Integrand size = 79, antiderivative size = 31 \[ \int \frac {e^{-x} \left (2048 x-128 x^2+\left (1920 x^2-128 x^3\right ) \log (x)+\left (8192+7168 x-256 x^2-288 x^3+33 x^4-x^5\right ) \log ^2(x)\right )}{\left (256 x^2-32 x^3+x^4\right ) \log ^2(x)} \, dx=e^{-x} \left (x+\frac {8 \left (4+\frac {x}{\log (x)}\right )}{-x+\frac {x^2}{16}}\right ) \] Output:
(8*(4+x/ln(x))/(1/16*x^2-x)+x)/exp(x)
Time = 3.66 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {e^{-x} \left (2048 x-128 x^2+\left (1920 x^2-128 x^3\right ) \log (x)+\left (8192+7168 x-256 x^2-288 x^3+33 x^4-x^5\right ) \log ^2(x)\right )}{\left (256 x^2-32 x^3+x^4\right ) \log ^2(x)} \, dx=e^{-x} \left (\frac {32}{-16+x}-\frac {32}{x}+x\right )+\frac {128 e^{-x}}{(-16+x) \log (x)} \] Input:
Integrate[(2048*x - 128*x^2 + (1920*x^2 - 128*x^3)*Log[x] + (8192 + 7168*x - 256*x^2 - 288*x^3 + 33*x^4 - x^5)*Log[x]^2)/(E^x*(256*x^2 - 32*x^3 + x^ 4)*Log[x]^2),x]
Output:
(32/(-16 + x) - 32/x + x)/E^x + 128/(E^x*(-16 + x)*Log[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.
\(\displaystyle \int \frac {e^{-x} \left (-128 x^2+\left (1920 x^2-128 x^3\right ) \log (x)+\left (-x^5+33 x^4-288 x^3-256 x^2+7168 x+8192\right ) \log ^2(x)+2048 x\right )}{\left (x^4-32 x^3+256 x^2\right ) \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {e^{-x} \left (-128 x^2+\left (1920 x^2-128 x^3\right ) \log (x)+\left (-x^5+33 x^4-288 x^3-256 x^2+7168 x+8192\right ) \log ^2(x)+2048 x\right )}{x^2 \left (x^2-32 x+256\right ) \log ^2(x)}dx\) |
\(\Big \downarrow \) 7277 |
\(\displaystyle 4 \int \frac {e^{-x} \left (-128 x^2+2048 x+\left (-x^5+33 x^4-288 x^3-256 x^2+7168 x+8192\right ) \log ^2(x)+128 \left (15 x^2-x^3\right ) \log (x)\right )}{4 (16-x)^2 x^2 \log ^2(x)}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {e^{-x} \left (-128 x^2+128 \left (15 x^2-x^3\right ) \log (x)+\left (-x^5+33 x^4-288 x^3-256 x^2+7168 x+8192\right ) \log ^2(x)+2048 x\right )}{(16-x)^2 x^2 \log ^2(x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^{-x} \left (-x^5+33 x^4-288 x^3-256 x^2+7168 x+8192\right )}{(x-16)^2 x^2}-\frac {128 e^{-x}}{(x-16) x \log ^2(x)}-\frac {128 e^{-x} (x-15)}{(x-16)^2 \log (x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -8 \int \frac {e^{-x}}{(x-16) \log ^2(x)}dx+8 \int \frac {e^{-x}}{x \log ^2(x)}dx-128 \int \frac {e^{-x}}{(x-16)^2 \log (x)}dx-128 \int \frac {e^{-x}}{(x-16) \log (x)}dx+e^{-x} x-\frac {32 e^{-x}}{16-x}-\frac {32 e^{-x}}{x}\) |
Input:
Int[(2048*x - 128*x^2 + (1920*x^2 - 128*x^3)*Log[x] + (8192 + 7168*x - 256 *x^2 - 288*x^3 + 33*x^4 - x^5)*Log[x]^2)/(E^x*(256*x^2 - 32*x^3 + x^4)*Log [x]^2),x]
Output:
$Aborted
Time = 0.42 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29
method | result | size |
risch | \(\frac {\left (x^{3}-16 x^{2}+512\right ) {\mathrm e}^{-x}}{x \left (x -16\right )}+\frac {128 \,{\mathrm e}^{-x}}{\left (x -16\right ) \ln \left (x \right )}\) | \(40\) |
parallelrisch | \(\frac {\left (16 x^{3} \ln \left (x \right )-256 x^{2} \ln \left (x \right )+2048 x +8192 \ln \left (x \right )\right ) {\mathrm e}^{-x}}{16 x \ln \left (x \right ) \left (x -16\right )}\) | \(41\) |
Input:
int(((-x^5+33*x^4-288*x^3-256*x^2+7168*x+8192)*ln(x)^2+(-128*x^3+1920*x^2) *ln(x)-128*x^2+2048*x)/(x^4-32*x^3+256*x^2)/exp(x)/ln(x)^2,x,method=_RETUR NVERBOSE)
Output:
(x^3-16*x^2+512)/x/(x-16)*exp(-x)+128*exp(-x)/(x-16)/ln(x)
Time = 0.09 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \frac {e^{-x} \left (2048 x-128 x^2+\left (1920 x^2-128 x^3\right ) \log (x)+\left (8192+7168 x-256 x^2-288 x^3+33 x^4-x^5\right ) \log ^2(x)\right )}{\left (256 x^2-32 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {{\left (x^{3} - 16 \, x^{2} + 512\right )} e^{\left (-x\right )} \log \left (x\right ) + 128 \, x e^{\left (-x\right )}}{{\left (x^{2} - 16 \, x\right )} \log \left (x\right )} \] Input:
integrate(((-x^5+33*x^4-288*x^3-256*x^2+7168*x+8192)*log(x)^2+(-128*x^3+19 20*x^2)*log(x)-128*x^2+2048*x)/(x^4-32*x^3+256*x^2)/exp(x)/log(x)^2,x, alg orithm="fricas")
Output:
((x^3 - 16*x^2 + 512)*e^(-x)*log(x) + 128*x*e^(-x))/((x^2 - 16*x)*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (19) = 38\).
