Integrand size = 162, antiderivative size = 19 \[ \int \frac {-18-83 x-15 e^{2 x} x-e^{3 x} x-12 x^2+e^x \left (-51 x-14 x^2+2 x^3\right )+\left (69 x+30 e^x x+3 e^{2 x} x+2 x^2\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)}{-125 x-75 e^x x-15 e^{2 x} x-e^{3 x} x+\left (75 x+30 e^x x+3 e^{2 x} x\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)} \, dx=x+\frac {(-3+x)^2}{\left (5+e^x-\log (x)\right )^2} \] Output:
(-3+x)^2/(5+exp(x)-ln(x))^2+x
Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-18-83 x-15 e^{2 x} x-e^{3 x} x-12 x^2+e^x \left (-51 x-14 x^2+2 x^3\right )+\left (69 x+30 e^x x+3 e^{2 x} x+2 x^2\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)}{-125 x-75 e^x x-15 e^{2 x} x-e^{3 x} x+\left (75 x+30 e^x x+3 e^{2 x} x\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)} \, dx=x+\frac {(-3+x)^2}{\left (-5-e^x+\log (x)\right )^2} \] Input:
Integrate[(-18 - 83*x - 15*E^(2*x)*x - E^(3*x)*x - 12*x^2 + E^x*(-51*x - 1 4*x^2 + 2*x^3) + (69*x + 30*E^x*x + 3*E^(2*x)*x + 2*x^2)*Log[x] + (-15*x - 3*E^x*x)*Log[x]^2 + x*Log[x]^3)/(-125*x - 75*E^x*x - 15*E^(2*x)*x - E^(3* x)*x + (75*x + 30*E^x*x + 3*E^(2*x)*x)*Log[x] + (-15*x - 3*E^x*x)*Log[x]^2 + x*Log[x]^3),x]
Output:
x + (-3 + x)^2/(-5 - E^x + Log[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.
| \(\displaystyle \int \frac {-12 x^2+\left (2 x^2+30 e^x x+3 e^{2 x} x+69 x\right ) \log (x)+e^x \left (2 x^3-14 x^2-51 x\right )-15 e^{2 x} x-e^{3 x} x-83 x+x \log ^3(x)+\left (-3 e^x x-15 x\right ) \log ^2(x)-18}{-75 e^x x-15 e^{2 x} x-e^{3 x} x-125 x+x \log ^3(x)+\left (-3 e^x x-15 x\right ) \log ^2(x)+\left (30 e^x x+3 e^{2 x} x+75 x\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {12 x^2-\left (2 x^2+30 e^x x+3 e^{2 x} x+69 x\right ) \log (x)-e^x \left (2 x^3-14 x^2-51 x\right )+15 e^{2 x} x+e^{3 x} x+83 x-x \log ^3(x)-\left (-3 e^x x-15 x\right ) \log ^2(x)+18}{x \left (e^x-\log (x)+5\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 \left (x^2-7 x+12\right )}{\left (e^x-\log (x)+5\right )^2}-\frac {2 (x-3)^2 (-5 x+x \log (x)-1)}{x \left (e^x-\log (x)+5\right )^3}+1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 10 \int \frac {x^2}{\left (-\log (x)+e^x+5\right )^3}dx-2 \int \frac {x^2}{\left (-\log (x)+e^x+5\right )^2}dx-2 \int \frac {x^2 \log (x)}{\left (-\log (x)+e^x+5\right )^3}dx+78 \int \frac {1}{\left (-\log (x)+e^x+5\right )^3}dx+18 \int \frac {1}{x \left (-\log (x)+e^x+5\right )^3}dx-58 \int \frac {x}{\left (-\log (x)+e^x+5\right )^3}dx-24 \int \frac {1}{\left (-\log (x)+e^x+5\right )^2}dx+14 \int \frac {x}{\left (-\log (x)+e^x+5\right )^2}dx-18 \int \frac {\log (x)}{\left (-\log (x)+e^x+5\right )^3}dx+12 \int \frac {x \log (x)}{\left (-\log (x)+e^x+5\right )^3}dx+x\) |
Input:
Int[(-18 - 83*x - 15*E^(2*x)*x - E^(3*x)*x - 12*x^2 + E^x*(-51*x - 14*x^2 + 2*x^3) + (69*x + 30*E^x*x + 3*E^(2*x)*x + 2*x^2)*Log[x] + (-15*x - 3*E^x *x)*Log[x]^2 + x*Log[x]^3)/(-125*x - 75*E^x*x - 15*E^(2*x)*x - E^(3*x)*x + (75*x + 30*E^x*x + 3*E^(2*x)*x)*Log[x] + (-15*x - 3*E^x*x)*Log[x]^2 + x*L og[x]^3),x]
Output:
$Aborted
Time = 13.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16
