\(\int \frac {e^{\frac {4 x+4 x^2+x^3+(4+4 x+x^2) \log (x)}{x^2}} (4 x^2-2 x^3+3 x^4+x^5+(-8 x-6 x^2-8 x^3-2 x^4) \log (x)+(4+16 x+9 x^2+x^3) \log ^2(x)+(-8-4 x) \log ^3(x))}{3 x^3} \, dx\) [2584]

Optimal result

Integrand size = 105, antiderivative size = 27 \[ \int \frac {e^{\frac {4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log (x)}{x^2}} \left (4 x^2-2 x^3+3 x^4+x^5+\left (-8 x-6 x^2-8 x^3-2 x^4\right ) \log (x)+\left (4+16 x+9 x^2+x^3\right ) \log ^2(x)+(-8-4 x) \log ^3(x)\right )}{3 x^3} \, dx=\frac {1}{3} e^{\frac {(2+x)^2 (x+\log (x))}{x^2}} (x-\log (x))^2 \] Output:

1/3*(x-ln(x))^2*exp((2+x)^2*(x+ln(x))/x^2)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {e^{\frac {4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log (x)}{x^2}} \left (4 x^2-2 x^3+3 x^4+x^5+\left (-8 x-6 x^2-8 x^3-2 x^4\right ) \log (x)+\left (4+16 x+9 x^2+x^3\right ) \log ^2(x)+(-8-4 x) \log ^3(x)\right )}{3 x^3} \, dx=\frac {1}{3} e^{4+\frac {4}{x}+x} x^{\frac {(2+x)^2}{x^2}} (x-\log (x))^2 \] Input:

Integrate[(E^((4*x + 4*x^2 + x^3 + (4 + 4*x + x^2)*Log[x])/x^2)*(4*x^2 - 2 
*x^3 + 3*x^4 + x^5 + (-8*x - 6*x^2 - 8*x^3 - 2*x^4)*Log[x] + (4 + 16*x + 9 
*x^2 + x^3)*Log[x]^2 + (-8 - 4*x)*Log[x]^3))/(3*x^3),x]
 

Output:

(E^(4 + 4/x + x)*x^((2 + x)^2/x^2)*(x - Log[x])^2)/3
 

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.

