Integrand size = 174, antiderivative size = 23 \[ \int \frac {-128-384 x-256 x^2+e^{2 x} \left (-128 x^2+640 x^4+512 x^5\right )+e^{4 x} \left (-32 x^4+96 x^5+128 x^6\right )+\left (-256-256 x+e^{2 x} \left (128 x^2+1152 x^3+1024 x^4\right )+e^{4 x} \left (128 x^4+128 x^5\right )\right ) \log (x)+e^{2 x} \left (512 x^2+512 x^3\right ) \log ^2(x)}{e^{4 x} x^8+3 e^{4 x} x^7 \log (x)+3 e^{4 x} x^6 \log ^2(x)+e^{4 x} x^5 \log ^3(x)} \, dx=\left (-16+\frac {4+\frac {8 e^{-2 x}}{x^2}}{x+\log (x)}\right )^2 \] Output:
((8/exp(x)^2/x^2+4)/(x+ln(x))-16)^2
Time = 0.08 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83 \[ \int \frac {-128-384 x-256 x^2+e^{2 x} \left (-128 x^2+640 x^4+512 x^5\right )+e^{4 x} \left (-32 x^4+96 x^5+128 x^6\right )+\left (-256-256 x+e^{2 x} \left (128 x^2+1152 x^3+1024 x^4\right )+e^{4 x} \left (128 x^4+128 x^5\right )\right ) \log (x)+e^{2 x} \left (512 x^2+512 x^3\right ) \log ^2(x)}{e^{4 x} x^8+3 e^{4 x} x^7 \log (x)+3 e^{4 x} x^6 \log ^2(x)+e^{4 x} x^5 \log ^3(x)} \, dx=\frac {16 \left (2 e^{-2 x}+x^2\right ) \left (2 e^{-2 x}+x^2-8 x^2 (x+\log (x))\right )}{x^4 (x+\log (x))^2} \] Input:
Integrate[(-128 - 384*x - 256*x^2 + E^(2*x)*(-128*x^2 + 640*x^4 + 512*x^5) + E^(4*x)*(-32*x^4 + 96*x^5 + 128*x^6) + (-256 - 256*x + E^(2*x)*(128*x^2 + 1152*x^3 + 1024*x^4) + E^(4*x)*(128*x^4 + 128*x^5))*Log[x] + E^(2*x)*(5 12*x^2 + 512*x^3)*Log[x]^2)/(E^(4*x)*x^8 + 3*E^(4*x)*x^7*Log[x] + 3*E^(4*x )*x^6*Log[x]^2 + E^(4*x)*x^5*Log[x]^3),x]
Output:
(16*(2/E^(2*x) + x^2)*(2/E^(2*x) + x^2 - 8*x^2*(x + Log[x])))/(x^4*(x + Lo g[x])^2)
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.
| \(\displaystyle \int \frac {-256 x^2+e^{2 x} \left (512 x^3+512 x^2\right ) \log ^2(x)+e^{4 x} \left (128 x^6+96 x^5-32 x^4\right )+e^{2 x} \left (512 x^5+640 x^4-128 x^2\right )+\left (e^{4 x} \left (128 x^5+128 x^4\right )+e^{2 x} \left (1024 x^4+1152 x^3+128 x^2\right )-256 x-256\right ) \log (x)-384 x-128}{e^{4 x} x^8+3 e^{4 x} x^7 \log (x)+3 e^{4 x} x^6 \log ^2(x)+e^{4 x} x^5 \log ^3(x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {32 e^{-4 x} (x+1) \left (4 e^{4 x} x^5+16 e^{2 x} x^4-e^{4 