| method | result | size |
| risch | \(x +\frac {x^{2}-6 x +9}{\left (5+{\mathrm e}^{x}-\ln \left (x \right )\right )^{2}}\) | \(22\) |
| parallelrisch | \(\frac {x \,{\mathrm e}^{2 x}-2 x \,{\mathrm e}^{x} \ln \left (x \right )+x \ln \left (x \right )^{2}+10 \,{\mathrm e}^{x} x -10 x \ln \left (x \right )+x^{2}+19 x +9}{\ln \left (x \right )^{2}-2 \,{\mathrm e}^{x} \ln \left (x \right )+{\mathrm e}^{2 x}-10 \ln \left (x \right )+10 \,{\mathrm e}^{x}+25}\) | \(65\) |
Input:
int((x*ln(x)^3+(-3*exp(x)*x-15*x)*ln(x)^2+(3*x*exp(x)^2+30*exp(x)*x+2*x^2+ 69*x)*ln(x)-x*exp(x)^3-15*x*exp(x)^2+(2*x^3-14*x^2-51*x)*exp(x)-12*x^2-83* x-18)/(x*ln(x)^3+(-3*exp(x)*x-15*x)*ln(x)^2+(3*x*exp(x)^2+30*exp(x)*x+75*x )*ln(x)-x*exp(x)^3-15*x*exp(x)^2-75*exp(x)*x-125*x),x,method=_RETURNVERBOS E)
Output:
x+(x^2-6*x+9)/(5+exp(x)-ln(x))^2
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (18) = 36\).
Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 3.53 \[ \int \frac {-18-83 x-15 e^{2 x} x-e^{3 x} x-12 x^2+e^x \left (-51 x-14 x^2+2 x^3\right )+\left (69 x+30 e^x x+3 e^{2 x} x+2 x^2\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)}{-125 x-75 e^x x-15 e^{2 x} x-e^{3 x} x+\left (75 x+30 e^x x+3 e^{2 x} x\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)} \, dx=-\frac {x \log \left (x\right )^{2} + x^{2} + x e^{\left (2 \, x\right )} + 10 \, x e^{x} - 2 \, {\left (x e^{x} + 5 \, x\right )} \log \left (x\right ) + 19 \, x + 9}{2 \, {\left (e^{x} + 5\right )} \log \left (x\right ) - \log \left (x\right )^{2} - e^{\left (2 \, x\right )} - 10 \, e^{x} - 25} \] Input:
integrate((x*log(x)^3+(-3*exp(x)*x-15*x)*log(x)^2+(3*x*exp(x)^2+30*exp(x)* x+2*x^2+69*x)*log(x)-x*exp(x)^3-15*x*exp(x)^2+(2*x^3-14*x^2-51*x)*exp(x)-1 2*x^2-83*x-18)/(x*log(x)^3+(-3*exp(x)*x-15*x)*log(x)^2+(3*x*exp(x)^2+30*ex p(x)*x+75*x)*log(x)-x*exp(x)^3-15*x*exp(x)^2-75*exp(x)*x-125*x),x, algorit hm="fricas")
Output:
-(x*log(x)^2 + x^2 + x*e^(2*x) + 10*x*e^x - 2*(x*e^x + 5*x)*log(x) + 19*x + 9)/(2*(e^x + 5)*log(x) - log(x)^2 - e^(2*x) - 10*e^x - 25)
Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (15) = 30\).
Time = 0.12 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int \frac {-18-83 x-15 e^{2 x} x-e^{3 x} x-12 x^2+e^x \left (-51 x-14 x^2+2 x^3\right )+\left (69 x+30 e^x x+3 e^{2 x} x+2 x^2\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)}{-125 x-75 e^x x-15 e^{2 x} x-e^{3 x} x+\left (75 x+30 e^x x+3 e^{2 x} x\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)} \, dx=x + \frac {x^{2} - 6 x + 9}{\left (10 - 2 \log {\left (x \right )}\right ) e^{x} + e^{2 x} + \log {\left (x \right )}^{2} - 10 \log {\left (x \right )} + 25} \] Input:
integrate((x*ln(x)**3+(-3*exp(x)*x-15*x)*ln(x)**2+(3*x*exp(x)**2+30*exp(x) *x+2*x**2+69*x)*ln(x)-x*exp(x)**3-15*x*exp(x)**2+(2*x**3-14*x**2-51*x)*exp (x)-12*x**2-83*x-18)/(x*ln(x)**3+(-3*exp(x)*x-15*x)*ln(x)**2+(3*x*exp(x)** 2+30*exp(x)*x+75*x)*ln(x)-x*exp(x)**3-15*x*exp(x)**2-75*exp(x)*x-125*x),x)
Output:
x + (x**2 - 6*x + 9)/((10 - 2*log(x))*exp(x) + exp(2*x) + log(x)**2 - 10*l og(x) + 25)
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (18) = 36\).
