\(\int \frac {-3 x+e^{2 x} (4 e^5+2 x)+e^x (-x+x^2+e^5 (4+x))+(24 x+e^{2 x} (-2 e^5+18 x)+e^x (e^5 (-2-4 x)+42 x-4 x^2)) \log (x)+(-54 x-53 e^{2 x} x+e^x (-107 x+e^5 x+x^2)) \log ^2(x)+(24 x+48 e^x x+24 e^{2 x} x) \log ^3(x)+(-3 x-6 e^x x-3 e^{2 x} x) \log ^4(x)}{x+2 e^x x+e^{2 x} x+(-8 x-16 e^x x-8 e^{2 x} x) \log (x)+(18 x+36 e^x x+18 e^{2 x} x) \log ^2(x)+(-8 x-16 e^x x-8 e^{2 x} x) \log ^3(x)+(x+2 e^x x+e^{2 x} x) \log ^4(x)} \, dx\) [323]

Optimal result

Integrand size = 265, antiderivative size = 30 \[ \int \frac {-3 x+e^{2 x} \left (4 e^5+2 x\right )+e^x \left (-x+x^2+e^5 (4+x)\right )+\left (24 x+e^{2 x} \left (-2 e^5+18 x\right )+e^x \left (e^5 (-2-4 x)+42 x-4 x^2\right )\right ) \log (x)+\left (-54 x-53 e^{2 x} x+e^x \left (-107 x+e^5 x+x^2\right )\right ) \log ^2(x)+\left (24 x+48 e^x x+24 e^{2 x} x\right ) \log ^3(x)+\left (-3 x-6 e^x x-3 e^{2 x} x\right ) \log ^4(x)}{x+2 e^x x+e^{2 x} x+\left (-8 x-16 e^x x-8 e^{2 x} x\right ) \log (x)+\left (18 x+36 e^x x+18 e^{2 x} x\right ) \log ^2(x)+\left (-8 x-16 e^x x-8 e^{2 x} x\right ) \log ^3(x)+\left (x+2 e^x x+e^{2 x} x\right ) \log ^4(x)} \, dx=-3 x+\frac {e^x \left (e^5+x\right )}{\left (1+e^x\right ) \left (-3+(-2+\log (x))^2\right )} \] Output:

-3*x+exp(x)*(exp(5)+x)/((ln(x)-2)^2-3)/(1+exp(x))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {-3 x+e^{2 x} \left (4 e^5+2 x\right )+e^x \left (-x+x^2+e^5 (4+x)\right )+\left (24 x+e^{2 x} \left (-2 e^5+18 x\right )+e^x \left (e^5 (-2-4 x)+42 x-4 x^2\right )\right ) \log (x)+\left (-54 x-53 e^{2 x} x+e^x \left (-107 x+e^5 x+x^2\right )\right ) \log ^2(x)+\left (24 x+48 e^x x+24 e^{2 x} x\right ) \log ^3(x)+\left (-3 x-6 e^x x-3 e^{2 x} x\right ) \log ^4(x)}{x+2 e^x x+e^{2 x} x+\left (-8 x-16 e^x x-8 e^{2 x} x\right ) \log (x)+\left (18 x+36 e^x x+18 e^{2 x} x\right ) \log ^2(x)+\left (-8 x-16 e^x x-8 e^{2 x} x\right ) \log ^3(x)+\left (x+2 e^x x+e^{2 x} x\right ) \log ^4(x)} \, dx=-3 x+\frac {e^x \left (e^5+x\right )}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )} \] Input:

Integrate[(-3*x + E^(2*x)*(4*E^5 + 2*x) + E^x*(-x + x^2 + E^5*(4 + x)) + ( 
24*x + E^(2*x)*(-2*E^5 + 18*x) + E^x*(E^5*(-2 - 4*x) + 42*x - 4*x^2))*Log[ 
x] + (-54*x - 53*E^(2*x)*x + E^x*(-107*x + E^5*x + x^2))*Log[x]^2 + (24*x 
+ 48*E^x*x + 24*E^(2*x)*x)*Log[x]^3 + (-3*x - 6*E^x*x - 3*E^(2*x)*x)*Log[x 
]^4)/(x + 2*E^x*x + E^(2*x)*x + (-8*x - 16*E^x*x - 8*E^(2*x)*x)*Log[x] + ( 
18*x + 36*E^x*x + 18*E^(2*x)*x)*Log[x]^2 + (-8*x - 16*E^x*x - 8*E^(2*x)*x) 
*Log[x]^3 + (x + 2*E^x*x + E^(2*x)*x)*Log[x]^4),x]
 

Output:

-3*x + (E^x*(E^5 + x))/((1 + E^x)*(1 - 4*Log[x] + 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.

