{2 x+1} (x+1)-e^x \left (2 x^2+7 x+5\right )-\left (-2 x+2 e^{x+1}-5\right ) \log \left (4 x-4 e^{x+1}+10\right )\right )}{\left (2 x-2 e^{x+1}+5\right ) \left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 20 \int \left (-\frac {8 e^3 (2 x+3) \left (-2 x+2 e \log \left (4 x-4 e^{x+1}+10\right )-5\right ) x^2}{\left (-2 x+2 e^{x+1}-5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2}+\frac {4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7-4 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^7+4 e^x x^6-8 e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6+8 e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+20 \left (1+\frac {e}{5}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6-20 \left (1+\frac {e}{5}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^6-4 \log \left (4 x-4 e^{x+1}+10\right ) x^6+10 e^x x^5+4 e^2 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5-4 e^{x+2} \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-20 \left (1+\frac {e}{5}\right ) e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5+20 \left (1+\frac {e}{5}\right ) e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+25 \left (1+\frac {14 e}{25}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5-25 \left (1+\frac {14 e}{25}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^5-10 \log \left (4 x-4 e^{x+1}+10\right ) x^5+20 x^5-40 e \log \left (4 x-4 e^{x+1}+10\right ) x^4+120 \left (1+\frac {e}{6}\right ) x^4+20 e^{x+2} x^3+40 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3-20 e^{x+2} \log \left (4 x-4 e^{x+1}+10\right ) x^3-140 \left (1+\frac {e}{7}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^3+225 \left (1+\frac {2 e}{15}\right ) x^3+20 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-100 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+125 x^2+100 e^2 x+100 e^2}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2 \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right ) x}+\frac {-16 x^9+64 e \log \left (4 x-4 e^{x+1}+10\right ) x^8-176 \left (1+\frac {e}{11}\right ) x^8-16 e^{x+2} x^7-96 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7+544 \left (1+\frac {3 e}{34}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^7-760 \left (1+\frac {13 e}{95}\right ) x^7-64 e^{x+2} x^6+64 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6-576 \left (1+\frac {e}{18}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+16 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^6+1680 \left (1+\frac {4 e}{35}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^6-1600 \left (1+\frac {11 e}{80}\right ) x^6-60 e^{x+2} x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5+224 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-1080 \left (1+\frac {2 e}{45}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+24 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^5+2200 \left (1+\frac {9 e}{110}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^5-1625 \left (1+\frac {2}{325} e (15+16 e)\right ) x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^4+160 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^4-600 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^4+1000 e \left (1+\frac {8 e^2}{25}\right ) \log \left (4 x-4 e^{x+1}+10\right ) x^4-625 \left (1+\frac {192 e^2}{125}\right ) x^4-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3+1120 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^3-1800 e^2 x^3-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2+800 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^2-1000 e^2 x^2-400 e^4 x-400 e^4}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2 x}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 20 \int \frac {x \left (e^x-\log \left (4 x-4 e^{x+1}+10\right )\right ) \left (-2 e^{x+1} x+2 x+2 e^{2 x+1} (x+1)-e^x \left (2 x^2+7 x+5\right )-\left (-2 x+2 e^{x+1}-5\right ) \log \left (4 x-4 e^{x+1}+10\right )\right )}{\left (2 x-2 e^{x+1}+5\right ) \left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 20 \int \left (-\frac {8 e^3 (2 x+3) \left (-2 x+2 e \log \left (4 x-4 e^{x+1}+10\right )-5\right ) x^2}{\left (-2 x+2 e^{x+1}-5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2}+\frac {4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7-4 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^7+4 e^x x^6-8 e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6+8 e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+20 \left (1+\frac {e}{5}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6-20 \left (1+\frac {e}{5}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^6-4 \log \left (4 x-4 e^{x+1}+10\right ) x^6+10 e^x x^5+4 e^2 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5-4 e^{x+2} \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-20 \left (1+\frac {e}{5}\right ) e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5+20 \left (1+\frac {e}{5}\right ) e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+25 \left (1+\frac {14 e}{25}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5-25 \left (1+\frac {14 e}{25}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^5-10 \log \left (4 x-4 e^{x+1}+10\right ) x^5+20 x^5-40 e \log \left (4 x-4 e^{x+1}+10\right ) x^4+120 \left (1+\frac {e}{6}\right ) x^4+20 e^{x+2} x^3+40 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3-20 e^{x+2} \log \left (4 x-4 e^{x+1}+10\right ) x^3-140 \left (1+\frac {e}{7}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^3+225 \left (1+\frac {2 e}{15}\right ) x^3+20 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-100 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+125 x^2+100 e^2 x+100 e^2}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2 \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right ) x}+\frac {-16 x^9+64 e \log \left (4 x-4 e^{x+1}+10\right ) x^8-176 \left (1+\frac {e}{11}\right ) x^8-16 e^{x+2} x^7-96 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7+544 \left (1+\frac {3 e}{34}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^7-760 \left (1+\frac {13 e}{95}\right ) x^7-64 e^{x+2} x^6+64 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6-576 \left (1+\frac {e}{18}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+16 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^6+1680 \left (1+\frac {4 e}{35}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^6-1600 \left (1+\frac {11 e}{80}\right ) x^6-60 e^{x+2} x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5+224 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-1080 \left (1+\frac {2 e}{45}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+24 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^5+2200 \left (1+\frac {9 e}{110}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^5-1625 \left (1+\frac {2}{325} e (15+16 e)\right ) x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^4+160 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^4-600 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^4+1000 e \left (1+\frac {8 e^2}{25}\right ) \log \left (4 x-4 e^{x+1}+10\right ) x^4-625 \left (1+\frac {192 e^2}{125}\right ) x^4-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3+1120 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^3-1800 e^2 x^3-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2+800 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^2-1000 e^2 x^2-400 e^4 x-400 e^4}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2 x}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 20 \int \frac {x \left (e^x-\log \left (4 x-4 e^{x+1}+10\right )\right ) \left (-2 e^{x+1} x+2 x+2 e^{2 x+1} (x+1)-e^x \left (2 x^2+7 x+5\right )-\left (-2 x+2 e^{x+1}-5\right ) \log \left (4 x-4 e^{x+1}+10\right )\right )}{\left (2 x-2 e^{x+1}+5\right ) \left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 20 \int \left (-\frac {8 e^3 (2 x+3) \left (-2 x+2 e \log \left (4 x-4 e^{x+1}+10\right )-5\right ) x^2}{\left (-2 x+2 e^{x+1}-5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2}+\frac {4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7-4 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^7+4 e^x x^6-8 e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6+8 e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+20 \left (1+\frac {e}{5}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6-20 \left (1+\frac {e}{5}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^6-4 \log \left (4 x-4 e^{x+1}+10\right ) x^6+10 e^x x^5+4 e^2 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5-4 e^{x+2} \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-20 \left (1+\frac {e}{5}\right ) e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5+20 \left (1+\frac {e}{5}\right ) e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+25 \left (1+\frac {14 e}{25}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5-25 \left (1+\frac {14 e}{25}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^5-10 \log \left (4 x-4 e^{x+1}+10\right ) x^5+20 x^5-40 e \log \left (4 x-4 e^{x+1}+10\right ) x^4+120 \left (1+\frac {e}{6}\right ) x^4+20 e^{x+2} x^3+40 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3-20 e^{x+2} \log \left (4 x-4 e^{x+1}+10\right ) x^3-140 \left (1+\frac {e}{7}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^3+225 \left (1+\frac {2 e}{15}\right ) x^3+20 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-100 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+125 x^2+100 e^2 x+100 e^2}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2 \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right ) x}+\frac {-16 x^9+64 e \log \left (4 x-4 e^{x+1}+10\right ) x^8-176 \left (1+\frac {e}{11}\right ) x^8-16 e^{x+2} x^7-96 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7+544 \left (1+\frac {3 e}{34}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^7-760 \left (1+\frac {13 e}{95}\right ) x^7-64 e^{x+2} x^6+64 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6-576 \left (1+\frac {e}{18}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+16 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^6+1680 \left (1+\frac {4 e}{35}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^6-1600 \left (1+\frac {11 e}{80}\right ) x^6-60 e^{x+2} x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5+224 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-1080 \left (1+\frac {2 e}{45}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+24 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^5+2200 \left (1+\frac {9 e}{110}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^5-1625 \left (1+\frac {2}{325} e (15+16 e)\right ) x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^4+160 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^4-600 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^4+1000 e \left (1+\frac {8 e^2}{25}\right ) \log \left (4 x-4 e^{x+1}+10\right ) x^4-625 \left (1+\frac {192 e^2}{125}\right ) x^4-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3+1120 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^3-1800 e^2 x^3-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2+800 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^2-1000 e^2 x^2-400 e^4 x-400 e^4}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2 x}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 20 \int \frac {x \left (e^x-\log \left (4 x-4 e^{x+1}+10\right )\right ) \left (-2 e^{x+1} x+2 x+2 e^{2 x+1} (x+1)-e^x \left (2 x^2+7 x+5\right )-\left (-2 x+2 e^{x+1}-5\right ) \log \left (4 x-4 e^{x+1}+10\right )\right )}{\left (2 x-2 e^{x+1}+5\right ) \left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 20 \int \left (-\frac {8 e^3 (2 x+3) \left (-2 x+2 e \log \left (4 x-4 e^{x+1}+10\right )-5\right ) x^2}{\left (-2 x+2 e^{x+1}-5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2}+\frac {4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7-4 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^7+4 e^x x^6-8 e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6+8 e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+20 \left (1+\frac {e}{5}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6-20 \left (1+\frac {e}{5}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^6-4 \log \left (4 x-4 e^{x+1}+10\right ) x^6+10 e^x x^5+4 e^2 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5-4 e^{x+2} \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-20 \left (1+\frac {e}{5}\right ) e \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5+20 \left (1+\frac {e}{5}\right ) e^{x+1} \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+25 \left (1+\frac {14 e}{25}\right ) \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5-25 \left (1+\frac {14 e}{25}\right ) e^x \log \left (4 x-4 e^{x+1}+10\right ) x^5-10 \log \left (4 x-4 e^{x+1}+10\right ) x^5+20 x^5-40 e \log \left (4 x-4 e^{x+1}+10\right ) x^4+120 \left (1+\frac {e}{6}\right ) x^4+20 e^{x+2} x^3+40 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3-20 e^{x+2} \log \left (4 x-4 e^{x+1}+10\right ) x^3-140 \left (1+\frac {e}{7}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^3+225 \left (1+\frac {2 e}{15}\right ) x^3+20 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-100 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+125 x^2+100 e^2 x+100 e^2}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2 \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right ) x}+\frac {-16 x^9+64 e \log \left (4 x-4 e^{x+1}+10\right ) x^8-176 \left (1+\frac {e}{11}\right ) x^8-16 e^{x+2} x^7-96 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^7+544 \left (1+\frac {3 e}{34}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^7-760 \left (1+\frac {13 e}{95}\right ) x^7-64 e^{x+2} x^6+64 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^6-576 \left (1+\frac {e}{18}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^6+16 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^6+1680 \left (1+\frac {4 e}{35}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^6-1600 \left (1+\frac {11 e}{80}\right ) x^6-60 e^{x+2} x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^5+224 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^5-1080 \left (1+\frac {2 e}{45}\right ) e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^5+24 e^{x+3} \log \left (4 x-4 e^{x+1}+10\right ) x^5+2200 \left (1+\frac {9 e}{110}\right ) e \log \left (4 x-4 e^{x+1}+10\right ) x^5-1625 \left (1+\frac {2}{325} e (15+16 e)\right ) x^5-16 e^4 \log ^4\left (4 x-4 e^{x+1}+10\right ) x^4+160 e^3 \log ^3\left (4 x-4 e^{x+1}+10\right ) x^4-600 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^4+1000 e \left (1+\frac {8 e^2}{25}\right ) \log \left (4 x-4 e^{x+1}+10\right ) x^4-625 \left (1+\frac {192 e^2}{125}\right ) x^4-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^3+1120 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^3-1800 e^2 x^3-160 e^4 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2+800 e^3 \log \left (4 x-4 e^{x+1}+10\right ) x^2-1000 e^2 x^2-400 e^4 x-400 e^4}{\left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right ) \left (4 x^4-8 e \log \left (4 x-4 e^{x+1}+10\right ) x^3+20 x^3+4 e^2 \log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-20 e \log \left (4 x-4 e^{x+1}+10\right ) x^2+25 x^2+20 e^2\right )^2 x}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 20 \int \frac {x \left (e^x-\log \left (4 x-4 e^{x+1}+10\right )\right ) \left (-2 e^{x+1} x+2 x+2 e^{2 x+1} (x+1)-e^x \left (2 x^2+7 x+5\right )-\left (-2 x+2 e^{x+1}-5\right ) \log \left (4 x-4 e^{x+1}+10\right )\right )}{\left (2 x-2 e^{x+1}+5\right ) \left (e^{2 x} x^2+\log ^2\left (4 x-4 e^{x+1}+10\right ) x^2-2 e^x \log \left (4 x-4 e^{x+1}+10\right ) x^2+5\right )^2}dx\)

