\(\int \frac {e^x (162-5832 x+1458 x^2+729 x^3)+e^x (-81 x+2916 x^2-729 x^3) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+(3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7) \log (x)+(-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7) \log ^2(x)+(x+27 x^3+243 x^5+729 x^7) \log ^3(x)} \, dx\) [1509]

Optimal result

Integrand size = 151, antiderivative size = 25 \[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=-\frac {e^x}{\left (\frac {\log (x)}{9}+x (-4-x+x \log (x))\right )^2} \] Output:

-exp(x)/(1/9*ln(x)+x*(x*ln(x)-4-x))^2
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=-\frac {81 e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \] Input:

Integrate[(E^x*(162 - 5832*x + 1458*x^2 + 729*x^3) + E^x*(-81*x + 2916*x^2 
 - 729*x^3)*Log[x])/(-46656*x^4 - 34992*x^5 - 8748*x^6 - 729*x^7 + (3888*x 
^3 + 1944*x^4 + 35235*x^5 + 17496*x^6 + 2187*x^7)*Log[x] + (-108*x^2 - 27* 
x^3 - 1944*x^4 - 486*x^5 - 8748*x^6 - 2187*x^7)*Log[x]^2 + (x + 27*x^3 + 2 
43*x^5 + 729*x^7)*Log[x]^3),x]
 

Output:

(-81*E^x)/(-36*x - 9*x^2 + Log[x] + 9*x^2*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.

\(\displaystyle \int \frac {e^x \left (729 x^3+1458 x^2-5832 x+162\right )+e^x \left (-729 x^3+2916 x^2-81 x\right ) \log (x)}{-729 x^7-8748 x^6-34992 x^5-46656 x^4+\left (729 x^7+243 x^5+27 x^3+x\right ) \log ^3(x)+\left (2187 x^7+17496 x^6+35235 x^5+1944 x^4+3888 x^3\right ) \log (x)+\left (-2187 x^7-8748 x^6-486 x^5-1944 x^4-27 x^3-108 x^2\right ) \log ^2(x)} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {81 e^x \left (-9 x^3-18 x^2+\left (9 x^2-36 x+1\right ) x \log (x)+72 x-2\right )}{x \left (9 x (x+4)-\left (9 x^2+1\right ) \log (x)\right )^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 81 \int -\frac {e^x \left (9 x^3+18 x^2-\left (9 x^2-36 x+1\right ) \log (x) x-72 x+2\right )}{x \left (9 x (x+4)-\left (9 x^2+1\right ) \log (x)\right )^3}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -81 \int \frac {e^x \left (9 x^3+18 x^2-\left (9 x^2-36 x+1\right ) \log (x) x-72 x+2\right )}{x \left (9 x (x+4)-\left (9 x^2+1\right ) \log (x)\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -81 \int \left (\frac {e^x \left (9 x^2-36 x+1\right )}{\left (9 x^2+1\right ) \left (9 \log (x) x^2-9 x^2-36 x+\log (x)\right )^2}-\frac {2 e^x \left (81 x^4+324 x^3-36 x+1\right )}{x \left (9 x^2+1\right ) \left (9 \log (x) x^2-9 x^2-36 x+\log (x)\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -81 \left (-72 \int \frac {e^x}{\left (9 \log (x) x^2-9 x^2-36 x+\log (x)\right )^3}dx-(6-72 i) \int \frac {e^x}{(i-3 x) \left (9 \log (x) x^2-9 x^2-36 x+\log (x)\right )^3}dx-2 \int \frac {e^x}{x \left (9 \log (x) x^2-9 x^2-36 x+\log (x)\right )^3}dx-18 \int \frac {e^x x}{\left (9 \log (x) x^2-9 x^2-36 x+\log (x)\right )^3}dx+(6+72 i) \int \frac {e^x}{(3 x+i) \left (9 \log (x) x^2-9 x^2-36 x+\log (x)\right )^3}dx+\int \frac {e^x}{\left (9 \log (x) x^2-9 x^2-36 x+\log (x)\right )^2}dx+6 \int \frac {e^x}{(i-3 x) \left (9 \log (x) x^2-9 x^2-36 x+\log (x)\right )^2}dx-6 \int \frac {e^x}{(3 x+i) \left (9 \log (x) x^2-9 x^2-36 x+\log (x)\right )^2}dx\right )\)

Input:

Int[(E^x*(162 - 5832*x + 1458*x^2 + 729*x^3) + E^x*(-81*x + 2916*x^2 - 729 
*x^3)*Log[x])/(-46656*x^4 - 34992*x^5 - 8748*x^6 - 729*x^7 + (3888*x^3 + 1 
944*x^4 + 35235*x^5 + 17496*x^6 + 2187*x^7)*Log[x] + (-108*x^2 - 27*x^3 - 
1944*x^4 - 486*x^5 - 8748*x^6 - 2187*x^7)*Log[x]^2 + (x + 27*x^3 + 243*x^5 
 + 729*x^7)*Log[x]^3),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 3.73 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

method	result	size
risch	\(-\frac {81 \,{\mathrm e}^{x}}{\left (9 x^{2} \ln \left (x \right )-9 x^{2}+\ln \left (x \right )-36 x \right )^{2}}\)	\(25\)
parallelrisch	\(-\frac {81 \,{\mathrm e}^{x}}{81 x^{4} \ln \left (x \right )^{2}-162 x^{4} \ln \left (x \right )+81 x^{4}-648 x^{3} \ln \left (x \right )+18 x^{2} \ln \left (x \right )^{2}+648 x^{3}-18 x^{2} \ln \left (x \right )+1296 x^{2}-72 x \ln \left (x \right )+\ln \left (x \right )^{2}}\)	\(71\)