\(\displaystyle \int \frac {e^x \left (x^2-x+e^5 (x+4)\right )+\left (e^x \left (x^2+e^5 x-107 x\right )-53 e^{2 x} x-54 x\right ) \log ^2(x)+\left (e^x \left (-4 x^2+42 x+e^5 (-4 x-2)\right )+24 x+e^{2 x} \left (18 x-2 e^5\right )\right ) \log (x)-3 x+e^{2 x} \left (2 x+4 e^5\right )+\left (-6 e^x x-3 e^{2 x} x-3 x\right ) \log ^4(x)+\left (48 e^x x+24 e^{2 x} x+24 x\right ) \log ^3(x)}{2 e^x x+e^{2 x} x+x+\left (2 e^x x+e^{2 x} x+x\right ) \log ^4(x)+\left (-16 e^x x-8 e^{2 x} x-8 x\right ) \log ^3(x)+\left (36 e^x x+18 e^{2 x} x+18 x\right ) \log ^2(x)+\left (-16 e^x x-8 e^{2 x} x-8 x\right ) \log (x)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^x \left (x^2-x+e^5 (x+4)\right )+\left (e^x \left (x^2+e^5 x-107 x\right )-53 e^{2 x} x-54 x\right ) \log ^2(x)+\left (e^x \left (-4 x^2+42 x+e^5 (-4 x-2)\right )+24 x+e^{2 x} \left (18 x-2 e^5\right )\right ) \log (x)-3 x+e^{2 x} \left (2 x+4 e^5\right )+\left (-6 e^x x-3 e^{2 x} x-3 x\right ) \log ^4(x)+\left (48 e^x x+24 e^{2 x} x+24 x\right ) \log ^3(x)}{\left (e^x+1\right )^2 x \left (\log ^2(x)-4 \log (x)+1\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x^2+x^2 \log ^2(x)-4 x^2 \log (x)-5 \left (1-\frac {e^5}{5}\right ) x-\left (1-e^5\right ) x \log ^2(x)+6 \left (1-\frac {2 e^5}{3}\right ) x \log (x)+2 e^5 \log (x)-4 e^5}{\left (e^x+1\right ) x \left (\log ^2(x)-4 \log (x)+1\right )^2}-\frac {x+e^5}{\left (e^x+1\right )^2 \left (\log ^2(x)-4 \log (x)+1\right )}+\frac {2 x-3 x \log ^4(x)+24 x \log ^3(x)-53 x \log ^2(x)+18 x \log (x)-2 e^5 \log (x)+4 e^5}{x \left (\log ^2(x)-4 \log (x)+1\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\left (5-e^5\right ) \int \frac {1}{\left (1+e^x\right ) \left (\log ^2(x)-4 \log (x)+1\right )^2}dx-4 e^5 \int \frac {1}{\left (1+e^x\right ) x \left (\log ^2(x)-4 \log (x)+1\right )^2}dx+\int \frac {x}{\left (1+e^x\right ) \left (\log ^2(x)-4 \log (x)+1\right )^2}dx+2 \left (3-2 e^5\right ) \int \frac {\log (x)}{\left (1+e^x\right ) \left (\log ^2(x)-4 \log (x)+1\right )^2}dx+2 e^5 \int \frac {\log (x)}{\left (1+e^x\right ) x \left (\log ^2(x)-4 \log (x)+1\right )^2}dx-4 \int \frac {x \log (x)}{\left (1+e^x\right ) \left (\log ^2(x)-4 \log (x)+1\right )^2}dx-\left (1-e^5\right ) \int \frac {\log ^2(x)}{\left (1+e^x\right ) \left (\log ^2(x)-4 \log (x)+1\right )^2}dx+\int \frac {x \log ^2(x)}{\left (1+e^x\right ) \left (\log ^2(x)-4 \log (x)+1\right )^2}dx-e^5 \int \frac {1}{\left (1+e^x\right )^2 \left (\log ^2(x)-4 \log (x)+1\right )}dx-\int \frac {x}{\left (1+e^x\right )^2 \left (\log ^2(x)-4 \log (x)+1\right )}dx-\frac {1}{6} \left (2+\sqrt {3}\right ) e^{2+\sqrt {3}} \operatorname {ExpIntegralEi}\left (\log (x)-\sqrt {3}-2\right )+\frac {e^{2+\sqrt {3}} \operatorname {ExpIntegralEi}\left (\log (x)-\sqrt {3}-2\right )}{2 \sqrt {3}}+\frac {1}{3} e^{2+\sqrt {3}} \operatorname {ExpIntegralEi}\left (\log (x)-\sqrt {3}-2\right )-\frac {1}{6} \left (2-\sqrt {3}\right ) e^{2-\sqrt {3}} \operatorname {ExpIntegralEi}\left (\log (x)+\sqrt {3}-2\right )-\frac {e^{2-\sqrt {3}} \operatorname {ExpIntegralEi}\left (\log (x)+\sqrt {3}-2\right )}{2 \sqrt {3}}+\frac {1}{3} e^{2-\sqrt {3}} \operatorname {ExpIntegralEi}\left (\log (x)+\sqrt {3}-2\right )-3 x+\frac {e^5}{\log ^2(x)-4 \log (x)+1}-\frac {\left (2-\sqrt {3}\right ) x}{6 \left (-\log (x)-\sqrt {3}+2\right )}+\frac {x}{3 \left (-\log (x)-\sqrt {3}+2\right )}-\frac {\left (2+\sqrt {3}\right ) x}{6 \left (-\log (x)+\sqrt {3}+2\right )}+\frac {x}{3 \left (-\log (x)+\sqrt {3}+2\right )}\)