Input:

Int[(-40*E^x*x^2 + E^(2*x)*(100*x + 140*x^2 + 40*x^3) + E^(1 + x)*(40*E^x* 
x^2 + E^(2*x)*(-40*x - 40*x^2)) + (40*x^2 + E^x*(-200*x - 180*x^2 - 40*x^3 
) + E^(1 + x)*(-40*x^2 + E^x*(80*x + 40*x^2)))*Log[10 - 4*E^(1 + x) + 4*x] 
 + (100*x - 40*E^(1 + x)*x + 40*x^2)*Log[10 - 4*E^(1 + x) + 4*x]^2)/(-125 
- 50*x + E^(2*x)*(-50*x^2 - 20*x^3) + E^(1 + x)*(50 + 20*E^(2*x)*x^2 + 2*E 
^(4*x)*x^4) + E^(4*x)*(-5*x^4 - 2*x^5) + (E^x*(100*x^2 + 40*x^3) + E^(1 + 
x)*(-40*E^x*x^2 - 8*E^(3*x)*x^4) + E^(3*x)*(20*x^4 + 8*x^5))*Log[10 - 4*E^ 
(1 + x) + 4*x] + (-50*x^2 - 20*x^3 + E^(1 + x)*(20*x^2 + 12*E^(2*x)*x^4) + 
 E^(2*x)*(-30*x^4 - 12*x^5))*Log[10 - 4*E^(1 + x) + 4*x]^2 + (-8*E^(1 + 2* 
x)*x^4 + E^x*(20*x^4 + 8*x^5))*Log[10 - 4*E^(1 + x) + 4*x]^3 + (-5*x^4 + 2 
*E^(1 + x)*x^4 - 2*x^5)*Log[10 - 4*E^(1 + x) + 4*x]^4),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 56.36 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.49

method	result	size
risch	\(\frac {10}{{\mathrm e}^{2 x} x^{2}-2 \,{\mathrm e}^{x} \ln \left (-4 \,{\mathrm e}^{1+x}+4 x +10\right ) x^{2}+\ln \left (-4 \,{\mathrm e}^{1+x}+4 x +10\right )^{2} x^{2}+5}\)	\(52\)
parallelrisch	\(\frac {10}{{\mathrm e}^{2 x} x^{2}-2 \,{\mathrm e}^{x} \ln \left (-4 \,{\mathrm e}^{1+x}+4 x +10\right ) x^{2}+\ln \left (-4 \,{\mathrm e}^{1+x}+4 x +10\right )^{2} x^{2}+5}\)	\(52\)

Input:

int(((-40*x*exp(1+x)+40*x^2+100*x)*ln(-4*exp(1+x)+4*x+10)^2+(((40*x^2+80*x 
)*exp(x)-40*x^2)*exp(1+x)+(-40*x^3-180*x^2-200*x)*exp(x)+40*x^2)*ln(-4*exp 
(1+x)+4*x+10)+((-40*x^2-40*x)*exp(x)^2+40*exp(x)*x^2)*exp(1+x)+(40*x^3+140 
*x^2+100*x)*exp(x)^2-40*exp(x)*x^2)/((2*x^4*exp(1+x)-2*x^5-5*x^4)*ln(-4*ex 
p(1+x)+4*x+10)^4+(-8*x^4*exp(x)*exp(1+x)+(8*x^5+20*x^4)*exp(x))*ln(-4*exp( 
1+x)+4*x+10)^3+((12*exp(x)^2*x^4+20*x^2)*exp(1+x)+(-12*x^5-30*x^4)*exp(x)^ 
2-20*x^3-50*x^2)*ln(-4*exp(1+x)+4*x+10)^2+((-8*x^4*exp(x)^3-40*exp(x)*x^2) 
*exp(1+x)+(8*x^5+20*x^4)*exp(x)^3+(40*x^3+100*x^2)*exp(x))*ln(-4*exp(1+x)+ 
4*x+10)+(2*x^4*exp(x)^4+20*exp(x)^2*x^2+50)*exp(1+x)+(-2*x^5-5*x^4)*exp(x) 
^4+(-20*x^3-50*x^2)*exp(x)^2-50*x-125),x,method=_RETURNVERBOSE)
 

Output:

10/(exp(2*x)*x^2-2*exp(x)*ln(-4*exp(1+x)+4*x+10)*x^2+ln(-4*exp(1+x)+4*x+10 
)^2*x^2+5)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (29) = 58\).

