Integrand size = 172, antiderivative size = 22 \[ \int \frac {-14+12 x-6 x^2+x^3+\left (-8+12 x-6 x^2+x^3\right ) \log (x)+e^{-3 e^x+3 x} (1+\log (x))+e^{-2 e^x+2 x} (-6+3 x+(-6+3 x) \log (x))+e^{-e^x+x} \left (6+6 e^x-12 x+3 x^2+\left (12-12 x+3 x^2\right ) \log (x)\right )}{-8+e^{-3 e^x+3 x}+12 x-6 x^2+x^3+e^{-2 e^x+2 x} (-6+3 x)+e^{-e^x+x} \left (12-12 x+3 x^2\right )} \, dx=-3+\frac {3}{\left (-2+e^{-e^x+x}+x\right )^2}+x \log (x) \] Output:
3/(x+exp(x-exp(x))-2)^2-3+x*ln(x)
Leaf count is larger than twice the leaf count of optimal. \(45\) vs. \(2(22)=44\).
Time = 0.12 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.05 \[ \int \frac {-14+12 x-6 x^2+x^3+\left (-8+12 x-6 x^2+x^3\right ) \log (x)+e^{-3 e^x+3 x} (1+\log (x))+e^{-2 e^x+2 x} (-6+3 x+(-6+3 x) \log (x))+e^{-e^x+x} \left (6+6 e^x-12 x+3 x^2+\left (12-12 x+3 x^2\right ) \log (x)\right )}{-8+e^{-3 e^x+3 x}+12 x-6 x^2+x^3+e^{-2 e^x+2 x} (-6+3 x)+e^{-e^x+x} \left (12-12 x+3 x^2\right )} \, dx=\frac {3 e^{2 e^x}+\left (e^x+e^{e^x} (-2+x)\right )^2 x \log (x)}{\left (e^x+e^{e^x} (-2+x)\right )^2} \] Input:
Integrate[(-14 + 12*x - 6*x^2 + x^3 + (-8 + 12*x - 6*x^2 + x^3)*Log[x] + E ^(-3*E^x + 3*x)*(1 + Log[x]) + E^(-2*E^x + 2*x)*(-6 + 3*x + (-6 + 3*x)*Log [x]) + E^(-E^x + x)*(6 + 6*E^x - 12*x + 3*x^2 + (12 - 12*x + 3*x^2)*Log[x] ))/(-8 + E^(-3*E^x + 3*x) + 12*x - 6*x^2 + x^3 + E^(-2*E^x + 2*x)*(-6 + 3* x) + E^(-E^x + x)*(12 - 12*x + 3*x^2)),x]
Output:
(3*E^(2*E^x) + (E^x + E^E^x*(-2 + x))^2*x*Log[x])/(E^x + E^E^x*(-2 + 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.
| \(\displaystyle \int \frac {x^3-6 x^2+e^{x-e^x} \left (3 x^2+\left (3 x^2-12 x+12\right ) \log (x)-12 x+6 e^x+6\right )+\left (x^3-6 x^2+12 x-8\right ) \log (x)+12 x+e^{3 x-3 e^x} (\log (x)+1)+e^{2 x-2 e^x} (3 x+(3 x-6) \log (x)-6)-14}{x^3-6 x^2+e^{x-e^x} \left (3 x^2-12 x+12\right )+12 x+e^{3 x-3 e^x}+e^{2 x-2 e^x} (3 x-6)-8} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{3 e^x} \left (-x^3+6 x^2-e^{x-e^x} \left (3 x^2+\left (3 x^2-12 x+12\right ) \log (x)-12 x+6 e^x+6\right )-\left (x^3-6 x^2+12 x-8\right ) \log (x)-12 x-e^{3 x-3 e^x} (\log (x)+1)-e^{2 x-2 e^x} (3 x+(3 x-6) \log (x)-6)+14\right )}{\left (-e^{e^x} x+2 e^{e^x}-e^x\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {6 e^{3 e^x} \left (e^{e^x} x^2-4 e^{e^x} x+x+4 e^{e^x}-3\right )}{\left (e^{e^x} x-2 e^{e^x}+e^x\right )^3}-\frac {6 e^{2 e^x} \left (2 e^{e^x} x-4 e^{e^x}+1\right )}{\left (e^{e^x} x-2 e^{e^x}+e^x\right )^2}+\frac {6 e^{2 e^x}}{e^{e^x} x-2 e^{e^x}+e^x}+\log (x)+1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 6 \int \frac {e^{4 e^x} x^2}{\left (e^{e^x} x-2 e^{e^x}+e^x\right )^3}dx-18 \int \frac {e^{3 e^x}}{\left (e^{e^x} x-2 e^{e^x}+e^x\right )^3}dx+24 \int \frac {e^{4 e^x}}{\left (e^{e^x} x-2 e^{e^x}+e^x\right )^3}dx+6 \int \frac {e^{3 e^x} x}{\left (e^{e^x} x-2 e^{e^x}+e^x\right )^3}dx-24 \int \frac {e^{4 e^x} x}{\left (e^{e^x} x-2 e^{e^x}+e^x\right )^3}dx-6 \int \frac {e^{2 e^x}}{\left (e^{e^x} x-2 e^{e^x}+e^x\right )^2}dx+24 \int \frac {e^{3 e^x}}{\left (e^{e^x} x-2 e^{e^x}+e^x\right )^2}dx-12 \int \frac {e^{3 e^x} x}{\left (e^{e^x} x-2 e^{e^x}+e^x\right )^2}dx+6 \int \frac {e^{2 e^x}}{e^{e^x} x-2 e^{e^x}+e^x}dx+x \log (x)\) |
Input:
Int[(-14 + 12*x - 6*x^2 + x^3 + (-8 + 12*x - 6*x^2 + x^3)*Log[x] + E^(-3*E ^x + 3*x)*(1 + Log[x]) + E^(-2*E^x + 2*x)*(-6 + 3*x + (-6 + 3*x)*Log[x]) + E^(-E^x + x)*(6 + 6*E^x - 12*x + 3*x^2 + (12 - 12*x + 3*x^2)*Log[x]))/(-8 + E^(-3*E^x + 3*x) + 12*x - 6*x^2 + x^3 + E^(-2*E^x + 2*x)*(-6 + 3*x) + E ^(-E^x + x)*(12 - 12*x + 3*x^2)),x]
Output:
$Aborted
Time = 0.82 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91
| method | result | size |
| risch | \(x \ln \left (x \right )+\frac {3}{\left (x +{\mathrm e}^{x -{\mathrm e}^{x}}-2\right )^{2}}\) | \(20\) |
| parallelrisch | \(-\frac {-6-8 x \ln \left (x \right )-2 x^{3} \ln \left (x \right )+8 x^{2} \ln \left (x \right )-2 \,{\mathrm e}^{2 x -2 \,{\mathrm e}^{x}} \ln \left (x \right ) x +8 \ln \left (x \right ) {\mathrm e}^{x -{\mathrm e}^{x}} x -4 \ln \left (x \right ) {\mathrm e}^{x -{\mathrm e}^{x}} x^{2}}{2 \left (x^{2}+2 \,{\mathrm e}^{x -{\mathrm e}^{x}} x +{\mathrm e}^{2 x -2 \,{\mathrm e}^{x}}-4 x -4 \,{\mathrm e}^{x -{\mathrm e}^{x}}+4\right )}\) | \(102\) |
Input:
int(((ln(x)+1)*exp(x-exp(x))^3+((-6+3*x)*ln(x)+3*x-6)*exp(x-exp(x))^2+((3* x^2-12*x+12)*ln(x)+6*exp(x)+3*x^2-12*x+6)*exp(x-exp(x))+(x^3-6*x^2+12*x-8) *ln(x)+x^3-6*x^2+12*x-14)/(exp(x-exp(x))^3+(-6+3*x)*exp(x-exp(x))^2+(3*x^2 -12*x+12)*exp(x-exp(x))+x^3-6*x^2+12*x-8),x,method=_RETURNVERBOSE)
Output:
x*ln(x)+3/(x+exp(x-exp(x))-2)^2
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (20) = 40\).
Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.64 \[ \int \frac {-14+12 x-6 x^2+x^3+\left (-8+12 x-6 x^2+x^3\right ) \log (x)+e^{-3 e^x+3 x} (1+\log (x))+e^{-2 e^x+2 x} (-6+3 x+(-6+3 x) \log (x))+e^{-e^x+x} \left (6+6 e^x-12 x+3 x^2+\left (12-12 x+3 x^2\right ) \log (x)\right )}{-8+e^{-3 e^x+3 x}+12 x-6 x^2+x^3+e^{-2 e^x+2 x} (-6+3 x)+e^{-e^x+x} \left (12-12 x+3 x^2\right )} \, dx=\frac {x e^{\left (2 \, x - 2 \, e^{x}\right )} \log \left (x\right ) + 2 \, {\left (x^{2} - 2 \, x\right )} e^{\left (x - e^{x}\right )} \log \left (x\right ) + {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} \log \left (x\right ) + 3}{x^{2} + 2 \, {\left (x - 2\right )} e^{\left (x - e^{x}\right )} - 4 \, x + e^{\left (2 \, x - 2 \, e^{x}\right )} + 4} \] Input:
integrate(((1+log(x))*exp(x-exp(x))^3+((-6+3*x)*log(x)+3*x-6)*exp(x-exp(x) )^2+((3*x^2-12*x+12)*log(x)+6*exp(x)+3*x^2-12*x+6)*exp(x-exp(x))+(x^3-6*x^ 2+12*x-8)*log(x)+x^3-6*x^2+12*x-14)/(exp(x-exp(x))^3+(-6+3*x)*exp(x-exp(x) )^2+(3*x^2-12*x+12)*exp(x-exp(x))+x^3-6*x^2+12*x-8),x, algorithm="fricas")
Output:
(x*e^(2*x - 2*e^x)*log(x) + 2*(x^2 - 2*x)*e^(x - e^x)*log(x) + (x^3 - 4*x^ 2 + 4*x)*log(x) + 3)/(x^2 + 2*(x - 2)*e^(x - e^x) - 4*x + e^(2*x - 2*e^x) + 4)
Time = 0.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64 \[ \int \frac {-14+12 x-6 x^2+x^3+\left (-8+12 x-6 x^2+x^3\right ) \log (x)+e^{-3 e^x+3 x} (1+\log (x))+e^{-2 e^x+2 x} (-6+3 x+(-6+3 x) \log (x))+e^{-e^x+x} \left (6+6 e^x-12 x+3 x^2+\left (12-12 x+3 x^2\right ) \log (x)\right )}{-8+e^{-3 e^x+3 x}+12 x-6 x^2+x^3+e^{-2 e^x+2 x} (-6+3 x)+e^{-e^x+x} \left (12-12 x+3 x^2\right )} \, dx=x \log {\left (x \right )} + \frac {3}{x^{2} - 4 x + \left (2 x - 4\right ) e^{x - e^{x}} + e^{2 x - 2 e^{x}} + 4} \] Input:
integrate(((1+ln(x))*exp(x-exp(x))**3+((-6+3*x)*ln(x)+3*x-6)*exp(x-exp(x)) **2+((3*x**2-12*x+12)*ln(x)+6*exp(x)+3*x**2-12*x+6)*exp(x-exp(x))+(x**3-6* x**2+12*x-8)*ln(x)+x**3-6*x**2+12*x-14)/(exp(x-exp(x))**3+(-6+3*x)*exp(x-e xp(x))**2+(3*x**2-12*x+12)*exp(x-exp(x))+x**3-6*x**2+12*x-8),x)
Output:
x*log(x) + 3/(x**2 - 4*x + (2*x - 4)*exp(x - exp(x)) + exp(2*x - 2*exp(x)) + 4)
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (20) = 40\).
