Integrand size = 89, antiderivative size = 27 \[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=e^{-x+\frac {x \left (5+x+(4-2 x-\log (x))^2\right )}{-3+x}} \] Output:
exp(x/(-3+x)*(5+x+(4-2*x-ln(x))^2))/exp(x)
Time = 5.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=e^{\frac {x \left (4 \left (6-4 x+x^2\right )+\log ^2(x)\right )}{-3+x}} x^{\frac {4 (-2+x) x}{-3+x}} \] Input:
Integrate[(E^(-x + (21*x - 15*x^2 + 4*x^3 + (-8*x + 4*x^2)*Log[x] + x*Log[ x]^2)/(-3 + x))*(-48 + 76*x - 48*x^2 + 8*x^3 + (18 - 22*x + 4*x^2)*Log[x] - 3*Log[x]^2))/(9 - 6*x + x^2),x]
Output:
E^((x*(4*(6 - 4*x + x^2) + Log[x]^2))/(-3 + x))*x^((4*(-2 + x)*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.
\(\displaystyle \int \frac {\left (8 x^3-48 x^2+\left (4 x^2-22 x+18\right ) \log (x)+76 x-3 \log ^2(x)-48\right ) \exp \left (\frac {4 x^3-15 x^2+\left (4 x^2-8 x\right ) \log (x)+21 x+x \log ^2(x)}{x-3}-x\right )}{x^2-6 x+9} \, dx\) |
\(\Big \downarrow \) 7277 |
\(\displaystyle 4 \int -\frac {\exp \left (-x-\frac {4 x^3-15 x^2+\log ^2(x) x+21 x-4 \left (2 x-x^2\right ) \log (x)}{3-x}\right ) \left (-8 x^3+48 x^2-76 x+3 \log ^2(x)-2 \left (2 x^2-11 x+9\right ) \log (x)+48\right )}{4 (3-x)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\int \frac {\exp \left (-x-\frac {4 x^3-15 x^2+\log ^2(x) x+21 x-4 \left (2 x-x^2\right ) \log (x)}{3-x}\right ) \left (-8 x^3+48 x^2-76 x+3 \log ^2(x)-2 \left (2 x^2-11 x+9\right ) \log (x)+48\right )}{(3-x)^2}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle -\int \frac {\exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right ) \left (-8 x^3+48 x^2-76 x+3 \log ^2(x)-2 \left (2 x^2-11 x+9\right ) \log (x)+48\right )}{(3-x)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\int \left (-\frac {8 \exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right ) x^3}{(x-3)^2}+\frac {48 \exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right ) x^2}{(x-3)^2}-\frac {76 \exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right ) x}{(x-3)^2}+\frac {3 \exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right ) \log ^2(x)}{(x-3)^2}-\frac {2 \exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right ) (x-1) (2 x-9) \log (x)}{(x-3)^2}+\frac {48 \exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right )}{(x-3)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -36 \int \frac {\exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right )}{(x-3)^2}dx+4 \int \frac {\exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right )}{x-3}dx+8 \int \exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right ) xdx+4 \int \exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right ) \log (x)dx-12 \int \frac {\exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right ) \log (x)}{(x-3)^2}dx+2 \int \frac {\exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right ) \log (x)}{x-3}dx-3 \int \frac {\exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right ) \log ^2(x)}{(x-3)^2}dx\) |
Input:
Int[(E^(-x + (21*x - 15*x^2 + 4*x^3 + (-8*x + 4*x^2)*Log[x] + x*Log[x]^2)/ (-3 + x))*(-48 + 76*x - 48*x^2 + 8*x^3 + (18 - 22*x + 4*x^2)*Log[x] - 3*Lo g[x]^2))/(9 - 6*x + x^2),x]
Output:
$Aborted
Time = 2.41 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19
method | result | size |
risch | \({\mathrm e}^{\frac {x \left (\ln \left (x \right )^{2}+4 x \ln \left (x \right )+4 x^{2}-8 \ln \left (x \right )-16 x +24\right )}{-3+x}}\) | \(32\) |
parallelrisch | \({\mathrm e}^{-x} {\mathrm e}^{\frac {x \left (\ln \left (x \right )^{2}+4 x \ln \left (x \right )+4 x^{2}-8 \ln \left (x \right )-15 x +21\right )}{-3+x}}\) | \(37\) |
Input:
int((-3*ln(x)^2+(4*x^2-22*x+18)*ln(x)+8*x^3-48*x^2+76*x-48)*exp((x*ln(x)^2 +(4*x^2-8*x)*ln(x)+4*x^3-15*x^2+21*x)/(-3+x))/(x^2-6*x+9)/exp(x),x,method= _RETURNVERBOSE)
Output:
exp(x*(ln(x)^2+4*x*ln(x)+4*x^2-8*ln(x)-16*x+24)/(-3+x))
Time = 0.11 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=e^{\left (\frac {4 \, x^{3} + x \log \left (x\right )^{2} - 16 \, x^{2} + 4 \, {\left (x^{2} - 2 \, x\right )} \log \left (x\right ) + 24 \, x}{x - 3}\right )} \] Input:
integrate((-3*log(x)^2+(4*x^2-22*x+18)*log(x)+8*x^3-48*x^2+76*x-48)*exp((x *log(x)^2+(4*x^2-8*x)*log(x)+4*x^3-15*x^2+21*x)/(-3+x))/(x^2-6*x+9)/exp(x) ,x, algorithm="fricas")
Output:
e^((4*x^3 + x*log(x)^2 - 16*x^2 + 4*(x^2 - 2*x)*log(x) + 24*x)/(x - 3))
Time = 36.44 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=e^{- x} e^{\frac {4 x^{3} - 15 x^{2} + x \log {\left (x \right )}^{2} + 21 x + \left (4 x^{2} - 8 x\right ) \log {\left (x \right )}}{x - 3}} \] Input:
integrate((-3*ln(x)**2+(4*x**2-22*x+18)*ln(x)+8*x**3-48*x**2+76*x-48)*exp( (x*ln(x)**2+(4*x**2-8*x)*ln(x)+4*x**3-15*x**2+21*x)/(-3+x))/(x**2-6*x+9)/e xp(x),x)
Output:
exp(-x)*exp((4*x**3 - 15*x**2 + x*log(x)**2 + 21*x + (4*x**2 - 8*x)*log(x) )/(x - 3))
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).