Input:

Int[(-3*x + E^(2*x)*(4*E^5 + 2*x) + E^x*(-x + x^2 + E^5*(4 + x)) + (24*x + 
 E^(2*x)*(-2*E^5 + 18*x) + E^x*(E^5*(-2 - 4*x) + 42*x - 4*x^2))*Log[x] + ( 
-54*x - 53*E^(2*x)*x + E^x*(-107*x + E^5*x + x^2))*Log[x]^2 + (24*x + 48*E 
^x*x + 24*E^(2*x)*x)*Log[x]^3 + (-3*x - 6*E^x*x - 3*E^(2*x)*x)*Log[x]^4)/( 
x + 2*E^x*x + E^(2*x)*x + (-8*x - 16*E^x*x - 8*E^(2*x)*x)*Log[x] + (18*x + 
 36*E^x*x + 18*E^(2*x)*x)*Log[x]^2 + (-8*x - 16*E^x*x - 8*E^(2*x)*x)*Log[x 
]^3 + (x + 2*E^x*x + E^(2*x)*x)*Log[x]^4),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00

\[-3 x +\frac {\left ({\mathrm e}^{5}+x \right ) {\mathrm e}^{x}}{\left ({\mathrm e}^{x}+1\right ) \left (\ln \left (x \right )^{2}-4 \ln \left (x \right )+1\right )}\]

Input:

int(((-3*x*exp(x)^2-6*exp(x)*x-3*x)*ln(x)^4+(24*x*exp(x)^2+48*exp(x)*x+24* 
x)*ln(x)^3+(-53*x*exp(x)^2+(x*exp(5)+x^2-107*x)*exp(x)-54*x)*ln(x)^2+((-2* 
exp(5)+18*x)*exp(x)^2+((-4*x-2)*exp(5)-4*x^2+42*x)*exp(x)+24*x)*ln(x)+(4*e 
xp(5)+2*x)*exp(x)^2+((4+x)*exp(5)+x^2-x)*exp(x)-3*x)/((x*exp(x)^2+2*exp(x) 
*x+x)*ln(x)^4+(-8*x*exp(x)^2-16*exp(x)*x-8*x)*ln(x)^3+(18*x*exp(x)^2+36*ex 
p(x)*x+18*x)*ln(x)^2+(-8*x*exp(x)^2-16*exp(x)*x-8*x)*ln(x)+x*exp(x)^2+2*ex 
p(x)*x+x),x)
 

Output:

-3*x+(exp(5)+x)*exp(x)/(exp(x)+1)/(ln(x)^2-4*ln(x)+1)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (27) = 54\).