x} x^4+4 e^{4 x} x^4 \log (x)+4 e^{2 x} x^3+32 e^{2 x} x^3 \log (x)-4 e^{2 x} x^2+16 e^{2 x} x^2 \log ^2(x)+4 e^{2 x} x^2 \log (x)-8 x-8 \log (x)-4\right )}{x^5 (x+\log (x))^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 32 \int -\frac {e^{-4 x} (x+1) \left (-4 e^{4 x} x^5-16 e^{2 x} x^4+e^{4 x} x^4-4 e^{4 x} \log (x) x^4-4 e^{2 x} x^3-32 e^{2 x} \log (x) x^3+4 e^{2 x} x^2-16 e^{2 x} \log ^2(x) x^2-4 e^{2 x} \log (x) x^2+8 x+8 \log (x)+4\right )}{x^5 (x+\log (x))^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -32 \int \frac {e^{-4 x} (x+1) \left (-4 e^{4 x} x^5-16 e^{2 x} x^4+e^{4 x} x^4-4 e^{4 x} \log (x) x^4-4 e^{2 x} x^3-32 e^{2 x} \log (x) x^3+4 e^{2 x} x^2-16 e^{2 x} \log ^2(x) x^2-4 e^{2 x} \log (x) x^2+8 x+8 \log (x)+4\right )}{x^5 (x+\log (x))^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -32 \int \left (-\frac {(4 x+4 \log (x)-1) (x+1)}{x (x+\log (x))^3}-\frac {4 e^{-2 x} \left (4 x^2+8 \log (x) x+x+4 \log ^2(x)+\log (x)-1\right ) (x+1)}{x^3 (x+\log (x))^3}+\frac {8 e^{-4 x} \log (x) (x+1)}{x^5 (x+\log (x))^3}+\frac {8 e^{-4 x} (x+1)}{x^4 (x+\log (x))^3}+\frac {4 e^{-4 x} (x+1)}{x^5 (x+\log (x))^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -32 \left (4 \int \frac {e^{-4 x}}{x^5 (x+\log (x))^3}dx+8 \int \frac {e^{-4 x}}{x^5 (x+\log (x))^2}dx+4 \int \frac {e^{-4 x}}{x^4 (x+\log (x))^3}dx+8 \int \frac {e^{-4 x}}{x^4 (x+\log (x))^2}dx+4 \int \frac {e^{-2 x}}{x^3 (x+\log (x))^3}dx-4 \int \frac {e^{-2 x}}{x^3 (x+\log (x))^2}dx-16 \int \frac {e^{-2 x}}{x^3 (x+\log (x))}dx+4 \int \frac {e^{-2 x}}{x^2 (x+\log (x))^3}dx-4 \int \frac {e^{-2 x}}{x^2 (x+\log (x))^2}dx-16 \int \frac {e^{-2 x}}{x^2 (x+\log (x))}dx+\frac {4}{x+\log (x)}-\frac {1}{2 (x+\log (x))^2}\right )\) |
Input:
Int[(-128 - 384*x - 256*x^2 + E^(2*x)*(-128*x^2 + 640*x^4 + 512*x^5) + E^( 4*x)*(-32*x^4 + 96*x^5 + 128*x^6) + (-256 - 256*x + E^(2*x)*(128*x^2 + 115 2*x^3 + 1024*x^4) + E^(4*x)*(128*x^4 + 128*x^5))*Log[x] + E^(2*x)*(512*x^2 + 512*x^3)*Log[x]^2)/(E^(4*x)*x^8 + 3*E^(4*x)*x^7*Log[x] + 3*E^(4*x)*x^6* Log[x]^2 + E^(4*x)*x^5*Log[x]^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs. \(2(22)=44\).
