\(\int \frac {-20+60 x^2-80 x^3+60 x^4+e^{4 x} (-20+80 x)+e^{3 x} (80-240 x+240 x^2)+e^{2 x} (-120+240 x-280 x^2+240 x^3)+e^x (80-80 x+80 x^4)+(20-80 x+180 x^2-160 x^3+100 x^4+e^{4 x} (20+80 x)+e^{3 x} (-80-80 x+240 x^2)+e^{2 x} (120-160 x-40 x^2+240 x^3)+e^x (-80+240 x-320 x^2+160 x^3+80 x^4)) \log (x)}{1+3 \log (x)+3 \log ^2(x)+\log ^3(x)} \, dx\) [2453]

Optimal result

Integrand size = 193, antiderivative size = 23 \[ \int \frac {-20+60 x^2-80 x^3+60 x^4+e^{4 x} (-20+80 x)+e^{3 x} \left (80-240 x+240 x^2\right )+e^{2 x} \left (-120+240 x-280 x^2+240 x^3\right )+e^x \left (80-80 x+80 x^4\right )+\left (20-80 x+180 x^2-160 x^3+100 x^4+e^{4 x} (20+80 x)+e^{3 x} \left (-80-80 x+240 x^2\right )+e^{2 x} \left (120-160 x-40 x^2+240 x^3\right )+e^x \left (-80+240 x-320 x^2+160 x^3+80 x^4\right )\right ) \log (x)}{1+3 \log (x)+3 \log ^2(x)+\log ^3(x)} \, dx=5+\frac {20 x \left (x+\left (-1+e^x+x\right )^2\right )^2}{(1+\log (x))^2} \] Output:

5+20*((x+exp(x)-1)^2+x)^2/(1+ln(x))^2*x
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {-20+60 x^2-80 x^3+60 x^4+e^{4 x} (-20+80 x)+e^{3 x} \left (80-240 x+240 x^2\right )+e^{2 x} \left (-120+240 x-280 x^2+240 x^3\right )+e^x \left (80-80 x+80 x^4\right )+\left (20-80 x+180 x^2-160 x^3+100 x^4+e^{4 x} (20+80 x)+e^{3 x} \left (-80-80 x+240 x^2\right )+e^{2 x} \left (120-160 x-40 x^2+240 x^3\right )+e^x \left (-80+240 x-320 x^2+160 x^3+80 x^4\right )\right ) \log (x)}{1+3 \log (x)+3 \log ^2(x)+\log ^3(x)} \, dx=\frac {20 x \left (1+e^{2 x}+2 e^x (-1+x)-x+x^2\right )^2}{(1+\log (x))^2} \] Input:

Integrate[(-20 + 60*x^2 - 80*x^3 + 60*x^4 + E^(4*x)*(-20 + 80*x) + E^(3*x) 
*(80 - 240*x + 240*x^2) + E^(2*x)*(-120 + 240*x - 280*x^2 + 240*x^3) + E^x 
*(80 - 80*x + 80*x^4) + (20 - 80*x + 180*x^2 - 160*x^3 + 100*x^4 + E^(4*x) 
*(20 + 80*x) + E^(3*x)*(-80 - 80*x + 240*x^2) + E^(2*x)*(120 - 160*x - 40* 
x^2 + 240*x^3) + E^x*(-80 + 240*x - 320*x^2 + 160*x^3 + 80*x^4))*Log[x])/( 
1 + 3*Log[x] + 3*Log[x]^2 + Log[x]^3),x]
 

Output:

(20*x*(1 + E^(2*x) + 2*E^x*(-1 + x) - x + x^2)^2)/(1 + Log[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.