Time = 0.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.07 \[ \int \frac {-3 x+e^{2 x} \left (4 e^5+2 x\right )+e^x \left (-x+x^2+e^5 (4+x)\right )+\left (24 x+e^{2 x} \left (-2 e^5+18 x\right )+e^x \left (e^5 (-2-4 x)+42 x-4 x^2\right )\right ) \log (x)+\left (-54 x-53 e^{2 x} x+e^x \left (-107 x+e^5 x+x^2\right )\right ) \log ^2(x)+\left (24 x+48 e^x x+24 e^{2 x} x\right ) \log ^3(x)+\left (-3 x-6 e^x x-3 e^{2 x} x\right ) \log ^4(x)}{x+2 e^x x+e^{2 x} x+\left (-8 x-16 e^x x-8 e^{2 x} x\right ) \log (x)+\left (18 x+36 e^x x+18 e^{2 x} x\right ) \log ^2(x)+\left (-8 x-16 e^x x-8 e^{2 x} x\right ) \log ^3(x)+\left (x+2 e^x x+e^{2 x} x\right ) \log ^4(x)} \, dx=-\frac {3 \, {\left (x e^{x} + x\right )} \log \left (x\right )^{2} + {\left (2 \, x - e^{5}\right )} e^{x} - 12 \, {\left (x e^{x} + x\right )} \log \left (x\right ) + 3 \, x}{{\left (e^{x} + 1\right )} \log \left (x\right )^{2} - 4 \, {\left (e^{x} + 1\right )} \log \left (x\right ) + e^{x} + 1} \] Input:

integrate(((-3*x*exp(x)^2-6*exp(x)*x-3*x)*log(x)^4+(24*x*exp(x)^2+48*exp(x 
)*x+24*x)*log(x)^3+(-53*x*exp(x)^2+(x*exp(5)+x^2-107*x)*exp(x)-54*x)*log(x 
)^2+((-2*exp(5)+18*x)*exp(x)^2+((-4*x-2)*exp(5)-4*x^2+42*x)*exp(x)+24*x)*l 
og(x)+(4*exp(5)+2*x)*exp(x)^2+((4+x)*exp(5)+x^2-x)*exp(x)-3*x)/((x*exp(x)^ 
2+2*exp(x)*x+x)*log(x)^4+(-8*x*exp(x)^2-16*exp(x)*x-8*x)*log(x)^3+(18*x*ex 
p(x)^2+36*exp(x)*x+18*x)*log(x)^2+(-8*x*exp(x)^2-16*exp(x)*x-8*x)*log(x)+x 
*exp(x)^2+2*exp(x)*x+x),x, algorithm="fricas")
 

Output:

-(3*(x*e^x + x)*log(x)^2 + (2*x - e^5)*e^x - 12*(x*e^x + x)*log(x) + 3*x)/ 
((e^x + 1)*log(x)^2 - 4*(e^x + 1)*log(x) + e^x + 1)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).

Time = 0.16 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.77 \[ \int \frac {-3 x+e^{2 x} \left (4 e^5+2 x\right )+e^x \left (-x+x^2+e^5 (4+x)\right )+\left (24 x+e^{2 x} \left (-2 e^5+18 x\right )+e^x \left (e^5 (-2-4 x)+42 x-4 x^2\right )\right ) \log (x)+\left (-54 x-53 e^{2 x} x+e^x \left (-107 x+e^5 x+x^2\right )\right ) \log ^2(x)+\left (24 x+48 e^x x+24 e^{2 x} x\right ) \log ^3(x)+\left (-3 x-6 e^x x-3 e^{2 x} x\right ) \log ^4(x)}{x+2 e^x x+e^{2 x} x+\left (-8 x-16 e^x x-8 e^{2 x} x\right ) \log (x)+\left (18 x+36 e^x x+18 e^{2 x} x\right ) \log ^2(x)+\left (-8 x-16 e^x x-8 e^{2 x} x\right ) \log ^3(x)+\left (x+2 e^x x+e^{2 x} x\right ) \log ^4(x)} \, dx=- 3 x + \frac {- x - e^{5}}{\left (\log {\left (x \right )}^{2} - 4 \log {\left (x \right )} + 1\right ) e^{x} + \log {\left (x \right )}^{2} - 4 \log {\left (x \right )} + 1} + \frac {x + e^{5}}{\log {\left (x \right )}^{2} - 4 \log {\left (x \right )} + 1} \] Input:

integrate(((-3*x*exp(x)**2-6*exp(x)*x-3*x)*ln(x)**4+(24*x*exp(x)**2+48*exp 
(x)*x+24*x)*ln(x)**3+(-53*x*exp(x)**2+(x*exp(5)+x**2-107*x)*exp(x)-54*x)*l 
n(x)**2+((-2*exp(5)+18*x)*exp(x)**2+((-4*x-2)*exp(5)-4*x**2+42*x)*exp(x)+2 
4*x)*ln(x)+(4*exp(5)+2*x)*exp(x)**2+((4+x)*exp(5)+x**2-x)*exp(x)-3*x)/((x* 
exp(x)**2+2*exp(x)*x+x)*ln(x)**4+(-8*x*exp(x)**2-16*exp(x)*x-8*x)*ln(x)**3 
+(18*x*exp(x)**2+36*exp(x)*x+18*x)*ln(x)**2+(-8*x*exp(x)**2-16*exp(x)*x-8* 
x)*ln(x)+x*exp(x)**2+2*exp(x)*x+x),x)
 

Output:

-3*x + (-x - exp(5))/((log(x)**2 - 4*log(x) + 1)*exp(x) + log(x)**2 - 4*lo 
g(x) + 1) + (x + exp(5))/(log(x)**2 - 4*log(x) + 1)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (27) = 54\).

Time = 0.17 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.20 \[ \int \frac {-3 x+e^{2 x} \left (4 e^5+2 x\right )+e^x \left (-x+x^2+e^5 (4+x)\right )+\left (24 x+e^{2 x} \left (-2 e^5+18 x\right )+e^x \left (e^5 (-2-4 x)+42 x-4 x^2\right )\right ) \log (x)+\left (-54 x-53 e^{2 x} x+e^x \left (-107 x+e^5 x+x^2\right )\right ) \log ^2(x)+\left (24 x+48 e^x x+24 e^{2 x} x\right ) \log ^3(x)+\left (-3 x-6 e^x x-3 e^{2 x} x\right ) \log ^4(x)}{x+2 e^x x+e^{2 x} x+\left (-8 x-16 e^x x-8 e^{2 x} x\right ) \log (x)+\left (18 x+36 e^x x+18 e^{2 x} x\right ) \log ^2(x)+\left (-8 x-16 e^x x-8 e^{2 x} x\right ) \log ^3(x)+\left (x+2 e^x x+e^{2 x} x\right ) \log ^4(x)} \, dx=-\frac {3 \, x \log \left (x\right )^{2} + {\left (3 \, x \log \left (x\right )^{2} - 12 \, x \log \left (x\right ) + 2 \, x - e^{5}\right )} e^{x} - 12 \, x \log \left (x\right ) + 3 \, x}{{\left (\log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1\right )} e^{x} + \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1} \] Input:

integrate(((-3*x*exp(x)^2-6*exp(x)*x-3*x)*log(x)^4+(24*x*exp(x)^2+48*exp(x 
)*x+24*x)*log(x)^3+(-53*x*exp(x)^2+(x*exp(5)+x^2-107*x)*exp(x)-54*x)*log(x 
)^2+((-2*exp(5)+18*x)*exp(x)^2+((-4*x-2)*exp(5)-4*x^2+42*x)*exp(x)+24*x)*l 
og(x)+(4*exp(5)+2*x)*exp(x)^2+((4+x)*exp(5)+x^2-x)*exp(x)-3*x)/((x*exp(x)^ 
2+2*exp(x)*x+x)*log(x)^4+(-8*x*exp(x)^2-16*exp(x)*x-8*x)*log(x)^3+(18*x*ex 
p(x)^2+36*exp(x)*x+18*x)*log(x)^2+(-8*x*exp(x)^2-16*exp(x)*x-8*x)*log(x)+x 
*exp(x)^2+2*exp(x)*x+x),x, algorithm="maxima")
 

Output:

-(3*x*log(x)^2 + (3*x*log(x)^2 - 12*x*log(x) + 2*x - e^5)*e^x - 12*x*log(x 
) + 3*x)/((log(x)^2 - 4*log(x) + 1)*e^x + log(x)^2 - 4*log(x) + 1)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (27) = 54\).

