\(\int \frac {e^{\frac {1}{72} (-4-4 x+7 x^2-2 x^3+(4-4 x+x^2) \log (x))} (4-8 x+15 x^2-6 x^3+(-4 x+2 x^2) \log (x))}{72 x} \, dx\) [16]

Optimal result

Integrand size = 66, antiderivative size = 20 \[ \int \frac {e^{\frac {1}{72} \left (-4-4 x+7 x^2-2 x^3+\left (4-4 x+x^2\right ) \log (x)\right )} \left (4-8 x+15 x^2-6 x^3+\left (-4 x+2 x^2\right ) \log (x)\right )}{72 x} \, dx=e^{\frac {1}{72} (2-x)^2 (-1-2 x+\log (x))} \] Output:

exp(1/72*(ln(x)-2*x-1)*(2-x)^2)
 

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.40 \[ \int \frac {e^{\frac {1}{72} \left (-4-4 x+7 x^2-2 x^3+\left (4-4 x+x^2\right ) \log (x)\right )} \left (4-8 x+15 x^2-6 x^3+\left (-4 x+2 x^2\right ) \log (x)\right )}{72 x} \, dx=e^{-\frac {1}{72} (-2+x)^2 (1+2 x)} x^{\frac {1}{72} (-2+x)^2} \] Input:

Integrate[(E^((-4 - 4*x + 7*x^2 - 2*x^3 + (4 - 4*x + x^2)*Log[x])/72)*(4 - 
 8*x + 15*x^2 - 6*x^3 + (-4*x + 2*x^2)*Log[x]))/(72*x),x]
 

Output:

x^((-2 + x)^2/72)/E^(((-2 + x)^2*(1 + 2*x))/72)
 

Rubi [F]

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.

\(\displaystyle \int \frac {\left (-6 x^3+15 x^2+\left (2 x^2-4 x\right ) \log (x)-8 x+4\right ) \exp \left (\frac {1}{72} \left (-2 x^3+7 x^2+\left (x^2-4 x+4\right ) \log (x)-4 x-4\right )\right )}{72 x} \, dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{72} \int e^{\frac {1}{72} \left (-2 x^3+7 x^2-4 x-4\right )} x^{\frac {1}{72} \left (x^2-4 x+4\right )-1} \left (-6 x^3+15 x^2-8 x-2 \left (2 x-x^2\right ) \log (x)+4\right )dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \frac {1}{72} \int e^{\frac {1}{72} \left (-2 x^3+7 x^2-4 x-4\right )} (2-x) x^{\frac {x^2}{72}-\frac {x}{18}-\frac {17}{18}} \left (6 x^2-2 \log (x) x-3 x+2\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{72} \int \left (4 e^{\frac {1}{72} \left (-2 x^3+7 x^2-4 x-4\right )} x^{\frac {x^2}{72}-\frac {x}{18}-\frac {17}{18}}-8 e^{\frac {1}{72} \left (-2 x^3+7 x^2-4 x-4\right )} x^{\frac {x^2}{72}-\frac {x}{18}+\frac {1}{18}}+2 e^{\frac {1}{72} \left (-2 x^3+7 x^2-4 x-4\right )} (x-2) \log (x) x^{\frac {x^2}{72}-\frac {x}{18}+\frac {1}{18}}+15 e^{\frac {1}{72} \left (-2 x^3+7 x^2-4 x-4\right )} x^{\frac {x^2}{72}-\frac {x}{18}+\frac {19}{18}}-6 e^{\frac {1}{72} \left (-2 x^3+7 x^2-4 x-4\right )} x^{\frac {x^2}{72}-\frac {x}{18}+\frac {37}{18}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{72} \left (4 \int \frac {\int e^{-\frac {1}{72} (x-2)^2 (2 x+1)} x^{\frac {1}{72} (x-2)^2}dx}{x}dx-2 \int \frac {\int e^{-\frac {1}{72} (x-2)^2 (2 x+1)} x^{\frac {1}{72} \left (x^2-4 x+76\right )}dx}{x}dx+4 \int e^{\frac {1}{72} \left (-2 x^3+7 x^2-4 x-4\right )} x^{\frac {x^2}{72}-\frac {x}{18}-\frac {17}{18}}dx-8 \int e^{\frac {1}{72} \left (-2 x^3+7 x^2-4 x-4\right )} x^{\frac {x^2}{72}-\frac {x}{18}+\frac {1}{18}}dx+15 \int e^{\frac {1}{72} \left (-2 x^3+7 x^2-4 x-4\right )} x^{\frac {x^2}{72}-\frac {x}{18}+\frac {19}{18}}dx-6 \int e^{\frac {1}{72} \left (-2 x^3+7 x^2-4 x-4\right )} x^{\frac {x^2}{72}-\frac {x}{18}+\frac {37}{18}}dx-4 \log (x) \int e^{\frac {1}{72} \left (-2 x^3+7 x^2-4 x-4\right )} x^{\frac {x^2}{72}-\frac {x}{18}+\frac {1}{18}}dx+2 \log (x) \int e^{\frac {1}{72} \left (-2 x^3+7 x^2-4 x-4\right )} x^{\frac {x^2}{72}-\frac {x}{18}+\frac {19}{18}}dx\right )\)