Time = 0.16 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.64 \[ \int \frac {-14+12 x-6 x^2+x^3+\left (-8+12 x-6 x^2+x^3\right ) \log (x)+e^{-3 e^x+3 x} (1+\log (x))+e^{-2 e^x+2 x} (-6+3 x+(-6+3 x) \log (x))+e^{-e^x+x} \left (6+6 e^x-12 x+3 x^2+\left (12-12 x+3 x^2\right ) \log (x)\right )}{-8+e^{-3 e^x+3 x}+12 x-6 x^2+x^3+e^{-2 e^x+2 x} (-6+3 x)+e^{-e^x+x} \left (12-12 x+3 x^2\right )} \, dx=\frac {x e^{\left (2 \, x\right )} \log \left (x\right ) + 2 \, {\left (x^{2} - 2 \, x\right )} e^{\left (x + e^{x}\right )} \log \left (x\right ) + {\left ({\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} \log \left (x\right ) + 3\right )} e^{\left (2 \, e^{x}\right )}}{2 \, {\left (x - 2\right )} e^{\left (x + e^{x}\right )} + {\left (x^{2} - 4 \, x + 4\right )} e^{\left (2 \, e^{x}\right )} + e^{\left (2 \, x\right )}} \] Input:
integrate(((1+log(x))*exp(x-exp(x))^3+((-6+3*x)*log(x)+3*x-6)*exp(x-exp(x) )^2+((3*x^2-12*x+12)*log(x)+6*exp(x)+3*x^2-12*x+6)*exp(x-exp(x))+(x^3-6*x^ 2+12*x-8)*log(x)+x^3-6*x^2+12*x-14)/(exp(x-exp(x))^3+(-6+3*x)*exp(x-exp(x) )^2+(3*x^2-12*x+12)*exp(x-exp(x))+x^3-6*x^2+12*x-8),x, algorithm="maxima")
Output:
(x*e^(2*x)*log(x) + 2*(x^2 - 2*x)*e^(x + e^x)*log(x) + ((x^3 - 4*x^2 + 4*x )*log(x) + 3)*e^(2*e^x))/(2*(x - 2)*e^(x + e^x) + (x^2 - 4*x + 4)*e^(2*e^x ) + e^(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 6825 vs. \(2 (20) = 40\).
Time = 0.47 (sec) , antiderivative size = 6825, normalized size of antiderivative = 310.23 \[ \int \frac {-14+12 x-6 x^2+x^3+\left (-8+12 x-6 x^2+x^3\right ) \log (x)+e^{-3 e^x+3 x} (1+\log (x))+e^{-2 e^x+2 x} (-6+3 x+(-6+3 x) \log (x))+e^{-e^x+x} \left (6+6 e^x-12 x+3 x^2+\left (12-12 x+3 x^2\right ) \log (x)\right )}{-8+e^{-3 e^x+3 x}+12 x-6 x^2+x^3+e^{-2 e^x+2 x} (-6+3 x)+e^{-e^x+x} \left (12-12 x+3 x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate(((1+log(x))*exp(x-exp(x))^3+((-6+3*x)*log(x)+3*x-6)*exp(x-exp(x) )^2+((3*x^2-12*x+12)*log(x)+6*exp(x)+3*x^2-12*x+6)*exp(x-exp(x))+(x^3-6*x^ 2+12*x-8)*log(x)+x^3-6*x^2+12*x-14)/(exp(x-exp(x))^3+(-6+3*x)*exp(x-exp(x) )^2+(3*x^2-12*x+12)*exp(x-exp(x))+x^3-6*x^2+12*x-8),x, algorithm="giac")
Output:
(x^11*e^(5*x + 2*e^x)*log(x) - 3*x^11*e^(4*x + 2*e^x)*log(x) + 3*x^11*e^(3 *x + 2*e^x)*log(x) - x^11*e^(2*x + 2*e^x)*log(x) + 6*x^10*e^(6*x + e^x)*lo g(x) - 18*x^10*e^(5*x + 2*e^x)*log(x) - 18*x^10*e^(5*x + e^x)*log(x) + 57* x^10*e^(4*x + 2*e^x)*log(x) + 18*x^10*e^(4*x + e^x)*log(x) - 60*x^10*e^(3* x + 2*e^x)*log(x) - 6*x^10*e^(3*x + e^x)*log(x) + 21*x^10*e^(2*x + 2*e^x)* log(x) + 15*x^9*e^(7*x)*log(x) - 45*x^9*e^(6*x)*log(x) + 45*x^9*e^(5*x)*lo g(x) - 15*x^9*e^(4*x)*log(x) - 96*x^9*e^(6*x + e^x)*log(x) + 144*x^9*e^(5* x + 2*e^x)*log(x) + 306*x^9*e^(5*x + e^x)*log(x) - 480*x^9*e^(4*x + 2*e^x) *log(x) - 324*x^9*e^(4*x + e^x)*log(x) + 531*x^9*e^(3*x + 2*e^x)*log(x) + 114*x^9*e^(3*x + e^x)*log(x) - 195*x^9*e^(2*x + 2*e^x)*log(x) - 210*x^8*e^ (7*x)*log(x) + 675*x^8*e^(6*x)*log(x) - 720*x^8*e^(5*x)*log(x) + 255*x^8*e ^(4*x)*log(x) + 20*x^8*e^(8*x - e^x)*log(x) - 60*x^8*e^(7*x - e^x)*log(x) + 672*x^8*e^(6*x + e^x)*log(x) + 60*x^8*e^(6*x - e^x)*log(x) - 672*x^8*e^( 5*x + 2*e^x)*log(x) - 2268*x^8*e^(5*x + e^x)*log(x) - 20*x^8*e^(5*x - e^x) *log(x) + 2352*x^8*e^(4*x + 2*e^x)*log(x) + 2538*x^8*e^(4*x + e^x)*log(x) - 2730*x^8*e^(3*x + 2*e^x)*log(x) - 942*x^8*e^(3*x + e^x)*log(x) + 1051*x^ 8*e^(2*x + 2*e^x)*log(x) + 3*x^8*e^(5*x + 2*e^x) - 9*x^8*e^(4*x + 2*e^x) + 9*x^8*e^(3*x + 2*e^x) - 3*x^8*e^(2*x + 2*e^x) + 1260*x^7*e^(7*x)*log(x) - 4320*x^7*e^(6*x)*log(x) + 4905*x^7*e^(5*x)*log(x) - 1845*x^7*e^(4*x)*log( x) + 15*x^7*e^(9*x - 2*e^x)*log(x) - 240*x^7*e^(8*x - e^x)*log(x) - 45*...
Timed out. \[ \int \frac {-14+12 x-6 x^2+x^3+\left (-8+12 x-6 x^2+x^3\right ) \log (x)+e^{-3 e^x+3 x} (1+\log (x))+e^{-2 e^x+2 x} (-6+3 x+(-6+3 x) \log (x))+e^{-e^x+x} \left (6+6 e^x-12 x+3 x^2+\left (12-12 x+3 x^2\right ) \log (x)\right )}{-8+e^{-3 e^x+3 x}+12 x-6 x^2+x^3+e^{-2 e^x+2 x} (-6+3 x)+e^{-e^x+x} \left (12-12 x+3 x^2\right )} \, dx=\int \frac {12\,x+{\mathrm {e}}^{x-{\mathrm {e}}^x}\,\left (6\,{\mathrm {e}}^x-12\,x+\ln \left (x\right )\,\left (3\,x^2-12\,x+12\right )+3\,x^2+6\right )+{\mathrm {e}}^{3\,x-3\,{\mathrm {e}}^x}\,\left (\ln \left (x\right )+1\right )+\ln \left (x\right )\,\left (x^3-6\,x^2+12\,x-8\right )-6\,x^2+x^3+{\mathrm {e}}^{2\,x-2\,{\mathrm {e}}^x}\,\left (3\,x+\ln \left (x\right )\,\left (3\,x-6\right )-6\right )-14}{12\,x+{\mathrm {e}}^{3\,x-3\,{\mathrm {e}}^x}+{\mathrm {e}}^{x-{\mathrm {e}}^x}\,\left (3\,x^2-12\,x+12\right )+{\mathrm {e}}^{2\,x-2\,{\mathrm {e}}^x}\,\left (3\,x-6\right )-6\,x^2+x^3-8} \,d x \] Input:
int((12*x + exp(x - exp(x))*(6*exp(x) - 12*x + log(x)*(3*x^2 - 12*x + 12) + 3*x^2 + 6) + exp(3*x - 3*exp(x))*(log(x) + 1) + log(x)*(12*x - 6*x^2 + x ^3 - 8) - 6*x^2 + x^3 + exp(2*x - 2*exp(x))*(3*x + log(x)*(3*x - 6) - 6) - 14)/(12*x + exp(3*x - 3*exp(x)) + exp(x - exp(x))*(3*x^2 - 12*x + 12) + e xp(2*x - 2*exp(x))*(3*x - 6) - 6*x^2 + x^3 - 8),x)
Output:
int((12*x + exp(x - exp(x))*(6*exp(x) - 12*x + log(x)*(3*x^2 - 12*x + 12) + 3*x^2 + 6) + exp(3*x - 3*exp(x))*(log(x) + 1) + log(x)*(12*x - 6*x^2 + x ^3 - 8) - 6*x^2 + x^3 + exp(2*x - 2*exp(x))*(3*x + log(x)*(3*x - 6) - 6) - 14)/(12*x + exp(3*x - 3*exp(x)) + exp(x - exp(x))*(3*x^2 - 12*x + 12) + e xp(2*x - 2*exp(x))*(3*x - 6) - 6*x^2 + x^3 - 8), x)
Time = 0.36 (sec) , antiderivative size = 142, normalized size of antiderivative = 6.45 \[ \int \frac {-14+12 x-6 x^2+x^3+\left (-8+12 x-6 x^2+x^3\right ) \log (x)+e^{-3 e^x+3 x} (1+\log (x))+e^{-2 e^x+2 x} (-6+3 x+(-6+3 x) \log (x))+e^{-e^x+x} \left (6+6 e^x-12 x+3 x^2+\left (12-12 x+3 x^2\right ) \log (x)\right )}{-8+e^{-3 e^x+3 x}+12 x-6 x^2+x^3+e^{-2 e^x+2 x} (-6+3 x)+e^{-e^x+x} \left (12-12 x+3 x^2\right )} \, dx=\frac {e^{2 e^{x}} \mathrm {log}\left (x \right ) x^{3}-4 e^{2 e^{x}} \mathrm {log}\left (x \right ) x^{2}+4 e^{2 e^{x}} \mathrm {log}\left (x \right ) x +3 e^{2 e^{x}}+2 e^{e^{x}+x} \mathrm {log}\left (x \right ) x^{2}-4 e^{e^{x}+x} \mathrm {log}\left (x \right ) x +e^{2 x} \mathrm {log}\left (x \right ) x}{e^{2 e^{x}} x^{2}-4 e^{2 e^{x}} x +4 e^{2 e^{x}}+2 e^{e^{x}+x} x -4 e^{e^{x}+x}+e^{2 x}} \] Input:
int(((1+log(x))*exp(x-exp(x))^3+((-6+3*x)*log(x)+3*x-6)*exp(x-exp(x))^2+(( 3*x^2-12*x+12)*log(x)+6*exp(x)+3*x^2-12*x+6)*exp(x-exp(x))+(x^3-6*x^2+12*x -8)*log(x)+x^3-6*x^2+12*x-14)/(exp(x-exp(x))^3+(-6+3*x)*exp(x-exp(x))^2+(3 *x^2-12*x+12)*exp(x-exp(x))+x^3-6*x^2+12*x-8),x)
Output:
(e**(2*e**x)*log(x)*x**3 - 4*e**(2*e**x)*log(x)*x**2 + 4*e**(2*e**x)*log(x )*x + 3*e**(2*e**x) + 2*e**(e**x + x)*log(x)*x**2 - 4*e**(e**x + x)*log(x) *x + e**(2*x)*log(x)*x)/(e**(2*e**x)*x**2 - 4*e**(2*e**x)*x + 4*e**(2*e**x ) + 2*e**(e**x + x)*x - 4*e**(e**x + x) + e**(2*x))