Time = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.77 \[ \int \frac {-40 e^x x^2+e^{2 x} \left (100 x+140 x^2+40 x^3\right )+e^{1+x} \left (40 e^x x^2+e^{2 x} \left (-40 x-40 x^2\right )\right )+\left (40 x^2+e^x \left (-200 x-180 x^2-40 x^3\right )+e^{1+x} \left (-40 x^2+e^x \left (80 x+40 x^2\right )\right )\right ) \log \left (10-4 e^{1+x}+4 x\right )+\left (100 x-40 e^{1+x} x+40 x^2\right ) \log ^2\left (10-4 e^{1+x}+4 x\right )}{-125-50 x+e^{2 x} \left (-50 x^2-20 x^3\right )+e^{1+x} \left (50+20 e^{2 x} x^2+2 e^{4 x} x^4\right )+e^{4 x} \left (-5 x^4-2 x^5\right )+\left (e^x \left (100 x^2+40 x^3\right )+e^{1+x} \left (-40 e^x x^2-8 e^{3 x} x^4\right )+e^{3 x} \left (20 x^4+8 x^5\right )\right ) \log \left (10-4 e^{1+x}+4 x\right )+\left (-50 x^2-20 x^3+e^{1+x} \left (20 x^2+12 e^{2 x} x^4\right )+e^{2 x} \left (-30 x^4-12 x^5\right )\right ) \log ^2\left (10-4 e^{1+x}+4 x\right )+\left (-8 e^{1+2 x} x^4+e^x \left (20 x^4+8 x^5\right )\right ) \log ^3\left (10-4 e^{1+x}+4 x\right )+\left (-5 x^4+2 e^{1+x} x^4-2 x^5\right ) \log ^4\left (10-4 e^{1+x}+4 x\right )} \, dx=\frac {10 \, e^{2}}{x^{2} e^{2} \log \left (4 \, x - 4 \, e^{\left (x + 1\right )} + 10\right )^{2} - 2 \, x^{2} e^{\left (x + 2\right )} \log \left (4 \, x - 4 \, e^{\left (x + 1\right )} + 10\right ) + x^{2} e^{\left (2 \, x + 2\right )} + 5 \, e^{2}} \] Input:

integrate(((-40*x*exp(1+x)+40*x^2+100*x)*log(-4*exp(1+x)+4*x+10)^2+(((40*x 
^2+80*x)*exp(x)-40*x^2)*exp(1+x)+(-40*x^3-180*x^2-200*x)*exp(x)+40*x^2)*lo 
g(-4*exp(1+x)+4*x+10)+((-40*x^2-40*x)*exp(x)^2+40*exp(x)*x^2)*exp(1+x)+(40 
*x^3+140*x^2+100*x)*exp(x)^2-40*exp(x)*x^2)/((2*x^4*exp(1+x)-2*x^5-5*x^4)* 
log(-4*exp(1+x)+4*x+10)^4+(-8*x^4*exp(x)*exp(1+x)+(8*x^5+20*x^4)*exp(x))*l 
og(-4*exp(1+x)+4*x+10)^3+((12*exp(x)^2*x^4+20*x^2)*exp(1+x)+(-12*x^5-30*x^ 
4)*exp(x)^2-20*x^3-50*x^2)*log(-4*exp(1+x)+4*x+10)^2+((-8*x^4*exp(x)^3-40* 
exp(x)*x^2)*exp(1+x)+(8*x^5+20*x^4)*exp(x)^3+(40*x^3+100*x^2)*exp(x))*log( 
-4*exp(1+x)+4*x+10)+(2*x^4*exp(x)^4+20*exp(x)^2*x^2+50)*exp(1+x)+(-2*x^5-5 
*x^4)*exp(x)^4+(-20*x^3-50*x^2)*exp(x)^2-50*x-125),x, algorithm="fricas")
 

Output:

10*e^2/(x^2*e^2*log(4*x - 4*e^(x + 1) + 10)^2 - 2*x^2*e^(x + 2)*log(4*x - 
4*e^(x + 1) + 10) + x^2*e^(2*x + 2) + 5*e^2)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (26) = 52\).

Time = 0.43 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.54 \[ \int \frac {-40 e^x x^2+e^{2 x} \left (100 x+140 x^2+40 x^3\right )+e^{1+x} \left (40 e^x x^2+e^{2 x} \left (-40 x-40 x^2\right )\right )+\left (40 x^2+e^x \left (-200 x-180 x^2-40 x^3\right )+e^{1+x} \left (-40 x^2+e^x \left (80 x+40 x^2\right )\right )\right ) \log \left (10-4 e^{1+x}+4 x\right )+\left (100 x-40 e^{1+x} x+40 x^2\right ) \log ^2\left (10-4 e^{1+x}+4 x\right )}{-125-50 x+e^{2 x} \left (-50 x^2-20 x^3\right )+e^{1+x} \left (50+20 e^{2 x} x^2+2 e^{4 x} x^4\right )+e^{4 x} \left (-5 x^4-2 x^5\right )+\left (e^x \left (100 x^2+40 x^3\right )+e^{1+x} \left (-40 e^x x^2-8 e^{3 x} x^4\right )+e^{3 x} \left (20 x^4+8 x^5\right )\right ) \log \left (10-4 e^{1+x}+4 x\right )+\left (-50 x^2-20 x^3+e^{1+x} \left (20 x^2+12 e^{2 x} x^4\right )+e^{2 x} \left (-30 x^4-12 x^5\right )\right ) \log ^2\left (10-4 e^{1+x}+4 x\right )+\left (-8 e^{1+2 x} x^4+e^x \left (20 x^4+8 x^5\right )\right ) \log ^3\left (10-4 e^{1+x}+4 x\right )+\left (-5 x^4+2 e^{1+x} x^4-2 x^5\right ) \log ^4\left (10-4 e^{1+x}+4 x\right )} \, dx=\frac {10}{x^{2} e^{2 x} - 2 x^{2} e^{x} \log {\left (4 x - 4 e e^{x} + 10 \right )} + x^{2} \log {\left (4 x - 4 e e^{x} + 10 \right )}^{2} + 5} \] Input:

integrate(((-40*x*exp(1+x)+40*x**2+100*x)*ln(-4*exp(1+x)+4*x+10)**2+(((40* 
x**2+80*x)*exp(x)-40*x**2)*exp(1+x)+(-40*x**3-180*x**2-200*x)*exp(x)+40*x* 
*2)*ln(-4*exp(1+x)+4*x+10)+((-40*x**2-40*x)*exp(x)**2+40*exp(x)*x**2)*exp( 
1+x)+(40*x**3+140*x**2+100*x)*exp(x)**2-40*exp(x)*x**2)/((2*x**4*exp(1+x)- 
2*x**5-5*x**4)*ln(-4*exp(1+x)+4*x+10)**4+(-8*x**4*exp(x)*exp(1+x)+(8*x**5+ 
20*x**4)*exp(x))*ln(-4*exp(1+x)+4*x+10)**3+((12*exp(x)**2*x**4+20*x**2)*ex 
p(1+x)+(-12*x**5-30*x**4)*exp(x)**2-20*x**3-50*x**2)*ln(-4*exp(1+x)+4*x+10 
)**2+((-8*x**4*exp(x)**3-40*exp(x)*x**2)*exp(1+x)+(8*x**5+20*x**4)*exp(x)* 
*3+(40*x**3+100*x**2)*exp(x))*ln(-4*exp(1+x)+4*x+10)+(2*x**4*exp(x)**4+20* 
exp(x)**2*x**2+50)*exp(1+x)+(-2*x**5-5*x**4)*exp(x)**4+(-20*x**3-50*x**2)* 
exp(x)**2-50*x-125),x)
 

Output:

10/(x**2*exp(2*x) - 2*x**2*exp(x)*log(4*x - 4*E*exp(x) + 10) + x**2*log(4* 
x - 4*E*exp(x) + 10)**2 + 5)
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.55 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.83 \[ \int \frac {-40 e^x x^2+e^{2 x} \left (100 x+140 x^2+40 x^3\right )+e^{1+x} \left (40 e^x x^2+e^{2 x} \left (-40 x-40 x^2\right )\right )+\left (40 x^2+e^x \left (-200 x-180 x^2-40 x^3\right )+e^{1+x} \left (-40 x^2+e^x \left (80 x+40 x^2\right )\right )\right ) \log \left (10-4 e^{1+x}+4 x\right )+\left (100 x-40 e^{1+x} x+40 x^2\right ) \log ^2\left (10-4 e^{1+x}+4 x\right )}{-125-50 x+e^{2 x} \left (-50 x^2-20 x^3\right )+e^{1+x} \left (50+20 e^{2 x} x^2+2 e^{4 x} x^4\right )+e^{4 x} \left (-5 x^4-2 x^5\right )+\left (e^x \left (100 x^2+40 x^3\right )+e^{1+x} \left (-40 e^x x^2-8 e^{3 x} x^4\right )+e^{3 x} \left (20 x^4+8 x^5\right )\right ) \log \left (10-4 e^{1+x}+4 x\right )+\left (-50 x^2-20 x^3+e^{1+x} \left (20 x^2+12 e^{2 x} x^4\right )+e^{2 x} \left (-30 x^4-12 x^5\right )\right ) \log ^2\left (10-4 e^{1+x}+4 x\right )+\left (-8 e^{1+2 x} x^4+e^x \left (20 x^4+8 x^5\right )\right ) \log ^3\left (10-4 e^{1+x}+4 x\right )+\left (-5 x^4+2 e^{1+x} x^4-2 x^5\right ) \log ^4\left (10-4 e^{1+x}+4 x\right )} \, dx=-\frac {10}{2 \, {\left (i \, \pi + \log \left (2\right )\right )} x^{2} e^{x} - x^{2} \log \left (-2 \, x + 2 \, e^{\left (x + 1\right )} - 5\right )^{2} + {\left (\pi ^{2} - 2 i \, \pi \log \left (2\right ) - \log \left (2\right )^{2}\right )} x^{2} - x^{2} e^{\left (2 \, x\right )} + 2 \, {\left ({\left (-i \, \pi - \log \left (2\right )\right )} x^{2} + x^{2} e^{x}\right )} \log \left (-2 \, x + 2 \, e^{\left (x + 1\right )} - 5\right ) - 5} \] Input:

integrate(((-40*x*exp(1+x)+40*x^2+100*x)*log(-4*exp(1+x)+4*x+10)^2+(((40*x 
^2+80*x)*exp(x)-40*x^2)*exp(1+x)+(-40*x^3-180*x^2-200*x)*exp(x)+40*x^2)*lo 
g(-4*exp(1+x)+4*x+10)+((-40*x^2-40*x)*exp(x)^2+40*exp(x)*x^2)*exp(1+x)+(40 
*x^3+140*x^2+100*x)*exp(x)^2-40*exp(x)*x^2)/((2*x^4*exp(1+x)-2*x^5-5*x^4)* 
log(-4*exp(1+x)+4*x+10)^4+(-8*x^4*exp(x)*exp(1+x)+(8*x^5+20*x^4)*exp(x))*l 
og(-4*exp(1+x)+4*x+10)^3+((12*exp(x)^2*x^4+20*x^2)*exp(1+x)+(-12*x^5-30*x^ 
4)*exp(x)^2-20*x^3-50*x^2)*log(-4*exp(1+x)+4*x+10)^2+((-8*x^4*exp(x)^3-40* 
exp(x)*x^2)*exp(1+x)+(8*x^5+20*x^4)*exp(x)^3+(40*x^3+100*x^2)*exp(x))*log( 
-4*exp(1+x)+4*x+10)+(2*x^4*exp(x)^4+20*exp(x)^2*x^2+50)*exp(1+x)+(-2*x^5-5 
*x^4)*exp(x)^4+(-20*x^3-50*x^2)*exp(x)^2-50*x-125),x, algorithm="maxima")
 