Input:

int(((-729*x^3+2916*x^2-81*x)*exp(x)*ln(x)+(729*x^3+1458*x^2-5832*x+162)*e 
xp(x))/((729*x^7+243*x^5+27*x^3+x)*ln(x)^3+(-2187*x^7-8748*x^6-486*x^5-194 
4*x^4-27*x^3-108*x^2)*ln(x)^2+(2187*x^7+17496*x^6+35235*x^5+1944*x^4+3888* 
x^3)*ln(x)-729*x^7-8748*x^6-34992*x^5-46656*x^4),x,method=_RETURNVERBOSE)
 

Output:

-81*exp(x)/(9*x^2*ln(x)-9*x^2+ln(x)-36*x)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (21) = 42\).

Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.40 \[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=-\frac {81 \, e^{x}}{81 \, x^{4} + 648 \, x^{3} + {\left (81 \, x^{4} + 18 \, x^{2} + 1\right )} \log \left (x\right )^{2} + 1296 \, x^{2} - 18 \, {\left (9 \, x^{4} + 36 \, x^{3} + x^{2} + 4 \, x\right )} \log \left (x\right )} \] Input:

integrate(((-729*x^3+2916*x^2-81*x)*exp(x)*log(x)+(729*x^3+1458*x^2-5832*x 
+162)*exp(x))/((729*x^7+243*x^5+27*x^3+x)*log(x)^3+(-2187*x^7-8748*x^6-486 
*x^5-1944*x^4-27*x^3-108*x^2)*log(x)^2+(2187*x^7+17496*x^6+35235*x^5+1944* 
x^4+3888*x^3)*log(x)-729*x^7-8748*x^6-34992*x^5-46656*x^4),x, algorithm="f 
ricas")
 

Output:

-81*e^x/(81*x^4 + 648*x^3 + (81*x^4 + 18*x^2 + 1)*log(x)^2 + 1296*x^2 - 18 
*(9*x^4 + 36*x^3 + x^2 + 4*x)*log(x))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (20) = 40\).

Time = 0.25 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.12 \[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=- \frac {81 e^{x}}{81 x^{4} \log {\left (x \right )}^{2} - 162 x^{4} \log {\left (x \right )} + 81 x^{4} - 648 x^{3} \log {\left (x \right )} + 648 x^{3} + 18 x^{2} \log {\left (x \right )}^{2} - 18 x^{2} \log {\left (x \right )} + 1296 x^{2} - 72 x \log {\left (x \right )} + \log {\left (x \right )}^{2}} \] Input:

integrate(((-729*x**3+2916*x**2-81*x)*exp(x)*ln(x)+(729*x**3+1458*x**2-583 
2*x+162)*exp(x))/((729*x**7+243*x**5+27*x**3+x)*ln(x)**3+(-2187*x**7-8748* 
x**6-486*x**5-1944*x**4-27*x**3-108*x**2)*ln(x)**2+(2187*x**7+17496*x**6+3 
5235*x**5+1944*x**4+3888*x**3)*ln(x)-729*x**7-8748*x**6-34992*x**5-46656*x 
**4),x)
 

Output:

-81*exp(x)/(81*x**4*log(x)**2 - 162*x**4*log(x) + 81*x**4 - 648*x**3*log(x 
) + 648*x**3 + 18*x**2*log(x)**2 - 18*x**2*log(x) + 1296*x**2 - 72*x*log(x 
) + log(x)**2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (21) = 42\).

Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.40 \[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=-\frac {81 \, e^{x}}{81 \, x^{4} + 648 \, x^{3} + {\left (81 \, x^{4} + 18 \, x^{2} + 1\right )} \log \left (x\right )^{2} + 1296 \, x^{2} - 18 \, {\left (9 \, x^{4} + 36 \, x^{3} + x^{2} + 4 \, x\right )} \log \left (x\right )} \] Input:

integrate(((-729*x^3+2916*x^2-81*x)*exp(x)*log(x)+(729*x^3+1458*x^2-5832*x 
+162)*exp(x))/((729*x^7+243*x^5+27*x^3+x)*log(x)^3+(-2187*x^7-8748*x^6-486 
*x^5-1944*x^4-27*x^3-108*x^2)*log(x)^2+(2187*x^7+17496*x^6+35235*x^5+1944* 
x^4+3888*x^3)*log(x)-729*x^7-8748*x^6-34992*x^5-46656*x^4),x, algorithm="m 
axima")
 

Output:

-81*e^x/(81*x^4 + 648*x^3 + (81*x^4 + 18*x^2 + 1)*log(x)^2 + 1296*x^2 - 18 
*(9*x^4 + 36*x^3 + x^2 + 4*x)*log(x))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (21) = 42\).