\(\displaystyle \int \frac {60 x^4+e^x \left (80 x^4-80 x+80\right )-80 x^3+60 x^2+e^{3 x} \left (240 x^2-240 x+80\right )+e^{2 x} \left (240 x^3-280 x^2+240 x-120\right )+\left (100 x^4-160 x^3+180 x^2+e^{3 x} \left (240 x^2-80 x-80\right )+e^{2 x} \left (240 x^3-40 x^2-160 x+120\right )+e^x \left (80 x^4+160 x^3-320 x^2+240 x-80\right )-80 x+e^{4 x} (80 x+20)+20\right ) \log (x)+e^{4 x} (80 x-20)-20}{\log ^3(x)+3 \log ^2(x)+3 \log (x)+1} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {20 \left (x^2-x+e^{2 x}+2 e^x (x-1)+1\right ) \left (3 x^2+e^x \left (4 x^2-2 x+2\right )+\left (5 x^2+2 e^x \left (2 x^2+x-1\right )-3 x+e^{2 x} (4 x+1)+1\right ) \log (x)-x+e^{2 x} (4 x-1)-1\right )}{(\log (x)+1)^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 20 \int -\frac {\left (x^2-x+e^{2 x}-2 e^x (1-x)+1\right ) \left (-3 x^2+x+e^{2 x} (1-4 x)-2 e^x \left (2 x^2-x+1\right )-\left (5 x^2-3 x+e^{2 x} (4 x+1)-2 e^x \left (-2 x^2-x+1\right )+1\right ) \log (x)+1\right )}{(\log (x)+1)^3}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -20 \int \frac {\left (x^2-x+e^{2 x}-2 e^x (1-x)+1\right ) \left (-3 x^2+x+e^{2 x} (1-4 x)-2 e^x \left (2 x^2-x+1\right )-\left (5 x^2-3 x+e^{2 x} (4 x+1)-2 e^x \left (-2 x^2-x+1\right )+1\right ) \log (x)+1\right )}{(\log (x)+1)^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -20 \int \left (-\frac {5 \log (x) x^4}{(\log (x)+1)^3}-\frac {3 x^4}{(\log (x)+1)^3}+\frac {8 \log (x) x^3}{(\log (x)+1)^3}+\frac {4 x^3}{(\log (x)+1)^3}-\frac {9 \log (x) x^2}{(\log (x)+1)^3}-\frac {3 x^2}{(\log (x)+1)^3}+\frac {4 \log (x) x}{(\log (x)+1)^3}-\frac {e^{4 x} (4 \log (x) x+4 x+\log (x)-1)}{(\log (x)+1)^3}-\frac {4 e^{3 x} \left (3 \log (x) x^2+3 x^2-\log (x) x-3 x-\log (x)+1\right )}{(\log (x)+1)^3}-\frac {2 e^{2 x} \left (6 \log (x) x^3+6 x^3-\log (x) x^2-7 x^2-4 \log (x) x+6 x+3 \log (x)-3\right )}{(\log (x)+1)^3}-\frac {4 e^x \left (\log (x) x^4+x^4+2 \log (x) x^3-4 \log (x) x^2+3 \log (x) x-x-\log (x)+1\right )}{(\log (x)+1)^3}-\frac {\log (x)}{(\log (x)+1)^3}+\frac {1}{(\log (x)+1)^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -20 \left (\frac {5 (5 \log (x)+6) x^5}{2 (\log (x)+1)}+\frac {25 \log (x) x^5}{2 (\log (x)+1)}+\frac {15 x^5}{2 (\log (x)+1)}+\frac {5 \log (x) x^5}{2 (\log (x)+1)^2}+\frac {3 x^5}{2 (\log (x)+1)^2}-25 x^5-\frac {4 (4 \log (x)+5) x^4}{\log (x)+1}-\frac {16 \log (x) x^4}{\log (x)+1}-\frac {8 x^4}{\log (x)+1}-\frac {4 \log (x) x^4}{(\log (x)+1)^2}-\frac {2 