Time = 0.13 (sec) , antiderivative size = 67, normalized size of antiderivative = 3.53 \[ \int \frac {-18-83 x-15 e^{2 x} x-e^{3 x} x-12 x^2+e^x \left (-51 x-14 x^2+2 x^3\right )+\left (69 x+30 e^x x+3 e^{2 x} x+2 x^2\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)}{-125 x-75 e^x x-15 e^{2 x} x-e^{3 x} x+\left (75 x+30 e^x x+3 e^{2 x} x\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)} \, dx=-\frac {x \log \left (x\right )^{2} + x^{2} + x e^{\left (2 \, x\right )} - 2 \, {\left (x \log \left (x\right ) - 5 \, x\right )} e^{x} - 10 \, x \log \left (x\right ) + 19 \, x + 9}{2 \, {\left (\log \left (x\right ) - 5\right )} e^{x} - \log \left (x\right )^{2} - e^{\left (2 \, x\right )} + 10 \, \log \left (x\right ) - 25} \] Input:
integrate((x*log(x)^3+(-3*exp(x)*x-15*x)*log(x)^2+(3*x*exp(x)^2+30*exp(x)* x+2*x^2+69*x)*log(x)-x*exp(x)^3-15*x*exp(x)^2+(2*x^3-14*x^2-51*x)*exp(x)-1 2*x^2-83*x-18)/(x*log(x)^3+(-3*exp(x)*x-15*x)*log(x)^2+(3*x*exp(x)^2+30*ex p(x)*x+75*x)*log(x)-x*exp(x)^3-15*x*exp(x)^2-75*exp(x)*x-125*x),x, algorit hm="maxima")
Output:
-(x*log(x)^2 + x^2 + x*e^(2*x) - 2*(x*log(x) - 5*x)*e^x - 10*x*log(x) + 19 *x + 9)/(2*(log(x) - 5)*e^x - log(x)^2 - e^(2*x) + 10*log(x) - 25)
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (18) = 36\).
Time = 0.18 (sec) , antiderivative size = 99, normalized size of antiderivative = 5.21 \[ \int \frac {-18-83 x-15 e^{2 x} x-e^{3 x} x-12 x^2+e^x \left (-51 x-14 x^2+2 x^3\right )+\left (69 x+30 e^x x+3 e^{2 x} x+2 x^2\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)}{-125 x-75 e^x x-15 e^{2 x} x-e^{3 x} x+\left (75 x+30 e^x x+3 e^{2 x} x\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)} \, dx=\frac {4 \, x e^{x} \log \left (x\right ) - 2 \, x \log \left (x\right )^{2} - 2 \, x^{2} - 2 \, x e^{\left (2 \, x\right )} - 20 \, x e^{x} + 20 \, x \log \left (x\right ) + 6 \, e^{x} \log \left (x\right ) - 3 \, \log \left (x\right )^{2} - 38 \, x - 3 \, e^{\left (2 \, x\right )} - 30 \, e^{x} + 30 \, \log \left (x\right ) - 93}{2 \, {\left (2 \, e^{x} \log \left (x\right ) - \log \left (x\right )^{2} - e^{\left (2 \, x\right )} - 10 \, e^{x} + 10 \, \log \left (x\right ) - 25\right )}} \] Input:
integrate((x*log(x)^3+(-3*exp(x)*x-15*x)*log(x)^2+(3*x*exp(x)^2+30*exp(x)* x+2*x^2+69*x)*log(x)-x*exp(x)^3-15*x*exp(x)^2+(2*x^3-14*x^2-51*x)*exp(x)-1 2*x^2-83*x-18)/(x*log(x)^3+(-3*exp(x)*x-15*x)*log(x)^2+(3*x*exp(x)^2+30*ex p(x)*x+75*x)*log(x)-x*exp(x)^3-15*x*exp(x)^2-75*exp(x)*x-125*x),x, algorit hm="giac")
Output:
1/2*(4*x*e^x*log(x) - 2*x*log(x)^2 - 2*x^2 - 2*x*e^(2*x) - 20*x*e^x + 20*x *log(x) + 6*e^x*log(x) - 3*log(x)^2 - 38*x - 3*e^(2*x) - 30*e^x + 30*log(x ) - 93)/(2*e^x*log(x) - log(x)^2 - e^(2*x) - 10*e^x + 10*log(x) - 25)