Time = 0.15 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.80 \[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=-\frac {81 \, e^{x}}{81 \, x^{4} \log \left (x\right )^{2} - 162 \, x^{4} \log \left (x\right ) + 81 \, x^{4} - 648 \, x^{3} \log \left (x\right ) + 18 \, x^{2} \log \left (x\right )^{2} + 648 \, x^{3} - 18 \, x^{2} \log \left (x\right ) + 1296 \, x^{2} - 72 \, x \log \left (x\right ) + \log \left (x\right )^{2}} \] Input:

integrate(((-729*x^3+2916*x^2-81*x)*exp(x)*log(x)+(729*x^3+1458*x^2-5832*x 
+162)*exp(x))/((729*x^7+243*x^5+27*x^3+x)*log(x)^3+(-2187*x^7-8748*x^6-486 
*x^5-1944*x^4-27*x^3-108*x^2)*log(x)^2+(2187*x^7+17496*x^6+35235*x^5+1944* 
x^4+3888*x^3)*log(x)-729*x^7-8748*x^6-34992*x^5-46656*x^4),x, algorithm="g 
iac")
 

Output:

-81*e^x/(81*x^4*log(x)^2 - 162*x^4*log(x) + 81*x^4 - 648*x^3*log(x) + 18*x 
^2*log(x)^2 + 648*x^3 - 18*x^2*log(x) + 1296*x^2 - 72*x*log(x) + log(x)^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=\int -\frac {{\mathrm {e}}^x\,\left (729\,x^3+1458\,x^2-5832\,x+162\right )-{\mathrm {e}}^x\,\ln \left (x\right )\,\left (729\,x^3-2916\,x^2+81\,x\right )}{{\ln \left (x\right )}^2\,\left (2187\,x^7+8748\,x^6+486\,x^5+1944\,x^4+27\,x^3+108\,x^2\right )-{\ln \left (x\right )}^3\,\left (729\,x^7+243\,x^5+27\,x^3+x\right )-\ln \left (x\right )\,\left (2187\,x^7+17496\,x^6+35235\,x^5+1944\,x^4+3888\,x^3\right )+46656\,x^4+34992\,x^5+8748\,x^6+729\,x^7} \,d x \] Input:

int(-(exp(x)*(1458*x^2 - 5832*x + 729*x^3 + 162) - exp(x)*log(x)*(81*x - 2 
916*x^2 + 729*x^3))/(log(x)^2*(108*x^2 + 27*x^3 + 1944*x^4 + 486*x^5 + 874 
8*x^6 + 2187*x^7) - log(x)^3*(x + 27*x^3 + 243*x^5 + 729*x^7) - log(x)*(38 
88*x^3 + 1944*x^4 + 35235*x^5 + 17496*x^6 + 2187*x^7) + 46656*x^4 + 34992* 
x^5 + 8748*x^6 + 729*x^7),x)
 

Output:

int(-(exp(x)*(1458*x^2 - 5832*x + 729*x^3 + 162) - exp(x)*log(x)*(81*x - 2 
916*x^2 + 729*x^3))/(log(x)^2*(108*x^2 + 27*x^3 + 1944*x^4 + 486*x^5 + 874 
8*x^6 + 2187*x^7) - log(x)^3*(x + 27*x^3 + 243*x^5 + 729*x^7) - log(x)*(38 
88*x^3 + 1944*x^4 + 35235*x^5 + 17496*x^6 + 2187*x^7) + 46656*x^4 + 34992* 
x^5 + 8748*x^6 + 729*x^7), x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.84 \[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=-\frac {81 e^{x}}{81 \mathrm {log}\left (x \right )^{2} x^{4}+18 \mathrm {log}\left (x \right )^{2} x^{2}+\mathrm {log}\left (x \right )^{2}-162 \,\mathrm {log}\left (x \right ) x^{4}-648 \,\mathrm {log}\left (x \right ) x^{3}-18 \,\mathrm {log}\left (x \right ) x^{2}-72 \,\mathrm {log}\left (x \right ) x +81 x^{4}+648 x^{3}+1296 x^{2}} \] Input:

int(((-729*x^3+2916*x^2-81*x)*exp(x)*log(x)+(729*x^3+1458*x^2-5832*x+162)* 
exp(x))/((729*x^7+243*x^5+27*x^3+x)*log(x)^3+(-2187*x^7-8748*x^6-486*x^5-1 
944*x^4-27*x^3-108*x^2)*log(x)^2+(2187*x^7+17496*x^6+35235*x^5+1944*x^4+38 
88*x^3)*log(x)-729*x^7-8748*x^6-34992*x^5-46656*x^4),x)
 

Output:

( - 81*e**x)/(81*log(x)**2*x**4 + 18*log(x)**2*x**2 + log(x)**2 - 162*log( 
x)*x**4 - 648*log(x)*x**3 - 18*log(x)*x**2 - 72*log(x)*x + 81*x**4 + 648*x 
**3 + 1296*x**2)