x^4}{(\log (x)+1)^2}+32 x^4+\frac {9 (3 \log (x)+4) x^3}{2 (\log (x)+1)}+\frac {27 \log (x) x^3}{2 (\log (x)+1)}+\frac {9 x^3}{2 (\log (x)+1)}+\frac {9 \log (x) x^3}{2 (\log (x)+1)^2}+\frac {3 x^3}{2 (\log (x)+1)^2}-27 x^3-\frac {2 (2 \log (x)+3) x^2}{\log (x)+1}-\frac {4 \log (x) x^2}{\log (x)+1}-\frac {2 \log (x) x^2}{(\log (x)+1)^2}+8 x^2-\frac {x}{(\log (x)+1)^2}+\frac {4 \operatorname {ExpIntegralEi}(2 (\log (x)+1))}{e^2}-\frac {27 \operatorname {ExpIntegralEi}(3 (\log (x)+1))}{e^3}+\frac {48 \operatorname {ExpIntegralEi}(4 (\log (x)+1))}{e^4}-\frac {50 \operatorname {ExpIntegralEi}(5 (\log (x)+1))}{e^5}+\frac {8 \operatorname {ExpIntegralEi}(2 (\log (x)+1)) \log (x)}{e^2}-\frac {81 \operatorname {ExpIntegralEi}(3 (\log (x)+1)) \log (x)}{2 e^3}+\frac {64 \operatorname {ExpIntegralEi}(4 (\log (x)+1)) \log (x)}{e^4}-\frac {125 \operatorname {ExpIntegralEi}(5 (\log (x)+1)) \log (x)}{2 e^5}-\frac {16 \operatorname {ExpIntegralEi}(2 (\log (x)+1)) (\log (x)+1)}{e^2}+\frac {81 \operatorname {ExpIntegralEi}(3 (\log (x)+1)) (\log (x)+1)}{e^3}-\frac {128 \operatorname {ExpIntegralEi}(4 (\log (x)+1)) (\log (x)+1)}{e^4}+\frac {125 \operatorname {ExpIntegralEi}(5 (\log (x)+1)) (\log (x)+1)}{e^5}+\frac {4 \operatorname {ExpIntegralEi}(2 (\log (x)+1)) (2 \log (x)+3)}{e^2}-\frac {27 \operatorname {ExpIntegralEi}(3 (\log (x)+1)) (3 \log (x)+4)}{2 e^3}+\frac {16 \operatorname {ExpIntegralEi}(4 (\log (x)+1)) (4 \log (x)+5)}{e^4}-\frac {25 \operatorname {ExpIntegralEi}(5 (\log (x)+1)) (5 \log (x)+6)}{2 e^5}-\frac {e^{4 x} (\log (x) x+x)}{(\log (x)+1)^3}-8 \int \frac {e^x}{(\log (x)+1)^3}dx+12 \int \frac {e^{2 x}}{(\log (x)+1)^3}dx-8 \int \frac {e^{3 x}}{(\log (x)+1)^3}dx+16 \int \frac {e^x x}{(\log (x)+1)^3}dx-20 \int \frac {e^{2 x} x}{(\log (x)+1)^3}dx+8 \int \frac {e^{3 x} x}{(\log (x)+1)^3}dx-16 \int \frac {e^x x^2}{(\log (x)+1)^3}dx+12 \int \frac {e^{2 x} x^2}{(\log (x)+1)^3}dx+8 \int \frac {e^x x^3}{(\log (x)+1)^3}dx+4 \int \frac {e^x}{(\log (x)+1)^2}dx-6 \int \frac {e^{2 x}}{(\log (x)+1)^2}dx+4 \int \frac {e^{3 x}}{(\log (x)+1)^2}dx-12 \int \frac {e^x x}{(\log (x)+1)^2}dx+8 \int \frac {e^{2 x} x}{(\log (x)+1)^2}dx+4 \int \frac {e^{3 x} x}{(\log (x)+1)^2}dx+16 \int \frac {e^x x^2}{(\log (x)+1)^2}dx+2 \int \frac {e^{2 x} x^2}{(\log (x)+1)^2}dx-12 \int \frac {e^{3 x} x^2}{(\log (x)+1)^2}dx-8 \int \frac {e^x x^3}{(\log (x)+1)^2}dx-12 \int \frac {e^{2 x} x^3}{(\log (x)+1)^2}dx-4 \int \frac {e^x x^4}{(\log (x)+1)^2}dx\right )\)