Time = 0.18 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=x^{4} e^{\left (4 \, x^{2} + 4 \, x \log \left (x\right ) + \log \left (x\right )^{2} - 4 \, x + \frac {3 \, \log \left (x\right )^{2}}{x - 3} + \frac {12 \, \log \left (x\right )}{x - 3} + \frac {36}{x - 3} + 12\right )} \] Input:
integrate((-3*log(x)^2+(4*x^2-22*x+18)*log(x)+8*x^3-48*x^2+76*x-48)*exp((x *log(x)^2+(4*x^2-8*x)*log(x)+4*x^3-15*x^2+21*x)/(-3+x))/(x^2-6*x+9)/exp(x) ,x, algorithm="maxima")
Output:
x^4*e^(4*x^2 + 4*x*log(x) + log(x)^2 - 4*x + 3*log(x)^2/(x - 3) + 12*log(x )/(x - 3) + 36/(x - 3) + 12)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (24) = 48\).
Time = 0.14 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.33 \[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=e^{\left (\frac {4 \, x^{3}}{x - 3} + \frac {4 \, x^{2} \log \left (x\right )}{x - 3} + \frac {x \log \left (x\right )^{2}}{x - 3} - \frac {16 \, x^{2}}{x - 3} - \frac {8 \, x \log \left (x\right )}{x - 3} + \frac {24 \, x}{x - 3}\right )} \] Input:
integrate((-3*log(x)^2+(4*x^2-22*x+18)*log(x)+8*x^3-48*x^2+76*x-48)*exp((x *log(x)^2+(4*x^2-8*x)*log(x)+4*x^3-15*x^2+21*x)/(-3+x))/(x^2-6*x+9)/exp(x) ,x, algorithm="giac")
Output:
e^(4*x^3/(x - 3) + 4*x^2*log(x)/(x - 3) + x*log(x)^2/(x - 3) - 16*x^2/(x - 3) - 8*x*log(x)/(x - 3) + 24*x/(x - 3))
Time = 2.88 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.52 \[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=\frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^{\frac {21\,x}{x-3}}\,{\mathrm {e}}^{\frac {x\,{\ln \left (x\right )}^2}{x-3}}\,{\mathrm {e}}^{\frac {4\,x^3}{x-3}}\,{\mathrm {e}}^{-\frac {15\,x^2}{x-3}}}{x^{\frac {4\,\left (2\,x-x^2\right )}{x-3}}} \] Input:
int((exp(-x)*exp((21*x + x*log(x)^2 - log(x)*(8*x - 4*x^2) - 15*x^2 + 4*x^ 3)/(x - 3))*(76*x - 3*log(x)^2 + log(x)*(4*x^2 - 22*x + 18) - 48*x^2 + 8*x ^3 - 48))/(x^2 - 6*x + 9),x)
Output:
(exp(-x)*exp((21*x)/(x - 3))*exp((x*log(x)^2)/(x - 3))*exp((4*x^3)/(x - 3) )*exp(-(15*x^2)/(x - 3)))/x^((4*(2*x - x^2))/(x - 3))
Time = 0.37 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=\frac {x^{4 x} e^{\frac {\mathrm {log}\left (x \right )^{2} x +12 \,\mathrm {log}\left (x \right )+4 x^{3}-12 x^{2}+36}{x -3}} e^{12} x^{4}}{e^{4 x}} \] Input:
int((-3*log(x)^2+(4*x^2-22*x+18)*log(x)+8*x^3-48*x^2+76*x-48)*exp((x*log(x )^2+(4*x^2-8*x)*log(x)+4*x^3-15*x^2+21*x)/(-3+x))/(x^2-6*x+9)/exp(x),x)
Output:
(x**(4*x)*e**((log(x)**2*x + 12*log(x) + 4*x**3 - 12*x**2 + 36)/(x - 3))*e **12*x**4)/e**(4*x)