Input:

Int[(E^((-4 - 4*x + 7*x^2 - 2*x^3 + (4 - 4*x + x^2)*Log[x])/72)*(4 - 8*x + 
 15*x^2 - 6*x^3 + (-4*x + 2*x^2)*Log[x]))/(72*x),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20

method	result	size
risch	\(x^{\frac {\left (-2+x \right )^{2}}{72}} {\mathrm e}^{-\frac {\left (1+2 x \right ) \left (-2+x \right )^{2}}{72}}\)	\(24\)
norman	\({\mathrm e}^{\frac {\left (x^{2}-4 x +4\right ) \ln \left (x \right )}{72}-\frac {x^{3}}{36}+\frac {7 x^{2}}{72}-\frac {x}{18}-\frac {1}{18}}\)	\(29\)
parallelrisch	\({\mathrm e}^{\frac {\left (x^{2}-4 x +4\right ) \ln \left (x \right )}{72}-\frac {x^{3}}{36}+\frac {7 x^{2}}{72}-\frac {x}{18}-\frac {1}{18}}\)	\(29\)

Input:

int(1/72*((2*x^2-4*x)*ln(x)-6*x^3+15*x^2-8*x+4)*exp(1/72*(x^2-4*x+4)*ln(x) 
-1/36*x^3+7/72*x^2-1/18*x-1/18)/x,x,method=_RETURNVERBOSE)
 

Output:

x^(1/72*(-2+x)^2)*exp(-1/72*(1+2*x)*(-2+x)^2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.40 \[ \int \frac {e^{\frac {1}{72} \left (-4-4 x+7 x^2-2 x^3+\left (4-4 x+x^2\right ) \log (x)\right )} \left (4-8 x+15 x^2-6 x^3+\left (-4 x+2 x^2\right ) \log (x)\right )}{72 x} \, dx=e^{\left (-\frac {1}{36} \, x^{3} + \frac {7}{72} \, x^{2} + \frac {1}{72} \, {\left (x^{2} - 4 \, x + 4\right )} \log \left (x\right ) - \frac {1}{18} \, x - \frac {1}{18}\right )} \] Input:

integrate(1/72*((2*x^2-4*x)*log(x)-6*x^3+15*x^2-8*x+4)*exp(1/72*(x^2-4*x+4 
)*log(x)-1/36*x^3+7/72*x^2-1/18*x-1/18)/x,x, algorithm="fricas")
 

Output:

e^(-1/36*x^3 + 7/72*x^2 + 1/72*(x^2 - 4*x + 4)*log(x) - 1/18*x - 1/18)
 

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.70 \[ \int \frac {e^{\frac {1}{72} \left (-4-4 x+7 x^2-2 x^3+\left (4-4 x+x^2\right ) \log (x)\right )} \left (4-8 x+15 x^2-6 x^3+\left (-4 x+2 x^2\right ) \log (x)\right )}{72 x} \, dx=e^{- \frac {x^{3}}{36} + \frac {7 x^{2}}{72} - \frac {x}{18} + \left (\frac {x^{2}}{72} - \frac {x}{18} + \frac {1}{18}\right ) \log {\left (x \right )} - \frac {1}{18}} \] Input:

integrate(1/72*((2*x**2-4*x)*ln(x)-6*x**3+15*x**2-8*x+4)*exp(1/72*(x**2-4* 
x+4)*ln(x)-1/36*x**3+7/72*x**2-1/18*x-1/18)/x,x)
 