Input:

Int[(-20 + 60*x^2 - 80*x^3 + 60*x^4 + E^(4*x)*(-20 + 80*x) + E^(3*x)*(80 - 
 240*x + 240*x^2) + E^(2*x)*(-120 + 240*x - 280*x^2 + 240*x^3) + E^x*(80 - 
 80*x + 80*x^4) + (20 - 80*x + 180*x^2 - 160*x^3 + 100*x^4 + E^(4*x)*(20 + 
 80*x) + E^(3*x)*(-80 - 80*x + 240*x^2) + E^(2*x)*(120 - 160*x - 40*x^2 + 
240*x^3) + E^x*(-80 + 240*x - 320*x^2 + 160*x^3 + 80*x^4))*Log[x])/(1 + 3* 
Log[x] + 3*Log[x]^2 + Log[x]^3),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(89\) vs. \(2(22)=44\).

Time = 0.47 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.91

\[\frac {20 x \left (x^{4}+4 \,{\mathrm e}^{x} x^{3}+6 \,{\mathrm e}^{2 x} x^{2}+4 x \,{\mathrm e}^{3 x}+{\mathrm e}^{4 x}-2 x^{3}-8 \,{\mathrm e}^{x} x^{2}-10 x \,{\mathrm e}^{2 x}-4 \,{\mathrm e}^{3 x}+3 x^{2}+8 \,{\mathrm e}^{x} x +6 \,{\mathrm e}^{2 x}-2 x -4 \,{\mathrm e}^{x}+1\right )}{\left (\ln \left (x \right )+1\right )^{2}}\]

Input:

int((((80*x+20)*exp(x)^4+(240*x^2-80*x-80)*exp(x)^3+(240*x^3-40*x^2-160*x+ 
120)*exp(x)^2+(80*x^4+160*x^3-320*x^2+240*x-80)*exp(x)+100*x^4-160*x^3+180 
*x^2-80*x+20)*ln(x)+(80*x-20)*exp(x)^4+(240*x^2-240*x+80)*exp(x)^3+(240*x^ 
3-280*x^2+240*x-120)*exp(x)^2+(80*x^4-80*x+80)*exp(x)+60*x^4-80*x^3+60*x^2 
-20)/(ln(x)^3+3*ln(x)^2+3*ln(x)+1),x)
 

Output:

20*x*(x^4+4*exp(x)*x^3+6*exp(x)^2*x^2+4*x*exp(x)^3+exp(x)^4-2*x^3-8*exp(x) 
*x^2-10*x*exp(x)^2-4*exp(x)^3+3*x^2+8*exp(x)*x+6*exp(x)^2-2*x-4*exp(x)+1)/ 
(ln(x)+1)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (22) = 44\).

Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 4.09 \[ \int \frac {-20+60 x^2-80 x^3+60 x^4+e^{4 x} (-20+80 x)+e^{3 x} \left (80-240 x+240 x^2\right )+e^{2 x} \left (-120+240 x-280 x^2+240 x^3\right )+e^x \left (80-80 x+80 x^4\right )+\left (20-80 x+180 x^2-160 x^3+100 x^4+e^{4 x} (20+80 x)+e^{3 x} \left (-80-80 x+240 x^2\right )+e^{2 x} \left (120-160 x-40 x^2+240 x^3\right )+e^x \left (-80+240 x-320 x^2+160 x^3+80 x^4\right )\right ) \log (x)}{1+3 \log (x)+3 \log ^2(x)+\log ^3(x)} \, dx=\frac {20 \, {\left (x^{5} - 2 \, x^{4} + 3 \, x^{3} - 2 \, x^{2} + x e^{\left (4 \, x\right )} + 4 \, {\left (x^{2} - x\right )} e^{\left (3 \, x\right )} + 2 \, {\left (3 \, x^{3} - 5 \, x^{2} + 3 \, x\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{4} - 2 \, x^{3} + 2 \, x^{2} - x\right )} e^{x} + x\right )}}{\log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1} \] Input:

integrate((((80*x+20)*exp(x)^4+(240*x^2-80*x-80)*exp(x)^3+(240*x^3-40*x^2- 
160*x+120)*exp(x)^2+(80*x^4+160*x^3-320*x^2+240*x-80)*exp(x)+100*x^4-160*x 
^3+180*x^2-80*x+20)*log(x)+(80*x-20)*exp(x)^4+(240*x^2-240*x+80)*exp(x)^3+ 
(240*x^3-280*x^2+240*x-120)*exp(x)^2+(80*x^4-80*x+80)*exp(x)+60*x^4-80*x^3 
+60*x^2-20)/(log(x)^3+3*log(x)^2+3*log(x)+1),x, algorithm="fricas")
 

Output:

20*(x^5 - 2*x^4 + 3*x^3 - 2*x^2 + x*e^(4*x) + 4*(x^2 - x)*e^(3*x) + 2*(3*x 
^3 - 5*x^2 + 3*x)*e^(2*x) + 4*(x^4 - 2*x^3 + 2*x^2 - x)*e^x + x)/(log(x)^2 
 + 2*log(x) + 1)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 704 vs. \(2 (22) = 44\).