Time = 227.67 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.30
| method | result | size |
| risch | \(-\frac {16 \left (8 x^{5} {\mathrm e}^{4 x}+8 \ln \left (x \right ) {\mathrm e}^{4 x} x^{4}-{\mathrm e}^{4 x} x^{4}+16 \,{\mathrm e}^{2 x} x^{3}+16 \,{\mathrm e}^{2 x} \ln \left (x \right ) x^{2}-4 \,{\mathrm e}^{2 x} x^{2}-4\right ) {\mathrm e}^{-4 x}}{x^{4} \left (x +\ln \left (x \right )\right )^{2}}\) | \(76\) |
| parallelrisch | \(\frac {\left (128-512 \,{\mathrm e}^{2 x} x^{3}+128 \,{\mathrm e}^{2 x} x^{2}-256 x^{5} {\mathrm e}^{4 x}+32 \,{\mathrm e}^{4 x} x^{4}-256 \ln \left (x \right ) {\mathrm e}^{4 x} x^{4}-512 \,{\mathrm e}^{2 x} \ln \left (x \right ) x^{2}\right ) {\mathrm e}^{-4 x}}{2 x^{4} \left (x^{2}+2 x \ln \left (x \right )+\ln \left (x \right )^{2}\right )}\) | \(85\) |
Input:
int(((512*x^3+512*x^2)*exp(x)^2*ln(x)^2+((128*x^5+128*x^4)*exp(x)^4+(1024* x^4+1152*x^3+128*x^2)*exp(x)^2-256*x-256)*ln(x)+(128*x^6+96*x^5-32*x^4)*ex p(x)^4+(512*x^5+640*x^4-128*x^2)*exp(x)^2-256*x^2-384*x-128)/(x^5*exp(x)^4 *ln(x)^3+3*x^6*exp(x)^4*ln(x)^2+3*x^7*exp(x)^4*ln(x)+x^8*exp(x)^4),x,metho d=_RETURNVERBOSE)
Output:
-16*(8*x^5*exp(4*x)+8*ln(x)*exp(4*x)*x^4-exp(4*x)*x^4+16*exp(2*x)*x^3+16*e xp(2*x)*ln(x)*x^2-4*exp(2*x)*x^2-4)/x^4*exp(-4*x)/(x+ln(x))^2
Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (24) = 48\).
Time = 0.10 (sec) , antiderivative size = 93, normalized size of antiderivative = 4.04 \[ \int \frac {-128-384 x-256 x^2+e^{2 x} \left (-128 x^2+640 x^4+512 x^5\right )+e^{4 x} \left (-32 x^4+96 x^5+128 x^6\right )+\left (-256-256 x+e^{2 x} \left (128 x^2+1152 x^3+1024 x^4\right )+e^{4 x} \left (128 x^4+128 x^5\right )\right ) \log (x)+e^{2 x} \left (512 x^2+512 x^3\right ) \log ^2(x)}{e^{4 x} x^8+3 e^{4 x} x^7 \log (x)+3 e^{4 x} x^6 \log ^2(x)+e^{4 x} x^5 \log ^3(x)} \, dx=-\frac {16 \, {\left ({\left (8 \, x^{5} - x^{4}\right )} e^{\left (4 \, x\right )} + 4 \, {\left (4 \, x^{3} - x^{2}\right )} e^{\left (2 \, x\right )} + 8 \, {\left (x^{4} e^{\left (4 \, x\right )} + 2 \, x^{2} e^{\left (2 \, x\right )}\right )} \log \left (x\right ) - 4\right )}}{x^{6} e^{\left (4 \, x\right )} + 2 \, x^{5} e^{\left (4 \, x\right )} \log \left (x\right ) + x^{4} e^{\left (4 \, x\right )} \log \left (x\right )^{2}} \] Input:
integrate(((512*x^3+512*x^2)*exp(x)^2*log(x)^2+((128*x^5+128*x^4)*exp(x)^4 +(1024*x^4+1152*x^3+128*x^2)*exp(x)^2-256*x-256)*log(x)+(128*x^6+96*x^5-32 *x^4)*exp(x)^4+(512*x^5+640*x^4-128*x^2)*exp(x)^2-256*x^2-384*x-128)/(x^5* exp(x)^4*log(x)^3+3*x^6*exp(x)^4*log(x)^2+3*x^7*exp(x)^4*log(x)+x^8*exp(x) ^4),x, algorithm="fricas")
Output:
-16*((8*x^5 - x^4)*e^(4*x) + 4*(4*x^3 - x^2)*e^(2*x) + 8*(x^4*e^(4*x) + 2* x^2*e^(2*x))*log(x) - 4)/(x^6*e^(4*x) + 2*x^5*e^(4*x)*log(x) + x^4*e^(4*x) *log(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (19) = 38\).