\(\displaystyle \int \frac {e^{\frac {x^3+4 x^2+\left (x^2+4 x+4\right ) \log (x)+4 x}{x^2}} \left (x^5+3 x^4-2 x^3+4 x^2+\left (x^3+9 x^2+16 x+4\right ) \log ^2(x)+\left (-2 x^4-8 x^3-6 x^2-8 x\right ) \log (x)+(-4 x-8) \log ^3(x)\right )}{3 x^3} \, dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int e^{\frac {x^3+4 x^2+4 x}{x^2}} x^{\frac {x^2+4 x+4}{x^2}-3} \left (x^5+3 x^4-2 x^3+4 x^2-4 (x+2) \log ^3(x)+\left (x^3+9 x^2+16 x+4\right ) \log ^2(x)-2 \left (x^4+4 x^3+3 x^2+4 x\right ) \log (x)\right )dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \frac {1}{3} \int e^{x+4+\frac {4}{x}} x^{-\frac {2 \left (x^2-2 x-2\right )}{x^2}} \left (x^5+3 x^4-2 x^3+4 x^2-4 (x+2) \log ^3(x)+\left (x^3+9 x^2+16 x+4\right ) \log ^2(x)-2 \left (x^4+4 x^3+3 x^2+4 x\right ) \log (x)\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{3} \int \left (-4 e^{x+4+\frac {4}{x}} (x+2) \log ^3(x) x^{-\frac {2 \left (x^2-2 x-2\right )}{x^2}}+e^{x+4+\frac {4}{x}} \left (x^3+9 x^2+16 x+4\right ) \log ^2(x) x^{-\frac {2 \left (x^2-2 x-2\right )}{x^2}}-2 e^{x+4+\frac {4}{x}} \left (x^3+4 x^2+3 x+4\right ) \log (x) x^{1-\frac {2 \left (x^2-2 x-2\right )}{x^2}}+4 e^{x+4+\frac {4}{x}} x^{2-\frac {2 \left (x^2-2 x-2\right )}{x^2}}-2 e^{x+4+\frac {4}{x}} x^{3-\frac {2 \left (x^2-2 x-2\right )}{x^2}}+3 e^{x+4+\frac {4}{x}} x^{4-\frac {2 \left (x^2-2 x-2\right )}{x^2}}+e^{x+4+\frac {4}{x}} x^{5-\frac {2 \left (x^2-2 x-2\right )}{x^2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} \left (\int e^{x+4+\frac {4}{x}} x^{5-\frac {2 \left (x^2-2 x-2\right )}{x^2}}dx+8 \int \frac {\int e^{x+4+\frac {4}{x}} x^{-1+\frac {4}{x}+\frac {4}{x^2}}dx}{x}dx-8 \int e^{x+4+\frac {4}{x}} x^{-\frac {2 \left (x^2-2 x-2\right )}{x^2}} \log ^3(x)dx-4 \int e^{x+4+\frac {4}{x}} x^{1-\frac {2 \left (x^2-2 x-2\right )}{x^2}} \log ^3(x)dx+4 \int e^{x+4+\frac {4}{x}} x^{-\frac {2 \left (x^2-2 x-2\right )}{x^2}} \log ^2(x)dx+16 \int e^{x+4+\frac {4}{x}} x^{1-\frac {2 \left (x^2-2 x-2\right )}{x^2}} \log ^2(x)dx-8 \log (x) \int e^{x+4+\frac {4}{x}} x^{1-\frac {2 \left (x^2-2 x-2\right )}{x^2}}dx+4 \int e^{x+4+\frac {4}{x}} x^{\frac {4 (x+1)}{x^2}}dx-2 \int e^{x+4+\frac {4}{x}} x^{\frac {(x+2)^2}{x^2}}dx+3 \int e^{x+4+\frac {4}{x}} x^{\frac {2 \left (x^2+2 x+2\right )}{x^2}}dx+6 \int \frac {\int e^{x+4+\frac {4}{x}} x^{\frac {4 (x+1)}{x^2}}dx}{x}dx+8 \int \frac {\int e^{x+4+\frac {4}{x}} x^{\frac {(x+2)^2}{x^2}}dx}{x}dx+2 \int \frac {\int e^{x+4+\frac {4}{x}} x^{\frac {2 \left (x^2+2 x+2\right )}{x^2}}dx}{x}dx+9 \int e^{x+4+\frac {4}{x}} x^{\frac {4 (x+1)}{x^2}} \log ^2(x)dx+\int e^{x+4+\frac {4}{x}} x^{\frac {(x+2)^2}{x^2}} \log ^2(x)dx-6 \log (x) \int e^{x+4+\frac {4}{x}} x^{\frac {4 (x+1)}{x^2}}dx-8 \log (x) \int e^{x+4+\frac {4}{x}} x^{\frac {(x+2)^2}{x^2}}dx-2 \log (x) \int e^{x+4+\frac {4}{x}} x^{\frac {2 \left (x^2+2 x+2\right )}{x^2}}dx\right )\)

Input:

Int[(E^((4*x + 4*x^2 + x^3 + (4 + 4*x + x^2)*Log[x])/x^2)*(4*x^2 - 2*x^3 + 
 3*x^4 + x^5 + (-8*x - 6*x^2 - 8*x^3 - 2*x^4)*Log[x] + (4 + 16*x + 9*x^2 + 
 x^3)*Log[x]^2 + (-8 - 4*x)*Log[x]^3))/(3*x^3),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 4.69 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11

method	result	size
risch	\(\frac {\left (x^{2}-2 x \ln \left (x \right )+\ln \left (x \right )^{2}\right ) {\mathrm e}^{\frac {\left (2+x \right )^{2} \left (x +\ln \left (x \right )\right )}{x^{2}}}}{3}\)	\(30\)
parallelrisch	\(\frac {x^{2} {\mathrm e}^{\frac {\left (x^{2}+4 x +4\right ) \ln \left (x \right )+x^{3}+4 x^{2}+4 x}{x^{2}}}}{3}-\frac {2 \ln \left (x \right ) x \,{\mathrm e}^{\frac {\left (x^{2}+4 x +4\right ) \ln \left (x \right )+x^{3}+4 x^{2}+4 x}{x^{2}}}}{3}+\frac {\ln \left (x \right )^{2} {\mathrm e}^{\frac {\left (x^{2}+4 x +4\right ) \ln \left (x \right )+x^{3}+4 x^{2}+4 x}{x^{2}}}}{3}\)	\(102\)

Input:

int(1/3*((-4*x-8)*ln(x)^3+(x^3+9*x^2+16*x+4)*ln(x)^2+(-2*x^4-8*x^3-6*x^2-8 
*x)*ln(x)+x^5+3*x^4-2*x^3+4*x^2)*exp(((x^2+4*x+4)*ln(x)+x^3+4*x^2+4*x)/x^2 
)/x^3,x,method=_RETURNVERBOSE)
 