Time = 0.34 (sec) , antiderivative size = 704, normalized size of antiderivative = 30.61 \[ \int \frac {-20+60 x^2-80 x^3+60 x^4+e^{4 x} (-20+80 x)+e^{3 x} \left (80-240 x+240 x^2\right )+e^{2 x} \left (-120+240 x-280 x^2+240 x^3\right )+e^x \left (80-80 x+80 x^4\right )+\left (20-80 x+180 x^2-160 x^3+100 x^4+e^{4 x} (20+80 x)+e^{3 x} \left (-80-80 x+240 x^2\right )+e^{2 x} \left (120-160 x-40 x^2+240 x^3\right )+e^x \left (-80+240 x-320 x^2+160 x^3+80 x^4\right )\right ) \log (x)}{1+3 \log (x)+3 \log ^2(x)+\log ^3(x)} \, dx =\text {Too large to display} \] Input:

integrate((((80*x+20)*exp(x)**4+(240*x**2-80*x-80)*exp(x)**3+(240*x**3-40* 
x**2-160*x+120)*exp(x)**2+(80*x**4+160*x**3-320*x**2+240*x-80)*exp(x)+100* 
x**4-160*x**3+180*x**2-80*x+20)*ln(x)+(80*x-20)*exp(x)**4+(240*x**2-240*x+ 
80)*exp(x)**3+(240*x**3-280*x**2+240*x-120)*exp(x)**2+(80*x**4-80*x+80)*ex 
p(x)+60*x**4-80*x**3+60*x**2-20)/(ln(x)**3+3*ln(x)**2+3*ln(x)+1),x)
 

Output:

((20*x*log(x)**6 + 120*x*log(x)**5 + 300*x*log(x)**4 + 400*x*log(x)**3 + 3 
00*x*log(x)**2 + 120*x*log(x) + 20*x)*exp(4*x) + (80*x**2*log(x)**6 + 480* 
x**2*log(x)**5 + 1200*x**2*log(x)**4 + 1600*x**2*log(x)**3 + 1200*x**2*log 
(x)**2 + 480*x**2*log(x) + 80*x**2 - 80*x*log(x)**6 - 480*x*log(x)**5 - 12 
00*x*log(x)**4 - 1600*x*log(x)**3 - 1200*x*log(x)**2 - 480*x*log(x) - 80*x 
)*exp(3*x) + (120*x**3*log(x)**6 + 720*x**3*log(x)**5 + 1800*x**3*log(x)** 
4 + 2400*x**3*log(x)**3 + 1800*x**3*log(x)**2 + 720*x**3*log(x) + 120*x**3 
 - 200*x**2*log(x)**6 - 1200*x**2*log(x)**5 - 3000*x**2*log(x)**4 - 4000*x 
**2*log(x)**3 - 3000*x**2*log(x)**2 - 1200*x**2*log(x) - 200*x**2 + 120*x* 
log(x)**6 + 720*x*log(x)**5 + 1800*x*log(x)**4 + 2400*x*log(x)**3 + 1800*x 
*log(x)**2 + 720*x*log(x) + 120*x)*exp(2*x) + (80*x**4*log(x)**6 + 480*x** 
4*log(x)**5 + 1200*x**4*log(x)**4 + 1600*x**4*log(x)**3 + 1200*x**4*log(x) 
**2 + 480*x**4*log(x) + 80*x**4 - 160*x**3*log(x)**6 - 960*x**3*log(x)**5 
- 2400*x**3*log(x)**4 - 3200*x**3*log(x)**3 - 2400*x**3*log(x)**2 - 960*x* 
*3*log(x) - 160*x**3 + 160*x**2*log(x)**6 + 960*x**2*log(x)**5 + 2400*x**2 
*log(x)**4 + 3200*x**2*log(x)**3 + 2400*x**2*log(x)**2 + 960*x**2*log(x) + 
 160*x**2 - 80*x*log(x)**6 - 480*x*log(x)**5 - 1200*x*log(x)**4 - 1600*x*l 
og(x)**3 - 1200*x*log(x)**2 - 480*x*log(x) - 80*x)*exp(x))/(log(x)**8 + 8* 
log(x)**7 + 28*log(x)**6 + 56*log(x)**5 + 70*log(x)**4 + 56*log(x)**3 + 28 
*log(x)**2 + 8*log(x) + 1) + (20*x**5 - 40*x**4 + 60*x**3 - 40*x**2 + 2...
 