Timed out. \[ \int \frac {-18-83 x-15 e^{2 x} x-e^{3 x} x-12 x^2+e^x \left (-51 x-14 x^2+2 x^3\right )+\left (69 x+30 e^x x+3 e^{2 x} x+2 x^2\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)}{-125 x-75 e^x x-15 e^{2 x} x-e^{3 x} x+\left (75 x+30 e^x x+3 e^{2 x} x\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)} \, dx=\int \frac {83\,x+15\,x\,{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^{3\,x}-x\,{\ln \left (x\right )}^3-\ln \left (x\right )\,\left (69\,x+3\,x\,{\mathrm {e}}^{2\,x}+30\,x\,{\mathrm {e}}^x+2\,x^2\right )+{\ln \left (x\right )}^2\,\left (15\,x+3\,x\,{\mathrm {e}}^x\right )+12\,x^2+{\mathrm {e}}^x\,\left (-2\,x^3+14\,x^2+51\,x\right )+18}{-x\,{\ln \left (x\right )}^3+\left (15\,x+3\,x\,{\mathrm {e}}^x\right )\,{\ln \left (x\right )}^2+\left (-75\,x-3\,x\,{\mathrm {e}}^{2\,x}-30\,x\,{\mathrm {e}}^x\right )\,\ln \left (x\right )+125\,x+15\,x\,{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^{3\,x}+75\,x\,{\mathrm {e}}^x} \,d x \] Input:
int((83*x + 15*x*exp(2*x) + x*exp(3*x) - x*log(x)^3 - log(x)*(69*x + 3*x*e xp(2*x) + 30*x*exp(x) + 2*x^2) + log(x)^2*(15*x + 3*x*exp(x)) + 12*x^2 + e xp(x)*(51*x + 14*x^2 - 2*x^3) + 18)/(125*x + 15*x*exp(2*x) + x*exp(3*x) - x*log(x)^3 - log(x)*(75*x + 3*x*exp(2*x) + 30*x*exp(x)) + log(x)^2*(15*x + 3*x*exp(x)) + 75*x*exp(x)),x)
Output:
int((83*x + 15*x*exp(2*x) + x*exp(3*x) - x*log(x)^3 - log(x)*(69*x + 3*x*e xp(2*x) + 30*x*exp(x) + 2*x^2) + log(x)^2*(15*x + 3*x*exp(x)) + 12*x^2 + e xp(x)*(51*x + 14*x^2 - 2*x^3) + 18)/(125*x + 15*x*exp(2*x) + x*exp(3*x) - x*log(x)^3 - log(x)*(75*x + 3*x*exp(2*x) + 30*x*exp(x)) + log(x)^2*(15*x + 3*x*exp(x)) + 75*x*exp(x)), x)
Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.68 \[ \int \frac {-18-83 x-15 e^{2 x} x-e^{3 x} x-12 x^2+e^x \left (-51 x-14 x^2+2 x^3\right )+\left (69 x+30 e^x x+3 e^{2 x} x+2 x^2\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)}{-125 x-75 e^x x-15 e^{2 x} x-e^{3 x} x+\left (75 x+30 e^x x+3 e^{2 x} x\right ) \log (x)+\left (-15 x-3 e^x x\right ) \log ^2(x)+x \log ^3(x)} \, dx=\frac {e^{2 x} x -2 e^{x} \mathrm {log}\left (x \right ) x +10 e^{x} x +\mathrm {log}\left (x \right )^{2} x -10 \,\mathrm {log}\left (x \right ) x +x^{2}+19 x +9}{e^{2 x}-2 e^{x} \mathrm {log}\left (x \right )+10 e^{x}+\mathrm {log}\left (x \right )^{2}-10 \,\mathrm {log}\left (x \right )+25} \] Input:
int((x*log(x)^3+(-3*exp(x)*x-15*x)*log(x)^2+(3*x*exp(x)^2+30*exp(x)*x+2*x^ 2+69*x)*log(x)-x*exp(x)^3-15*x*exp(x)^2+(2*x^3-14*x^2-51*x)*exp(x)-12*x^2- 83*x-18)/(x*log(x)^3+(-3*exp(x)*x-15*x)*log(x)^2+(3*x*exp(x)^2+30*exp(x)*x +75*x)*log(x)-x*exp(x)^3-15*x*exp(x)^2-75*exp(x)*x-125*x),x)
Output:
(e**(2*x)*x - 2*e**x*log(x)*x + 10*e**x*x + log(x)**2*x - 10*log(x)*x + x* *2 + 19*x + 9)/(e**(2*x) - 2*e**x*log(x) + 10*e**x + log(x)**2 - 10*log(x) + 25)