Time = 0.15 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.37 \[ \int \frac {-3 x+e^{2 x} \left (4 e^5+2 x\right )+e^x \left (-x+x^2+e^5 (4+x)\right )+\left (24 x+e^{2 x} \left (-2 e^5+18 x\right )+e^x \left (e^5 (-2-4 x)+42 x-4 x^2\right )\right ) \log (x)+\left (-54 x-53 e^{2 x} x+e^x \left (-107 x+e^5 x+x^2\right )\right ) \log ^2(x)+\left (24 x+48 e^x x+24 e^{2 x} x\right ) \log ^3(x)+\left (-3 x-6 e^x x-3 e^{2 x} x\right ) \log ^4(x)}{x+2 e^x x+e^{2 x} x+\left (-8 x-16 e^x x-8 e^{2 x} x\right ) \log (x)+\left (18 x+36 e^x x+18 e^{2 x} x\right ) \log ^2(x)+\left (-8 x-16 e^x x-8 e^{2 x} x\right ) \log ^3(x)+\left (x+2 e^x x+e^{2 x} x\right ) \log ^4(x)} \, dx=-\frac {3 \, x e^{x} \log \left (x\right )^{2} - 12 \, x e^{x} \log \left (x\right ) + 3 \, x \log \left (x\right )^{2} + x e^{x} - 12 \, x \log \left (x\right ) + 3 \, x - 2 \, e^{\left (x + 5\right )}}{e^{x} \log \left (x\right )^{2} - 4 \, e^{x} \log \left (x\right ) + \log \left (x\right )^{2} + e^{x} - 4 \, \log \left (x\right ) + 1} \] Input:

integrate(((-3*x*exp(x)^2-6*exp(x)*x-3*x)*log(x)^4+(24*x*exp(x)^2+48*exp(x 
)*x+24*x)*log(x)^3+(-53*x*exp(x)^2+(x*exp(5)+x^2-107*x)*exp(x)-54*x)*log(x 
)^2+((-2*exp(5)+18*x)*exp(x)^2+((-4*x-2)*exp(5)-4*x^2+42*x)*exp(x)+24*x)*l 
og(x)+(4*exp(5)+2*x)*exp(x)^2+((4+x)*exp(5)+x^2-x)*exp(x)-3*x)/((x*exp(x)^ 
2+2*exp(x)*x+x)*log(x)^4+(-8*x*exp(x)^2-16*exp(x)*x-8*x)*log(x)^3+(18*x*ex 
p(x)^2+36*exp(x)*x+18*x)*log(x)^2+(-8*x*exp(x)^2-16*exp(x)*x-8*x)*log(x)+x 
*exp(x)^2+2*exp(x)*x+x),x, algorithm="giac")
 

Output:

-(3*x*e^x*log(x)^2 - 12*x*e^x*log(x) + 3*x*log(x)^2 + x*e^x - 12*x*log(x) 
+ 3*x - 2*e^(x + 5))/(e^x*log(x)^2 - 4*e^x*log(x) + log(x)^2 + e^x - 4*log 
(x) + 1)
 

Mupad [B] (verification not implemented)

Time = 4.13 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {-3 x+e^{2 x} \left (4 e^5+2 x\right )+e^x \left (-x+x^2+e^5 (4+x)\right )+\left (24 x+e^{2 x} \left (-2 e^5+18 x\right )+e^x \left (e^5 (-2-4 x)+42 x-4 x^2\right )\right ) \log (x)+\left (-54 x-53 e^{2 x} x+e^x \left (-107 x+e^5 x+x^2\right )\right ) \log ^2(x)+\left (24 x+48 e^x x+24 e^{2 x} x\right ) \log ^3(x)+\left (-3 x-6 e^x x-3 e^{2 x} x\right ) \log ^4(x)}{x+2 e^x x+e^{2 x} x+\left (-8 x-16 e^x x-8 e^{2 x} x\right ) \log (x)+\left (18 x+36 e^x x+18 e^{2 x} x\right ) \log ^2(x)+\left (-8 x-16 e^x x-8 e^{2 x} x\right ) \log ^3(x)+\left (x+2 e^x x+e^{2 x} x\right ) \log ^4(x)} \, dx=\frac {{\mathrm {e}}^{x+5}+{\mathrm {e}}^{2\,x+5}+x\,{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^x}{{\left ({\mathrm {e}}^x+1\right )}^2\,\left ({\ln \left (x\right )}^2-4\,\ln \left (x\right )+1\right )}-3\,x \] Input:

int((exp(2*x)*(2*x + 4*exp(5)) - log(x)^2*(54*x + 53*x*exp(2*x) - exp(x)*( 
x*exp(5) - 107*x + x^2)) - 3*x - log(x)^4*(3*x + 3*x*exp(2*x) + 6*x*exp(x) 
) + log(x)^3*(24*x + 24*x*exp(2*x) + 48*x*exp(x)) + log(x)*(24*x + exp(2*x 
)*(18*x - 2*exp(5)) - exp(x)*(4*x^2 - 42*x + exp(5)*(4*x + 2))) + exp(x)*( 
exp(5)*(x + 4) - x + x^2))/(x + x*exp(2*x) - log(x)*(8*x + 8*x*exp(2*x) + 
16*x*exp(x)) + log(x)^4*(x + x*exp(2*x) + 2*x*exp(x)) - log(x)^3*(8*x + 8* 
x*exp(2*x) + 16*x*exp(x)) + log(x)^2*(18*x + 18*x*exp(2*x) + 36*x*exp(x)) 
+ 2*x*exp(x)),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.60 \[ \int \frac {-3 x+e^{2 x} \left (4 e^5+2 x\right )+e^x \left (-x+x^2+e^5 (4+x)\right )+\left (24 x+e^{2 x} \left (-2 e^5+18 x\right )+e^x \left (e^5 (-2-4 x)+42 x-4 x^2\right )\right ) \log (x)+\left (-54 x-53 e^{2 x} x+e^x \left (-107 x+e^5 x+x^2\right )\right ) \log ^2(x)+\left (24 x+48 e^x x+24 e^{2 x} x\right ) \log ^3(x)+\left (-3 x-6 e^x x-3 e^{2 x} x\right ) \log ^4(x)}{x+2 e^x x+e^{2 x} x+\left (-8 x-16 e^x x-8 e^{2 x} x\right ) \log (x)+\left (18 x+36 e^x x+18 e^{2 x} x\right ) \log ^2(x)+\left (-8 x-16 e^x x-8 e^{2 x} x\right ) \log ^3(x)+\left (x+2 e^x x+e^{2 x} x\right ) \log ^4(x)} \, dx=\frac {-3 e^{x} \mathrm {log}\left (x \right )^{2} x +12 e^{x} \mathrm {log}\left (x \right ) x +e^{x} e^{5}-2 e^{x} x -3 \mathrm {log}\left (x \right )^{2} x +12 \,\mathrm {log}\left (x \right ) x -3 x}{e^{x} \mathrm {log}\left (x \right )^{2}-4 e^{x} \mathrm {log}\left (x \right )+e^{x}+\mathrm {log}\left (x \right )^{2}-4 \,\mathrm {log}\left (x \right )+1} \] Input:

int(((-3*x*exp(x)^2-6*exp(x)*x-3*x)*log(x)^4+(24*x*exp(x)^2+48*exp(x)*x+24 
*x)*log(x)^3+(-53*x*exp(x)^2+(x*exp(5)+x^2-107*x)*exp(x)-54*x)*log(x)^2+(( 
-2*exp(5)+18*x)*exp(x)^2+((-4*x-2)*exp(5)-4*x^2+42*x)*exp(x)+24*x)*log(x)+ 
(4*exp(5)+2*x)*exp(x)^2+((4+x)*exp(5)+x^2-x)*exp(x)-3*x)/((x*exp(x)^2+2*ex 
p(x)*x+x)*log(x)^4+(-8*x*exp(x)^2-16*exp(x)*x-8*x)*log(x)^3+(18*x*exp(x)^2 
+36*exp(x)*x+18*x)*log(x)^2+(-8*x*exp(x)^2-16*exp(x)*x-8*x)*log(x)+x*exp(x 
)^2+2*exp(x)*x+x),x)
 

Output:

( - 3*e**x*log(x)**2*x + 12*e**x*log(x)*x + e**x*e**5 - 2*e**x*x - 3*log(x 
)**2*x + 12*log(x)*x - 3*x)/(e**x*log(x)**2 - 4*e**x*log(x) + e**x + log(x 
)**2 - 4*log(x) + 1)