Output:

-10/(2*(I*pi + log(2))*x^2*e^x - x^2*log(-2*x + 2*e^(x + 1) - 5)^2 + (pi^2 
 - 2*I*pi*log(2) - log(2)^2)*x^2 - x^2*e^(2*x) + 2*((-I*pi - log(2))*x^2 + 
 x^2*e^x)*log(-2*x + 2*e^(x + 1) - 5) - 5)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (29) = 58\).

Time = 18.38 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.57 \[ \int \frac {-40 e^x x^2+e^{2 x} \left (100 x+140 x^2+40 x^3\right )+e^{1+x} \left (40 e^x x^2+e^{2 x} \left (-40 x-40 x^2\right )\right )+\left (40 x^2+e^x \left (-200 x-180 x^2-40 x^3\right )+e^{1+x} \left (-40 x^2+e^x \left (80 x+40 x^2\right )\right )\right ) \log \left (10-4 e^{1+x}+4 x\right )+\left (100 x-40 e^{1+x} x+40 x^2\right ) \log ^2\left (10-4 e^{1+x}+4 x\right )}{-125-50 x+e^{2 x} \left (-50 x^2-20 x^3\right )+e^{1+x} \left (50+20 e^{2 x} x^2+2 e^{4 x} x^4\right )+e^{4 x} \left (-5 x^4-2 x^5\right )+\left (e^x \left (100 x^2+40 x^3\right )+e^{1+x} \left (-40 e^x x^2-8 e^{3 x} x^4\right )+e^{3 x} \left (20 x^4+8 x^5\right )\right ) \log \left (10-4 e^{1+x}+4 x\right )+\left (-50 x^2-20 x^3+e^{1+x} \left (20 x^2+12 e^{2 x} x^4\right )+e^{2 x} \left (-30 x^4-12 x^5\right )\right ) \log ^2\left (10-4 e^{1+x}+4 x\right )+\left (-8 e^{1+2 x} x^4+e^x \left (20 x^4+8 x^5\right )\right ) \log ^3\left (10-4 e^{1+x}+4 x\right )+\left (-5 x^4+2 e^{1+x} x^4-2 x^5\right ) \log ^4\left (10-4 e^{1+x}+4 x\right )} \, dx=-\frac {20}{2 \, x^{2} e^{x} \log \left (2\right ) - x^{2} \log \left (2\right )^{2} + 2 \, x^{2} e^{x} \log \left (2 \, x - 2 \, e^{\left (x + 1\right )} + 5\right ) - 2 \, x^{2} \log \left (2\right ) \log \left (2 \, x - 2 \, e^{\left (x + 1\right )} + 5\right ) - x^{2} \log \left (2 \, x - 2 \, e^{\left (x + 1\right )} + 5\right )^{2} - x^{2} e^{\left (2 \, x\right )} - 5} \] Input:

integrate(((-40*x*exp(1+x)+40*x^2+100*x)*log(-4*exp(1+x)+4*x+10)^2+(((40*x 
^2+80*x)*exp(x)-40*x^2)*exp(1+x)+(-40*x^3-180*x^2-200*x)*exp(x)+40*x^2)*lo 
g(-4*exp(1+x)+4*x+10)+((-40*x^2-40*x)*exp(x)^2+40*exp(x)*x^2)*exp(1+x)+(40 
*x^3+140*x^2+100*x)*exp(x)^2-40*exp(x)*x^2)/((2*x^4*exp(1+x)-2*x^5-5*x^4)* 
log(-4*exp(1+x)+4*x+10)^4+(-8*x^4*exp(x)*exp(1+x)+(8*x^5+20*x^4)*exp(x))*l 
og(-4*exp(1+x)+4*x+10)^3+((12*exp(x)^2*x^4+20*x^2)*exp(1+x)+(-12*x^5-30*x^ 
4)*exp(x)^2-20*x^3-50*x^2)*log(-4*exp(1+x)+4*x+10)^2+((-8*x^4*exp(x)^3-40* 
exp(x)*x^2)*exp(1+x)+(8*x^5+20*x^4)*exp(x)^3+(40*x^3+100*x^2)*exp(x))*log( 
-4*exp(1+x)+4*x+10)+(2*x^4*exp(x)^4+20*exp(x)^2*x^2+50)*exp(1+x)+(-2*x^5-5 
*x^4)*exp(x)^4+(-20*x^3-50*x^2)*exp(x)^2-50*x-125),x, algorithm="giac")
 

Output:

-20/(2*x^2*e^x*log(2) - x^2*log(2)^2 + 2*x^2*e^x*log(2*x - 2*e^(x + 1) + 5 
) - 2*x^2*log(2)*log(2*x - 2*e^(x + 1) + 5) - x^2*log(2*x - 2*e^(x + 1) + 
5)^2 - x^2*e^(2*x) - 5)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-40 e^x x^2+e^{2 x} \left (100 x+140 x^2+40 x^3\right )+e^{1+x} \left (40 e^x x^2+e^{2 x} \left (-40 x-40 x^2\right )\right )+\left (40 x^2+e^x \left (-200 x-180 x^2-40 x^3\right )+e^{1+x} \left (-40 x^2+e^x \left (80 x+40 x^2\right )\right )\right ) \log \left (10-4 e^{1+x}+4 x\right )+\left (100 x-40 e^{1+x} x+40 x^2\right ) \log ^2\left (10-4 e^{1+x}+4 x\right )}{-125-50 x+e^{2 x} \left (-50 x^2-20 x^3\right )+e^{1+x} \left (50+20 e^{2 x} x^2+2 e^{4 x} x^4\right )+e^{4 x} \left (-5 x^4-2 x^5\right )+\left (e^x \left (100 x^2+40 x^3\right )+e^{1+x} \left (-40 e^x x^2-8 e^{3 x} x^4\right )+e^{3 x} \left (20 x^4+8 x^5\right )\right ) \log \left (10-4 e^{1+x}+4 x\right )+\left (-50 x^2-20 x^3+e^{1+x} \left (20 x^2+12 e^{2 x} x^4\right )+e^{2 x} \left (-30 x^4-12 x^5\right )\right ) \log ^2\left (10-4 e^{1+x}+4 x\right )+\left (-8 e^{1+2 x} x^4+e^x \left (20 x^4+8 x^5\right )\right ) \log ^3\left (10-4 e^{1+x}+4 x\right )+\left (-5 x^4+2 e^{1+x} x^4-2 x^5\right ) \log ^4\left (10-4 e^{1+x}+4 x\right )} \, dx=\int -\frac {{\mathrm {e}}^{2\,x}\,\left (40\,x^3+140\,x^2+100\,x\right )-{\mathrm {e}}^{x+1}\,\left ({\mathrm {e}}^{2\,x}\,\left (40\,x^2+40\,x\right )-40\,x^2\,{\mathrm {e}}^x\right )-40\,x^2\,{\mathrm {e}}^x+{\ln \left (4\,x-4\,{\mathrm {e}}^{x+1}+10\right )}^2\,\left (100\,x-40\,x\,{\mathrm {e}}^{x+1}+40\,x^2\right )+\ln \left (4\,x-4\,{\mathrm {e}}^{x+1}+10\right )\,\left (40\,x^2+{\mathrm {e}}^{x+1}\,\left ({\mathrm {e}}^x\,\left (40\,x^2+80\,x\right )-40\,x^2\right )-{\mathrm {e}}^x\,\left (40\,x^3+180\,x^2+200\,x\right )\right )}{\left (5\,x^4-2\,x^4\,{\mathrm {e}}^{x+1}+2\,x^5\right )\,{\ln \left (4\,x-4\,{\mathrm {e}}^{x+1}+10\right )}^4+\left (8\,x^4\,{\mathrm {e}}^{x+1}\,{\mathrm {e}}^x-{\mathrm {e}}^x\,\left (8\,x^5+20\,x^4\right )\right )\,{\ln \left (4\,x-4\,{\mathrm {e}}^{x+1}+10\right )}^3+\left ({\mathrm {e}}^{2\,x}\,\left (12\,x^5+30\,x^4\right )-{\mathrm {e}}^{x+1}\,\left (12\,x^4\,{\mathrm {e}}^{2\,x}+20\,x^2\right )+50\,x^2+20\,x^3\right )\,{\ln \left (4\,x-4\,{\mathrm {e}}^{x+1}+10\right )}^2+\left ({\mathrm {e}}^{x+1}\,\left (40\,x^2\,{\mathrm {e}}^x+8\,x^4\,{\mathrm {e}}^{3\,x}\right )-{\mathrm {e}}^{3\,x}\,\left (8\,x^5+20\,x^4\right )-{\mathrm {e}}^x\,\left (40\,x^3+100\,x^2\right )\right )\,\ln \left (4\,x-4\,{\mathrm {e}}^{x+1}+10\right )+50\,x+{\mathrm {e}}^{4\,x}\,\left (2\,x^5+5\,x^4\right )+{\mathrm {e}}^{2\,x}\,\left (20\,x^3+50\,x^2\right )-{\mathrm {e}}^{x+1}\,\left (20\,x^2\,{\mathrm {e}}^{2\,x}+2\,x^4\,{\mathrm {e}}^{4\,x}+50\right )+125} \,d x \] Input:

int(-(exp(2*x)*(100*x + 140*x^2 + 40*x^3) - exp(x + 1)*(exp(2*x)*(40*x + 4 
0*x^2) - 40*x^2*exp(x)) - 40*x^2*exp(x) + log(4*x - 4*exp(x + 1) + 10)^2*( 
100*x - 40*x*exp(x + 1) + 40*x^2) + log(4*x - 4*exp(x + 1) + 10)*(40*x^2 + 
 exp(x + 1)*(exp(x)*(80*x + 40*x^2) - 40*x^2) - exp(x)*(200*x + 180*x^2 + 
40*x^3)))/(50*x + log(4*x - 4*exp(x + 1) + 10)^4*(5*x^4 - 2*x^4*exp(x + 1) 
 + 2*x^5) + exp(4*x)*(5*x^4 + 2*x^5) + exp(2*x)*(50*x^2 + 20*x^3) - log(4* 
x - 4*exp(x + 1) + 10)^3*(exp(x)*(20*x^4 + 8*x^5) - 8*x^4*exp(x + 1)*exp(x 
)) + log(4*x - 4*exp(x + 1) + 10)^2*(exp(2*x)*(30*x^4 + 12*x^5) - exp(x + 
1)*(12*x^4*exp(2*x) + 20*x^2) + 50*x^2 + 20*x^3) - exp(x + 1)*(20*x^2*exp( 
2*x) + 2*x^4*exp(4*x) + 50) - log(4*x - 4*exp(x + 1) + 10)*(exp(x)*(100*x^ 
2 + 40*x^3) + exp(3*x)*(20*x^4 + 8*x^5) - exp(x + 1)*(40*x^2*exp(x) + 8*x^ 
4*exp(3*x))) + 125),x)
 