Output:

exp(-x**3/36 + 7*x**2/72 - x/18 + (x**2/72 - x/18 + 1/18)*log(x) - 1/18)
 

Maxima [F]

\[ \int \frac {e^{\frac {1}{72} \left (-4-4 x+7 x^2-2 x^3+\left (4-4 x+x^2\right ) \log (x)\right )} \left (4-8 x+15 x^2-6 x^3+\left (-4 x+2 x^2\right ) \log (x)\right )}{72 x} \, dx=\int { -\frac {{\left (6 \, x^{3} - 15 \, x^{2} - 2 \, {\left (x^{2} - 2 \, x\right )} \log \left (x\right ) + 8 \, x - 4\right )} e^{\left (-\frac {1}{36} \, x^{3} + \frac {7}{72} \, x^{2} + \frac {1}{72} \, {\left (x^{2} - 4 \, x + 4\right )} \log \left (x\right ) - \frac {1}{18} \, x - \frac {1}{18}\right )}}{72 \, x} \,d x } \] Input:

integrate(1/72*((2*x^2-4*x)*log(x)-6*x^3+15*x^2-8*x+4)*exp(1/72*(x^2-4*x+4 
)*log(x)-1/36*x^3+7/72*x^2-1/18*x-1/18)/x,x, algorithm="maxima")
 

Output:

-1/72*integrate((6*x^3 - 15*x^2 - 2*(x^2 - 2*x)*log(x) + 8*x - 4)*e^(-1/36 
*x^3 + 7/72*x^2 + 1/72*(x^2 - 4*x + 4)*log(x) - 1/18*x - 1/18)/x, x)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.60 \[ \int \frac {e^{\frac {1}{72} \left (-4-4 x+7 x^2-2 x^3+\left (4-4 x+x^2\right ) \log (x)\right )} \left (4-8 x+15 x^2-6 x^3+\left (-4 x+2 x^2\right ) \log (x)\right )}{72 x} \, dx=e^{\left (-\frac {1}{36} \, x^{3} + \frac {1}{72} \, x^{2} \log \left (x\right ) + \frac {7}{72} \, x^{2} - \frac {1}{18} \, x \log \left (x\right ) - \frac {1}{18} \, x + \frac {1}{18} \, \log \left (x\right ) - \frac {1}{18}\right )} \] Input:

integrate(1/72*((2*x^2-4*x)*log(x)-6*x^3+15*x^2-8*x+4)*exp(1/72*(x^2-4*x+4 
)*log(x)-1/36*x^3+7/72*x^2-1/18*x-1/18)/x,x, algorithm="giac")
 

Output:

e^(-1/36*x^3 + 1/72*x^2*log(x) + 7/72*x^2 - 1/18*x*log(x) - 1/18*x + 1/18* 
log(x) - 1/18)
 

Mupad [B] (verification not implemented)

Time = 3.95 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.70 \[ \int \frac {e^{\frac {1}{72} \left (-4-4 x+7 x^2-2 x^3+\left (4-4 x+x^2\right ) \log (x)\right )} \left (4-8 x+15 x^2-6 x^3+\left (-4 x+2 x^2\right ) \log (x)\right )}{72 x} \, dx=x^{\frac {x^2}{72}+\frac {1}{18}}\,{\mathrm {e}}^{-\frac {x\,\ln \left (x\right )}{18}}\,{\mathrm {e}}^{-\frac {x}{18}}\,{\mathrm {e}}^{-\frac {1}{18}}\,{\mathrm {e}}^{-\frac {x^3}{36}}\,{\mathrm {e}}^{\frac {7\,x^2}{72}} \] Input:

int(-(exp((log(x)*(x^2 - 4*x + 4))/72 - x/18 + (7*x^2)/72 - x^3/36 - 1/18) 
*(8*x + log(x)*(4*x - 2*x^2) - 15*x^2 + 6*x^3 - 4))/(72*x),x)
 