Time = 0.20 (sec) , antiderivative size = 156, normalized size of antiderivative = 6.78 \[ \int \frac {-128-384 x-256 x^2+e^{2 x} \left (-128 x^2+640 x^4+512 x^5\right )+e^{4 x} \left (-32 x^4+96 x^5+128 x^6\right )+\left (-256-256 x+e^{2 x} \left (128 x^2+1152 x^3+1024 x^4\right )+e^{4 x} \left (128 x^4+128 x^5\right )\right ) \log (x)+e^{2 x} \left (512 x^2+512 x^3\right ) \log ^2(x)}{e^{4 x} x^8+3 e^{4 x} x^7 \log (x)+3 e^{4 x} x^6 \log ^2(x)+e^{4 x} x^5 \log ^3(x)} \, dx=\frac {\left (64 x^{4} + 128 x^{3} \log {\left (x \right )} + 64 x^{2} \log {\left (x \right )}^{2}\right ) e^{- 4 x} + \left (- 256 x^{7} - 768 x^{6} \log {\left (x \right )} + 64 x^{6} - 768 x^{5} \log {\left (x \right )}^{2} + 128 x^{5} \log {\left (x \right )} - 256 x^{4} \log {\left (x \right )}^{3} + 64 x^{4} \log {\left (x \right )}^{2}\right ) e^{- 2 x}}{x^{10} + 4 x^{9} \log {\left (x \right )} + 6 x^{8} \log {\left (x \right )}^{2} + 4 x^{7} \log {\left (x \right )}^{3} + x^{6} \log {\left (x \right )}^{4}} + \frac {- 128 x - 128 \log {\left (x \right )} + 16}{x^{2} + 2 x \log {\left (x \right )} + \log {\left (x \right )}^{2}} \] Input:
integrate(((512*x**3+512*x**2)*exp(x)**2*ln(x)**2+((128*x**5+128*x**4)*exp (x)**4+(1024*x**4+1152*x**3+128*x**2)*exp(x)**2-256*x-256)*ln(x)+(128*x**6 +96*x**5-32*x**4)*exp(x)**4+(512*x**5+640*x**4-128*x**2)*exp(x)**2-256*x** 2-384*x-128)/(x**5*exp(x)**4*ln(x)**3+3*x**6*exp(x)**4*ln(x)**2+3*x**7*exp (x)**4*ln(x)+x**8*exp(x)**4),x)
Output:
((64*x**4 + 128*x**3*log(x) + 64*x**2*log(x)**2)*exp(-4*x) + (-256*x**7 - 768*x**6*log(x) + 64*x**6 - 768*x**5*log(x)**2 + 128*x**5*log(x) - 256*x** 4*log(x)**3 + 64*x**4*log(x)**2)*exp(-2*x))/(x**10 + 4*x**9*log(x) + 6*x** 8*log(x)**2 + 4*x**7*log(x)**3 + x**6*log(x)**4) + (-128*x - 128*log(x) + 16)/(x**2 + 2*x*log(x) + log(x)**2)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (24) = 48\).
Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.09 \[ \int \frac {-128-384 x-256 x^2+e^{2 x} \left (-128 x^2+640 x^4+512 x^5\right )+e^{4 x} \left (-32 x^4+96 x^5+128 x^6\right )+\left (-256-256 x+e^{2 x} \left (128 x^2+1152 x^3+1024 x^4\right )+e^{4 x} \left (128 x^4+128 x^5\right )\right ) \log (x)+e^{2 x} \left (512 x^2+512 x^3\right ) \log ^2(x)}{e^{4 x} x^8+3 e^{4 x} x^7 \log (x)+3 e^{4 x} x^6 \log ^2(x)+e^{4 x} x^5 \log ^3(x)} \, dx=-\frac {16 \, {\left (8 \, x^{5} + 8 \, x^{4} \log \left (x\right ) - x^{4} + 4 \, {\left (4 \, x^{3} + 4 \, x^{2} \log \left (x\right ) - x^{2}\right )} e^{\left (-2 \, x\right )} - 4 \, e^{\left (-4 \, x\right )}\right )}}{x^{6} + 2 \, x^{5} \log \left (x\right ) + x^{4} \log \left (x\right )^{2}} \] Input:
integrate(((512*x^3+512*x^2)*exp(x)^2*log(x)^2+((128*x^5+128*x^4)*exp(x)^4 +(1024*x^4+1152*x^3+128*x^2)*exp(x)^2-256*x-256)*log(x)+(128*x^6+96*x^5-32 *x^4)*exp(x)^4+(512*x^5+640*x^4-128*x^2)*exp(x)^2-256*x^2-384*x-128)/(x^5* exp(x)^4*log(x)^3+3*x^6*exp(x)^4*log(x)^2+3*x^7*exp(x)^4*log(x)+x^8*exp(x) ^4),x, algorithm="maxima")
Output:
-16*(8*x^5 + 8*x^4*log(x) - x^4 + 4*(4*x^3 + 4*x^2*log(x) - x^2)*e^(-2*x) - 4*e^(-4*x))/(x^6 + 2*x^5*log(x) + x^4*log(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (24) = 48\).
