Integrand size = 151, antiderivative size = 25 \[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=-\frac {e^x}{\left (\frac {\log (x)}{9}+x (-4-x+x \log (x))\right )^2} \]
Time = 0.59 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=-\frac {81 e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \]
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]
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 |
\(\displaystyle -81 \left (-72 \int \frac {e^x}{\left (9 \log (x) x^2-9 x^2-36 x+\log (x)\right )^3}dx-(6-72 i) \int \frac {e^x}{(i-3 x) \left (9 \log (x) x^2-9 x^2-36 x+\log (x)\right )^3}dx-2 \int \frac {e^x}{x \left (9 \log (x) x^2-9 x^2-36 x+\log (x)\right )^3}dx-18 \int \frac {e^x x}{\left (9 \log (x) x^2-9 x^2-36 x+\log (x)\right )^3}dx+(6+72 i) \int \frac {e^x}{(3 x+i) \left (9 \log (x) x^2-9 x^2-36 x+\log (x)\right )^3}dx+\int \frac {e^x}{\left (9 \log (x) x^2-9 x^2-36 x+\log (x)\right )^2}dx+6 \int \frac {e^x}{(i-3 x) \left (9 \log (x) x^2-9 x^2-36 x+\log (x)\right )^2}dx-6 \int \frac {e^x}{(3 x+i) \left (9 \log (x) x^2-9 x^2-36 x+\log (x)\right )^2}dx\right )\) |
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]
3.16.9.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 3.35 (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 | \(-\frac {81 \,{\mathrm e}^{x}}{81 x^{4} \ln \left (x \right )^{2}-162 x^{4} \ln \left (x \right )+81 x^{4}-648 x^{3} \ln \left (x \right )+18 x^{2} \ln \left (x \right )^{2}+648 x^{3}-18 x^{2} \ln \left (x \right )+1296 x^{2}-72 x \ln \left (x \right )+\ln \left (x \right )^{2}}\) | \(71\) |
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)
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (21) = 42\).
Time = 0.24 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.40 \[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=-\frac {81 \, e^{x}}{81 \, x^{4} + 648 \, x^{3} + {\left (81 \, x^{4} + 18 \, x^{2} + 1\right )} \log \left (x\right )^{2} + 1296 \, x^{2} - 18 \, {\left (9 \, x^{4} + 36 \, x^{3} + x^{2} + 4 \, x\right )} \log \left (x\right )} \]
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=\
-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))
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (20) = 40\).
Time = 0.23 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.12 \[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=- \frac {81 e^{x}}{81 x^{4} \log {\left (x \right )}^{2} - 162 x^{4} \log {\left (x \right )} + 81 x^{4} - 648 x^{3} \log {\left (x \right )} + 648 x^{3} + 18 x^{2} \log {\left (x \right )}^{2} - 18 x^{2} \log {\left (x \right )} + 1296 x^{2} - 72 x \log {\left (x \right )} + \log {\left (x \right )}^{2}} \]
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)
-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)
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (21) = 42\).
Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.40 \[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=-\frac {81 \, e^{x}}{81 \, x^{4} + 648 \, x^{3} + {\left (81 \, x^{4} + 18 \, x^{2} + 1\right )} \log \left (x\right )^{2} + 1296 \, x^{2} - 18 \, {\left (9 \, x^{4} + 36 \, x^{3} + x^{2} + 4 \, x\right )} \log \left (x\right )} \]
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=\
-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))
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (21) = 42\).
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.80 \[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=-\frac {81 \, e^{x}}{81 \, x^{4} \log \left (x\right )^{2} - 162 \, x^{4} \log \left (x\right ) + 81 \, x^{4} - 648 \, x^{3} \log \left (x\right ) + 18 \, x^{2} \log \left (x\right )^{2} + 648 \, x^{3} - 18 \, x^{2} \log \left (x\right ) + 1296 \, x^{2} - 72 \, x \log \left (x\right ) + \log \left (x\right )^{2}} \]
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=\
-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)
Timed out. \[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=\int -\frac {{\mathrm {e}}^x\,\left (729\,x^3+1458\,x^2-5832\,x+162\right )-{\mathrm {e}}^x\,\ln \left (x\right )\,\left (729\,x^3-2916\,x^2+81\,x\right )}{{\ln \left (x\right )}^2\,\left (2187\,x^7+8748\,x^6+486\,x^5+1944\,x^4+27\,x^3+108\,x^2\right )-{\ln \left (x\right )}^3\,\left (729\,x^7+243\,x^5+27\,x^3+x\right )-\ln \left (x\right )\,\left (2187\,x^7+17496\,x^6+35235\,x^5+1944\,x^4+3888\,x^3\right )+46656\,x^4+34992\,x^5+8748\,x^6+729\,x^7} \,d x \]
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)
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)