Output:

x^(x^2/72 + 1/18)*exp(-(x*log(x))/18)*exp(-x/18)*exp(-1/18)*exp(-x^3/36)*e 
xp((7*x^2)/72)
 

Reduce [F]

\[ \int \frac {e^{\frac {1}{72} \left (-4-4 x+7 x^2-2 x^3+\left (4-4 x+x^2\right ) \log (x)\right )} \left (4-8 x+15 x^2-6 x^3+\left (-4 x+2 x^2\right ) \log (x)\right )}{72 x} \, dx=-\frac {\left (\int \frac {x^{\frac {x^{2}}{72}+\frac {2}{9}} e^{\frac {7 x^{2}}{72}} x^{2}}{x^{\frac {x}{18}+\frac {1}{6}} e^{\frac {1}{36} x^{3}+\frac {1}{18} x +\frac {1}{18}}}d x \right )}{12}+\frac {\left (\int \frac {x^{\frac {x^{2}}{72}+\frac {2}{9}} e^{\frac {7 x^{2}}{72}} \mathrm {log}\left (x \right ) x}{x^{\frac {x}{18}+\frac {1}{6}} e^{\frac {1}{36} x^{3}+\frac {1}{18} x +\frac {1}{18}}}d x \right )}{36}-\frac {\left (\int \frac {x^{\frac {x^{2}}{72}+\frac {2}{9}} e^{\frac {7 x^{2}}{72}} \mathrm {log}\left (x \right )}{x^{\frac {x}{18}+\frac {1}{6}} e^{\frac {1}{36} x^{3}+\frac {1}{18} x +\frac {1}{18}}}d x \right )}{18}+\frac {5 \left (\int \frac {x^{\frac {x^{2}}{72}+\frac {2}{9}} e^{\frac {7 x^{2}}{72}} x}{x^{\frac {x}{18}+\frac {1}{6}} e^{\frac {1}{36} x^{3}+\frac {1}{18} x +\frac {1}{18}}}d x \right )}{24}+\frac {\left (\int \frac {x^{\frac {x^{2}}{72}+\frac {2}{9}} e^{\frac {7 x^{2}}{72}}}{x^{\frac {x}{18}+\frac {1}{6}} e^{\frac {1}{36} x^{3}+\frac {1}{18} x +\frac {1}{18}} x}d x \right )}{18}-\frac {\left (\int \frac {x^{\frac {x^{2}}{72}+\frac {2}{9}} e^{\frac {7 x^{2}}{72}}}{x^{\frac {x}{18}+\frac {1}{6}} e^{\frac {1}{36} x^{3}+\frac {1}{18} x +\frac {1}{18}}}d x \right )}{9} \] Input:

int(1/72*((2*x^2-4*x)*log(x)-6*x^3+15*x^2-8*x+4)*exp(1/72*(x^2-4*x+4)*log( 
x)-1/36*x^3+7/72*x^2-1/18*x-1/18)/x,x)
 

Output:

( - 6*int((x**((x**2 + 16)/72)*e**((7*x**2)/72)*x**2)/(x**((x + 3)/18)*e** 
((x**3 + 2*x + 2)/36)),x) + 2*int((x**((x**2 + 16)/72)*e**((7*x**2)/72)*lo 
g(x)*x)/(x**((x + 3)/18)*e**((x**3 + 2*x + 2)/36)),x) - 4*int((x**((x**2 + 
 16)/72)*e**((7*x**2)/72)*log(x))/(x**((x + 3)/18)*e**((x**3 + 2*x + 2)/36 
)),x) + 15*int((x**((x**2 + 16)/72)*e**((7*x**2)/72)*x)/(x**((x + 3)/18)*e 
**((x**3 + 2*x + 2)/36)),x) + 4*int((x**((x**2 + 16)/72)*e**((7*x**2)/72)) 
/(x**((x + 3)/18)*e**((x**3 + 2*x + 2)/36)*x),x) - 8*int((x**((x**2 + 16)/ 
72)*e**((7*x**2)/72))/(x**((x + 3)/18)*e**((x**3 + 2*x + 2)/36)),x))/72