Time = 0.13 (sec) , antiderivative size = 96, normalized size of antiderivative = 4.17 \[ \int \frac {-128-384 x-256 x^2+e^{2 x} \left (-128 x^2+640 x^4+512 x^5\right )+e^{4 x} \left (-32 x^4+96 x^5+128 x^6\right )+\left (-256-256 x+e^{2 x} \left (128 x^2+1152 x^3+1024 x^4\right )+e^{4 x} \left (128 x^4+128 x^5\right )\right ) \log (x)+e^{2 x} \left (512 x^2+512 x^3\right ) \log ^2(x)}{e^{4 x} x^8+3 e^{4 x} x^7 \log (x)+3 e^{4 x} x^6 \log ^2(x)+e^{4 x} x^5 \log ^3(x)} \, dx=-\frac {16 \, {\left (8 \, x^{5} e^{\left (4 \, x\right )} + 8 \, x^{4} e^{\left (4 \, x\right )} \log \left (x\right ) - x^{4} e^{\left (4 \, x\right )} + 16 \, x^{3} e^{\left (2 \, x\right )} + 16 \, x^{2} e^{\left (2 \, x\right )} \log \left (x\right ) - 4 \, x^{2} e^{\left (2 \, x\right )} - 4\right )}}{x^{6} e^{\left (4 \, x\right )} + 2 \, x^{5} e^{\left (4 \, x\right )} \log \left (x\right ) + x^{4} e^{\left (4 \, x\right )} \log \left (x\right )^{2}} \] Input:
integrate(((512*x^3+512*x^2)*exp(x)^2*log(x)^2+((128*x^5+128*x^4)*exp(x)^4 +(1024*x^4+1152*x^3+128*x^2)*exp(x)^2-256*x-256)*log(x)+(128*x^6+96*x^5-32 *x^4)*exp(x)^4+(512*x^5+640*x^4-128*x^2)*exp(x)^2-256*x^2-384*x-128)/(x^5* exp(x)^4*log(x)^3+3*x^6*exp(x)^4*log(x)^2+3*x^7*exp(x)^4*log(x)+x^8*exp(x) ^4),x, algorithm="giac")
Output:
-16*(8*x^5*e^(4*x) + 8*x^4*e^(4*x)*log(x) - x^4*e^(4*x) + 16*x^3*e^(2*x) + 16*x^2*e^(2*x)*log(x) - 4*x^2*e^(2*x) - 4)/(x^6*e^(4*x) + 2*x^5*e^(4*x)*l og(x) + x^4*e^(4*x)*log(x)^2)
Timed out. \[ \int \frac {-128-384 x-256 x^2+e^{2 x} \left (-128 x^2+640 x^4+512 x^5\right )+e^{4 x} \left (-32 x^4+96 x^5+128 x^6\right )+\left (-256-256 x+e^{2 x} \left (128 x^2+1152 x^3+1024 x^4\right )+e^{4 x} \left (128 x^4+128 x^5\right )\right ) \log (x)+e^{2 x} \left (512 x^2+512 x^3\right ) \log ^2(x)}{e^{4 x} x^8+3 e^{4 x} x^7 \log (x)+3 e^{4 x} x^6 \log ^2(x)+e^{4 x} x^5 \log ^3(x)} \, dx=-\int \frac {384\,x+\ln \left (x\right )\,\left (256\,x-{\mathrm {e}}^{4\,x}\,\left (128\,x^5+128\,x^4\right )-{\mathrm {e}}^{2\,x}\,\left (1024\,x^4+1152\,x^3+128\,x^2\right )+256\right )-{\mathrm {e}}^{4\,x}\,\left (128\,x^6+96\,x^5-32\,x^4\right )-{\mathrm {e}}^{2\,x}\,\left (512\,x^5+640\,x^4-128\,x^2\right )+256\,x^2-{\mathrm {e}}^{2\,x}\,{\ln \left (x\right )}^2\,\left (512\,x^3+512\,x^2\right )+128}{x^8\,{\mathrm {e}}^{4\,x}+3\,x^7\,{\mathrm {e}}^{4\,x}\,\ln \left (x\right )+x^5\,{\mathrm {e}}^{4\,x}\,{\ln \left (x\right )}^3+3\,x^6\,{\mathrm {e}}^{4\,x}\,{\ln \left (x\right )}^2} \,d x \] Input:
int(-(384*x + log(x)*(256*x - exp(4*x)*(128*x^4 + 128*x^5) - exp(2*x)*(128 *x^2 + 1152*x^3 + 1024*x^4) + 256) - exp(4*x)*(96*x^5 - 32*x^4 + 128*x^6) - exp(2*x)*(640*x^4 - 128*x^2 + 512*x^5) + 256*x^2 - exp(2*x)*log(x)^2*(51 2*x^2 + 512*x^3) + 128)/(x^8*exp(4*x) + 3*x^7*exp(4*x)*log(x) + x^5*exp(4* x)*log(x)^3 + 3*x^6*exp(4*x)*log(x)^2),x)
Output:
-int((384*x + log(x)*(256*x - exp(4*x)*(128*x^4 + 128*x^5) - exp(2*x)*(128 *x^2 + 1152*x^3 + 1024*x^4) + 256) - exp(4*x)*(96*x^5 - 32*x^4 + 128*x^6) - exp(2*x)*(640*x^4 - 128*x^2 + 512*x^5) + 256*x^2 - exp(2*x)*log(x)^2*(51 2*x^2 + 512*x^3) + 128)/(x^8*exp(4*x) + 3*x^7*exp(4*x)*log(x) + x^5*exp(4* x)*log(x)^3 + 3*x^6*exp(4*x)*log(x)^2), x)
Time = 0.15 (sec) , antiderivative size = 92, normalized size of antiderivative = 4.00 \[ \int \frac {-128-384 x-256 x^2+e^{2 x} \left (-128 x^2+640 x^4+512 x^5\right )+e^{4 x} \left (-32 x^4+96 x^5+128 x^6\right )+\left (-256-256 x+e^{2 x} \left (128 x^2+1152 x^3+1024 x^4\right )+e^{4 x} \left (128 x^4+128 x^5\right )\right ) \log (x)+e^{2 x} \left (512 x^2+512 x^3\right ) \log ^2(x)}{e^{4 x} x^8+3 e^{4 x} x^7 \log (x)+3 e^{4 x} x^6 \log ^2(x)+e^{4 x} x^5 \log ^3(x)} \, dx=\frac {-128 e^{4 x} \mathrm {log}\left (x \right ) x^{4}-128 e^{4 x} x^{5}+16 e^{4 x} x^{4}-256 e^{2 x} \mathrm {log}\left (x \right ) x^{2}-256 e^{2 x} x^{3}+64 e^{2 x} x^{2}+64}{e^{4 x} x^{4} \left (\mathrm {log}\left (x \right )^{2}+2 \,\mathrm {log}\left (x \right ) x +x^{2}\right )} \] Input:
int(((512*x^3+512*x^2)*exp(x)^2*log(x)^2+((128*x^5+128*x^4)*exp(x)^4+(1024 *x^4+1152*x^3+128*x^2)*exp(x)^2-256*x-256)*log(x)+(128*x^6+96*x^5-32*x^4)* exp(x)^4+(512*x^5+640*x^4-128*x^2)*exp(x)^2-256*x^2-384*x-128)/(x^5*exp(x) ^4*log(x)^3+3*x^6*exp(x)^4*log(x)^2+3*x^7*exp(x)^4*log(x)+x^8*exp(x)^4),x)
Output:
(16*( - 8*e**(4*x)*log(x)*x**4 - 8*e**(4*x)*x**5 + e**(4*x)*x**4 - 16*e**( 2*x)*log(x)*x**2 - 16*e**(2*x)*x**3 + 4*e**(2*x)*x**2 + 4))/(e**(4*x)*x**4 *(log(x)**2 + 2*log(x)*x + x**2))