Output:

int(-(exp(2*x)*(100*x + 140*x^2 + 40*x^3) - exp(x + 1)*(exp(2*x)*(40*x + 4 
0*x^2) - 40*x^2*exp(x)) - 40*x^2*exp(x) + log(4*x - 4*exp(x + 1) + 10)^2*( 
100*x - 40*x*exp(x + 1) + 40*x^2) + log(4*x - 4*exp(x + 1) + 10)*(40*x^2 + 
 exp(x + 1)*(exp(x)*(80*x + 40*x^2) - 40*x^2) - exp(x)*(200*x + 180*x^2 + 
40*x^3)))/(50*x + log(4*x - 4*exp(x + 1) + 10)^4*(5*x^4 - 2*x^4*exp(x + 1) 
 + 2*x^5) + exp(4*x)*(5*x^4 + 2*x^5) + exp(2*x)*(50*x^2 + 20*x^3) - log(4* 
x - 4*exp(x + 1) + 10)^3*(exp(x)*(20*x^4 + 8*x^5) - 8*x^4*exp(x + 1)*exp(x 
)) + log(4*x - 4*exp(x + 1) + 10)^2*(exp(2*x)*(30*x^4 + 12*x^5) - exp(x + 
1)*(12*x^4*exp(2*x) + 20*x^2) + 50*x^2 + 20*x^3) - exp(x + 1)*(20*x^2*exp( 
2*x) + 2*x^4*exp(4*x) + 50) - log(4*x - 4*exp(x + 1) + 10)*(exp(x)*(100*x^ 
2 + 40*x^3) + exp(3*x)*(20*x^4 + 8*x^5) - exp(x + 1)*(40*x^2*exp(x) + 8*x^ 
4*exp(3*x))) + 125), x)
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.77 \[ \int \frac {-40 e^x x^2+e^{2 x} \left (100 x+140 x^2+40 x^3\right )+e^{1+x} \left (40 e^x x^2+e^{2 x} \left (-40 x-40 x^2\right )\right )+\left (40 x^2+e^x \left (-200 x-180 x^2-40 x^3\right )+e^{1+x} \left (-40 x^2+e^x \left (80 x+40 x^2\right )\right )\right ) \log \left (10-4 e^{1+x}+4 x\right )+\left (100 x-40 e^{1+x} x+40 x^2\right ) \log ^2\left (10-4 e^{1+x}+4 x\right )}{-125-50 x+e^{2 x} \left (-50 x^2-20 x^3\right )+e^{1+x} \left (50+20 e^{2 x} x^2+2 e^{4 x} x^4\right )+e^{4 x} \left (-5 x^4-2 x^5\right )+\left (e^x \left (100 x^2+40 x^3\right )+e^{1+x} \left (-40 e^x x^2-8 e^{3 x} x^4\right )+e^{3 x} \left (20 x^4+8 x^5\right )\right ) \log \left (10-4 e^{1+x}+4 x\right )+\left (-50 x^2-20 x^3+e^{1+x} \left (20 x^2+12 e^{2 x} x^4\right )+e^{2 x} \left (-30 x^4-12 x^5\right )\right ) \log ^2\left (10-4 e^{1+x}+4 x\right )+\left (-8 e^{1+2 x} x^4+e^x \left (20 x^4+8 x^5\right )\right ) \log ^3\left (10-4 e^{1+x}+4 x\right )+\left (-5 x^4+2 e^{1+x} x^4-2 x^5\right ) \log ^4\left (10-4 e^{1+x}+4 x\right )} \, dx=\frac {2 x^{2} \left (-e^{2 x}+2 e^{x} \mathrm {log}\left (-4 e^{x} e +4 x +10\right )-\mathrm {log}\left (-4 e^{x} e +4 x +10\right )^{2}\right )}{e^{2 x} x^{2}-2 e^{x} \mathrm {log}\left (-4 e^{x} e +4 x +10\right ) x^{2}+\mathrm {log}\left (-4 e^{x} e +4 x +10\right )^{2} x^{2}+5} \] Input:

int(((-40*x*exp(1+x)+40*x^2+100*x)*log(-4*exp(1+x)+4*x+10)^2+(((40*x^2+80* 
x)*exp(x)-40*x^2)*exp(1+x)+(-40*x^3-180*x^2-200*x)*exp(x)+40*x^2)*log(-4*e 
xp(1+x)+4*x+10)+((-40*x^2-40*x)*exp(x)^2+40*exp(x)*x^2)*exp(1+x)+(40*x^3+1 
40*x^2+100*x)*exp(x)^2-40*exp(x)*x^2)/((2*x^4*exp(1+x)-2*x^5-5*x^4)*log(-4 
*exp(1+x)+4*x+10)^4+(-8*x^4*exp(x)*exp(1+x)+(8*x^5+20*x^4)*exp(x))*log(-4* 
exp(1+x)+4*x+10)^3+((12*exp(x)^2*x^4+20*x^2)*exp(1+x)+(-12*x^5-30*x^4)*exp 
(x)^2-20*x^3-50*x^2)*log(-4*exp(1+x)+4*x+10)^2+((-8*x^4*exp(x)^3-40*exp(x) 
*x^2)*exp(1+x)+(8*x^5+20*x^4)*exp(x)^3+(40*x^3+100*x^2)*exp(x))*log(-4*exp 
(1+x)+4*x+10)+(2*x^4*exp(x)^4+20*exp(x)^2*x^2+50)*exp(1+x)+(-2*x^5-5*x^4)* 
exp(x)^4+(-20*x^3-50*x^2)*exp(x)^2-50*x-125),x)
 

Output:

(2*x**2*( - e**(2*x) + 2*e**x*log( - 4*e**x*e + 4*x + 10) - log( - 4*e**x* 
e + 4*x + 10)**2))/(e**(2*x)*x**2 - 2*e**x*log( - 4*e**x*e + 4*x + 10)*x** 
2 + log( - 4*e**x*e + 4*x + 10)**2*x**2 + 5)