Time = 0.12 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \frac {e^{-x} \left (2048 x-128 x^2+\left (1920 x^2-128 x^3\right ) \log (x)+\left (8192+7168 x-256 x^2-288 x^3+33 x^4-x^5\right ) \log ^2(x)\right )}{\left (256 x^2-32 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {\left (x^{3} \log {\left (x \right )} - 16 x^{2} \log {\left (x \right )} + 128 x + 512 \log {\left (x \right )}\right ) e^{- x}}{x^{2} \log {\left (x \right )} - 16 x \log {\left (x \right )}} \] Input:
integrate(((-x**5+33*x**4-288*x**3-256*x**2+7168*x+8192)*ln(x)**2+(-128*x* *3+1920*x**2)*ln(x)-128*x**2+2048*x)/(x**4-32*x**3+256*x**2)/exp(x)/ln(x)* *2,x)
Output:
(x**3*log(x) - 16*x**2*log(x) + 128*x + 512*log(x))*exp(-x)/(x**2*log(x) - 16*x*log(x))
Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {e^{-x} \left (2048 x-128 x^2+\left (1920 x^2-128 x^3\right ) \log (x)+\left (8192+7168 x-256 x^2-288 x^3+33 x^4-x^5\right ) \log ^2(x)\right )}{\left (256 x^2-32 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {{\left ({\left (x^{3} - 16 \, x^{2} + 512\right )} \log \left (x\right ) + 128 \, x\right )} e^{\left (-x\right )}}{{\left (x^{2} - 16 \, x\right )} \log \left (x\right )} \] Input:
integrate(((-x^5+33*x^4-288*x^3-256*x^2+7168*x+8192)*log(x)^2+(-128*x^3+19 20*x^2)*log(x)-128*x^2+2048*x)/(x^4-32*x^3+256*x^2)/exp(x)/log(x)^2,x, alg orithm="maxima")
Output:
((x^3 - 16*x^2 + 512)*log(x) + 128*x)*e^(-x)/((x^2 - 16*x)*log(x))
Time = 0.13 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68 \[ \int \frac {e^{-x} \left (2048 x-128 x^2+\left (1920 x^2-128 x^3\right ) \log (x)+\left (8192+7168 x-256 x^2-288 x^3+33 x^4-x^5\right ) \log ^2(x)\right )}{\left (256 x^2-32 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {x^{3} e^{\left (-x\right )} \log \left (x\right ) - 16 \, x^{2} e^{\left (-x\right )} \log \left (x\right ) + 128 \, x e^{\left (-x\right )} + 512 \, e^{\left (-x\right )} \log \left (x\right )}{x^{2} \log \left (x\right ) - 16 \, x \log \left (x\right )} \] Input:
integrate(((-x^5+33*x^4-288*x^3-256*x^2+7168*x+8192)*log(x)^2+(-128*x^3+19 20*x^2)*log(x)-128*x^2+2048*x)/(x^4-32*x^3+256*x^2)/exp(x)/log(x)^2,x, alg orithm="giac")
Output:
(x^3*e^(-x)*log(x) - 16*x^2*e^(-x)*log(x) + 128*x*e^(-x) + 512*e^(-x)*log( x))/(x^2*log(x) - 16*x*log(x))
Time = 7.35 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {e^{-x} \left (2048 x-128 x^2+\left (1920 x^2-128 x^3\right ) \log (x)+\left (8192+7168 x-256 x^2-288 x^3+33 x^4-x^5\right ) \log ^2(x)\right )}{\left (256 x^2-32 x^3+x^4\right ) \log ^2(x)} \, dx=x\,{\mathrm {e}}^{-x}+\frac {128\,x\,{\mathrm {e}}^{-x}+512\,{\mathrm {e}}^{-x}\,\ln \left (x\right )}{x\,\ln \left (x\right )\,\left (x-16\right )} \] Input:
int((exp(-x)*(2048*x + log(x)*(1920*x^2 - 128*x^3) + log(x)^2*(7168*x - 25 6*x^2 - 288*x^3 + 33*x^4 - x^5 + 8192) - 128*x^2))/(log(x)^2*(256*x^2 - 32 *x^3 + x^4)),x)
Output:
x*exp(-x) + (128*x*exp(-x) + 512*exp(-x)*log(x))/(x*log(x)*(x - 16))
Time = 0.18 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \frac {e^{-x} \left (2048 x-128 x^2+\left (1920 x^2-128 x^3\right ) \log (x)+\left (8192+7168 x-256 x^2-288 x^3+33 x^4-x^5\right ) \log ^2(x)\right )}{\left (256 x^2-32 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {\mathrm {log}\left (x \right ) x^{3}-16 \,\mathrm {log}\left (x \right ) x^{2}+512 \,\mathrm {log}\left (x \right )+128 x}{e^{x} \mathrm {log}\left (x \right ) x \left (x -16\right )} \] Input:
int(((-x^5+33*x^4-288*x^3-256*x^2+7168*x+8192)*log(x)^2+(-128*x^3+1920*x^2 )*log(x)-128*x^2+2048*x)/(x^4-32*x^3+256*x^2)/exp(x)/log(x)^2,x)
Output:
(log(x)*x**3 - 16*log(x)*x**2 + 512*log(x) + 128*x)/(e**x*log(x)*x*(x - 16 ))