Maxima [F]

\[ \int \frac {-20+60 x^2-80 x^3+60 x^4+e^{4 x} (-20+80 x)+e^{3 x} \left (80-240 x+240 x^2\right )+e^{2 x} \left (-120+240 x-280 x^2+240 x^3\right )+e^x \left (80-80 x+80 x^4\right )+\left (20-80 x+180 x^2-160 x^3+100 x^4+e^{4 x} (20+80 x)+e^{3 x} \left (-80-80 x+240 x^2\right )+e^{2 x} \left (120-160 x-40 x^2+240 x^3\right )+e^x \left (-80+240 x-320 x^2+160 x^3+80 x^4\right )\right ) \log (x)}{1+3 \log (x)+3 \log ^2(x)+\log ^3(x)} \, dx=\int { \frac {20 \, {\left (3 \, x^{4} - 4 \, x^{3} + 3 \, x^{2} + {\left (4 \, x - 1\right )} e^{\left (4 \, x\right )} + 4 \, {\left (3 \, x^{2} - 3 \, x + 1\right )} e^{\left (3 \, x\right )} + 2 \, {\left (6 \, x^{3} - 7 \, x^{2} + 6 \, x - 3\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{4} - x + 1\right )} e^{x} + {\left (5 \, x^{4} - 8 \, x^{3} + 9 \, x^{2} + {\left (4 \, x + 1\right )} e^{\left (4 \, x\right )} + 4 \, {\left (3 \, x^{2} - x - 1\right )} e^{\left (3 \, x\right )} + 2 \, {\left (6 \, x^{3} - x^{2} - 4 \, x + 3\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{4} + 2 \, x^{3} - 4 \, x^{2} + 3 \, x - 1\right )} e^{x} - 4 \, x + 1\right )} \log \left (x\right ) - 1\right )}}{\log \left (x\right )^{3} + 3 \, \log \left (x\right )^{2} + 3 \, \log \left (x\right ) + 1} \,d x } \] Input:

integrate((((80*x+20)*exp(x)^4+(240*x^2-80*x-80)*exp(x)^3+(240*x^3-40*x^2- 
160*x+120)*exp(x)^2+(80*x^4+160*x^3-320*x^2+240*x-80)*exp(x)+100*x^4-160*x 
^3+180*x^2-80*x+20)*log(x)+(80*x-20)*exp(x)^4+(240*x^2-240*x+80)*exp(x)^3+ 
(240*x^3-280*x^2+240*x-120)*exp(x)^2+(80*x^4-80*x+80)*exp(x)+60*x^4-80*x^3 
+60*x^2-20)/(log(x)^3+3*log(x)^2+3*log(x)+1),x, algorithm="maxima")
 

Output:

10*(20*x^5 - 24*x^4 + 18*x^3 - 4*x^2 + 2*x*e^(4*x) + 8*(x^2 - x)*e^(3*x) + 
 4*(3*x^3 - 5*x^2 + 3*x)*e^(2*x) + 8*(x^4 - 2*x^3 + 2*x^2 - x)*e^x + (15*x 
^5 - 16*x^4 + 9*x^3 - x)*log(x))/(log(x)^2 + 2*log(x) + 1) + 20*e^(-1)*exp 
_integral_e(3, -log(x) - 1)/(log(x) + 1)^2 - 60*e^(-3)*exp_integral_e(3, - 
3*log(x) - 3)/(log(x) + 1)^2 + 80*e^(-4)*exp_integral_e(3, -4*log(x) - 4)/ 
(log(x) + 1)^2 - 60*e^(-5)*exp_integral_e(3, -5*log(x) - 5)/(log(x) + 1)^2 
 - 20*integrate(1/2*(75*x^4 - 64*x^3 + 27*x^2 - 1)/(log(x) + 1), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (22) = 44\).

Time = 0.13 (sec) , antiderivative size = 107, normalized size of antiderivative = 4.65 \[ \int \frac {-20+60 x^2-80 x^3+60 x^4+e^{4 x} (-20+80 x)+e^{3 x} \left (80-240 x+240 x^2\right )+e^{2 x} \left (-120+240 x-280 x^2+240 x^3\right )+e^x \left (80-80 x+80 x^4\right )+\left (20-80 x+180 x^2-160 x^3+100 x^4+e^{4 x} (20+80 x)+e^{3 x} \left (-80-80 x+240 x^2\right )+e^{2 x} \left (120-160 x-40 x^2+240 x^3\right )+e^x \left (-80+240 x-320 x^2+160 x^3+80 x^4\right )\right ) \log (x)}{1+3 \log (x)+3 \log ^2(x)+\log ^3(x)} \, dx=\frac {20 \, {\left (x^{5} + 4 \, x^{4} e^{x} - 2 \, x^{4} + 6 \, x^{3} e^{\left (2 \, x\right )} - 8 \, x^{3} e^{x} + 3 \, x^{3} + 4 \, x^{2} e^{\left (3 \, x\right )} - 10 \, x^{2} e^{\left (2 \, x\right )} + 8 \, x^{2} e^{x} - 2 \, x^{2} + x e^{\left (4 \, x\right )} - 4 \, x e^{\left (3 \, x\right )} + 6 \, x e^{\left (2 \, x\right )} - 4 \, x e^{x} + x\right )}}{\log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1} \] Input:

integrate((((80*x+20)*exp(x)^4+(240*x^2-80*x-80)*exp(x)^3+(240*x^3-40*x^2- 
160*x+120)*exp(x)^2+(80*x^4+160*x^3-320*x^2+240*x-80)*exp(x)+100*x^4-160*x 
^3+180*x^2-80*x+20)*log(x)+(80*x-20)*exp(x)^4+(240*x^2-240*x+80)*exp(x)^3+ 
(240*x^3-280*x^2+240*x-120)*exp(x)^2+(80*x^4-80*x+80)*exp(x)+60*x^4-80*x^3 
+60*x^2-20)/(log(x)^3+3*log(x)^2+3*log(x)+1),x, algorithm="giac")
 

Output:

20*(x^5 + 4*x^4*e^x - 2*x^4 + 6*x^3*e^(2*x) - 8*x^3*e^x + 3*x^3 + 4*x^2*e^ 
(3*x) - 10*x^2*e^(2*x) + 8*x^2*e^x - 2*x^2 + x*e^(4*x) - 4*x*e^(3*x) + 6*x 
*e^(2*x) - 4*x*e^x + x)/(log(x)^2 + 2*log(x) + 1)
 

Mupad [B] (verification not implemented)

Time = 2.20 (sec) , antiderivative size = 657, normalized size of antiderivative = 28.57 \[ \int \frac {-20+60 x^2-80 x^3+60 x^4+e^{4 x} (-20+80 x)+e^{3 x} \left (80-240 x+240 x^2\right )+e^{2 x} \left (-120+240 x-280 x^2+240 x^3\right )+e^x \left (80-80 x+80 x^4\right )+\left (20-80 x+180 x^2-160 x^3+100 x^4+e^{4 x} (20+80 x)+e^{3 x} \left (-80-80 x+240 x^2\right )+e^{2 x} \left (120-160 x-40 x^2+240 x^3\right )+e^x \left (-80+240 x-320 x^2+160 x^3+80 x^4\right )\right ) \log (x)}{1+3 \log (x)+3 \log ^2(x)+\log ^3(x)} \, dx =\text {Too large to display} \] Input:

int((exp(3*x)*(240*x^2 - 240*x + 80) + exp(2*x)*(240*x - 280*x^2 + 240*x^3 
 - 120) + log(x)*(exp(x)*(240*x - 320*x^2 + 160*x^3 + 80*x^4 - 80) - exp(3 
*x)*(80*x - 240*x^2 + 80) - 80*x - exp(2*x)*(160*x + 40*x^2 - 240*x^3 - 12 
0) + exp(4*x)*(80*x + 20) + 180*x^2 - 160*x^3 + 100*x^4 + 20) + exp(x)*(80 
*x^4 - 80*x + 80) + exp(4*x)*(80*x - 20) + 60*x^2 - 80*x^3 + 60*x^4 - 20)/ 
(3*log(x) + 3*log(x)^2 + log(x)^3 + 1),x)
 

Output:

10*x - (10*x*(4*exp(3*x) - 6*exp(2*x) - exp(4*x) + 4*exp(x) + 12*x*exp(2*x 
) - 12*x*exp(3*x) + 4*x*exp(4*x) + 4*x^4*exp(x) - 14*x^2*exp(2*x) + 12*x^2 
*exp(3*x) + 12*x^3*exp(2*x) - 4*x*exp(x) + 3*x^2 - 4*x^3 + 3*x^4 - 1) + 10 
*x*log(x)*(6*exp(2*x) - 4*x - 4*exp(3*x) + exp(4*x) - 4*exp(x) - 8*x*exp(2 
*x) - 4*x*exp(3*x) + 4*x*exp(4*x) - 16*x^2*exp(x) + 8*x^3*exp(x) + 4*x^4*e 
xp(x) - 2*x^2*exp(2*x) + 12*x^2*exp(3*x) + 12*x^3*exp(2*x) + 12*x*exp(x) + 
 9*x^2 - 8*x^3 + 5*x^4 + 1))/(2*log(x) + log(x)^2 + 1) + exp(4*x)*(10*x + 
120*x^2 + 160*x^3) - (20*x*(2*x*exp(2*x) - 2*x - 8*x*exp(3*x) + 4*x*exp(4* 
x) - 10*x^2*exp(x) + 4*x^3*exp(x) + 12*x^4*exp(x) + 2*x^5*exp(x) - 10*x^2* 
exp(2*x) + 6*x^2*exp(3*x) + 16*x^3*exp(2*x) + 8*x^2*exp(4*x) + 18*x^3*exp( 
3*x) + 12*x^4*exp(2*x) + 4*x*exp(x) + 9*x^2 - 12*x^3 + 10*x^4) + 10*x*log( 
x)*(6*exp(2*x) - 8*x - 4*exp(3*x) + exp(4*x) - 4*exp(x) - 4*x*exp(2*x) - 2 
0*x*exp(3*x) + 12*x*exp(4*x) - 36*x^2*exp(x) + 16*x^3*exp(x) + 28*x^4*exp( 
x) + 4*x^5*exp(x) - 22*x^2*exp(2*x) + 24*x^2*exp(3*x) + 44*x^3*exp(2*x) + 
16*x^2*exp(4*x) + 36*x^3*exp(3*x) + 24*x^4*exp(2*x) + 20*x*exp(x) + 27*x^2 
 - 32*x^3 + 25*x^4 + 1))/(log(x) + 1) - exp(3*x)*(40*x + 200*x^2 - 240*x^3 
 - 360*x^4) + exp(x)*(200*x^2 - 40*x - 360*x^3 + 160*x^4 + 280*x^5 + 40*x^ 
6) + exp(2*x)*(60*x - 40*x^2 - 220*x^3 + 440*x^4 + 240*x^5) - 80*x^2 + 270 
*x^3 - 320*x^4 + 250*x^5
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 105, normalized size of antiderivative = 4.57 \[ \int \frac {-20+60 x^2-80 x^3+60 x^4+e^{4 x} (-20+80 x)+e^{3 x} \left (80-240 x+240 x^2\right )+e^{2 x} \left (-120+240 x-280 x^2+240 x^3\right )+e^x \left (80-80 x+80 x^4\right )+\left (20-80 x+180 x^2-160 x^3+100 x^4+e^{4 x} (20+80 x)+e^{3 x} \left (-80-80 x+240 x^2\right )+e^{2 x} \left (120-160 x-40 x^2+240 x^3\right )+e^x \left (-80+240 x-320 x^2+160 x^3+80 x^4\right )\right ) \log (x)}{1+3 \log (x)+3 \log ^2(x)+\log ^3(x)} \, dx=\frac {20 x \left (e^{4 x}+4 e^{3 x} x -4 e^{3 x}+6 e^{2 x} x^{2}-10 e^{2 x} x +6 e^{2 x}+4 e^{x} x^{3}-8 e^{x} x^{2}+8 e^{x} x -4 e^{x}+x^{4}-2 x^{3}+3 x^{2}-2 x +1\right )}{\mathrm {log}\left (x \right )^{2}+2 \,\mathrm {log}\left (x \right )+1} \] Input:

int((((80*x+20)*exp(x)^4+(240*x^2-80*x-80)*exp(x)^3+(240*x^3-40*x^2-160*x+ 
120)*exp(x)^2+(80*x^4+160*x^3-320*x^2+240*x-80)*exp(x)+100*x^4-160*x^3+180 
*x^2-80*x+20)*log(x)+(80*x-20)*exp(x)^4+(240*x^2-240*x+80)*exp(x)^3+(240*x 
^3-280*x^2+240*x-120)*exp(x)^2+(80*x^4-80*x+80)*exp(x)+60*x^4-80*x^3+60*x^ 
2-20)/(log(x)^3+3*log(x)^2+3*log(x)+1),x)
 

Output:

(20*x*(e**(4*x) + 4*e**(3*x)*x - 4*e**(3*x) + 6*e**(2*x)*x**2 - 10*e**(2*x 
)*x + 6*e**(2*x) + 4*e**x*x**3 - 8*e**x*x**2 + 8*e**x*x - 4*e**x + x**4 - 
2*x**3 + 3*x**2 - 2*x + 1))/(log(x)**2 + 2*log(x) + 1)