Output:

1/3*(x^2-2*x*ln(x)+ln(x)^2)*exp((2+x)^2*(x+ln(x))/x^2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59 \[ \int \frac {e^{\frac {4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log (x)}{x^2}} \left (4 x^2-2 x^3+3 x^4+x^5+\left (-8 x-6 x^2-8 x^3-2 x^4\right ) \log (x)+\left (4+16 x+9 x^2+x^3\right ) \log ^2(x)+(-8-4 x) \log ^3(x)\right )}{3 x^3} \, dx=\frac {1}{3} \, {\left (x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}\right )} e^{\left (\frac {x^{3} + 4 \, x^{2} + {\left (x^{2} + 4 \, x + 4\right )} \log \left (x\right ) + 4 \, x}{x^{2}}\right )} \] Input:

integrate(1/3*((-4*x-8)*log(x)^3+(x^3+9*x^2+16*x+4)*log(x)^2+(-2*x^4-8*x^3 
-6*x^2-8*x)*log(x)+x^5+3*x^4-2*x^3+4*x^2)*exp(((x^2+4*x+4)*log(x)+x^3+4*x^ 
2+4*x)/x^2)/x^3,x, algorithm="fricas")
 

Output:

1/3*(x^2 - 2*x*log(x) + log(x)^2)*e^((x^3 + 4*x^2 + (x^2 + 4*x + 4)*log(x) 
 + 4*x)/x^2)
 

Sympy [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {e^{\frac {4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log (x)}{x^2}} \left (4 x^2-2 x^3+3 x^4+x^5+\left (-8 x-6 x^2-8 x^3-2 x^4\right ) \log (x)+\left (4+16 x+9 x^2+x^3\right ) \log ^2(x)+(-8-4 x) \log ^3(x)\right )}{3 x^3} \, dx=\frac {\left (x^{2} - 2 x \log {\left (x \right )} + \log {\left (x \right )}^{2}\right ) e^{\frac {x^{3} + 4 x^{2} + 4 x + \left (x^{2} + 4 x + 4\right ) \log {\left (x \right )}}{x^{2}}}}{3} \] Input:

integrate(1/3*((-4*x-8)*ln(x)**3+(x**3+9*x**2+16*x+4)*ln(x)**2+(-2*x**4-8* 
x**3-6*x**2-8*x)*ln(x)+x**5+3*x**4-2*x**3+4*x**2)*exp(((x**2+4*x+4)*ln(x)+ 
x**3+4*x**2+4*x)/x**2)/x**3,x)
 

Output:

(x**2 - 2*x*log(x) + log(x)**2)*exp((x**3 + 4*x**2 + 4*x + (x**2 + 4*x + 4 
)*log(x))/x**2)/3
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {e^{\frac {4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log (x)}{x^2}} \left (4 x^2-2 x^3+3 x^4+x^5+\left (-8 x-6 x^2-8 x^3-2 x^4\right ) \log (x)+\left (4+16 x+9 x^2+x^3\right ) \log ^2(x)+(-8-4 x) \log ^3(x)\right )}{3 x^3} \, dx=\frac {1}{3} \, {\left (x^{3} e^{4} - 2 \, x^{2} e^{4} \log \left (x\right ) + x e^{4} \log \left (x\right )^{2}\right )} e^{\left (x + \frac {4 \, \log \left (x\right )}{x} + \frac {4}{x} + \frac {4 \, \log \left (x\right )}{x^{2}}\right )} \] Input:

integrate(1/3*((-4*x-8)*log(x)^3+(x^3+9*x^2+16*x+4)*log(x)^2+(-2*x^4-8*x^3 
-6*x^2-8*x)*log(x)+x^5+3*x^4-2*x^3+4*x^2)*exp(((x^2+4*x+4)*log(x)+x^3+4*x^ 
2+4*x)/x^2)/x^3,x, algorithm="maxima")
 

Output:

1/3*(x^3*e^4 - 2*x^2*e^4*log(x) + x*e^4*log(x)^2)*e^(x + 4*log(x)/x + 4/x 
+ 4*log(x)/x^2)
 

Giac [F]

\[ \int \frac {e^{\frac {4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log (x)}{x^2}} \left (4 x^2-2 x^3+3 x^4+x^5+\left (-8 x-6 x^2-8 x^3-2 x^4\right ) \log (x)+\left (4+16 x+9 x^2+x^3\right ) \log ^2(x)+(-8-4 x) \log ^3(x)\right )}{3 x^3} \, dx=\int { \frac {{\left (x^{5} + 3 \, x^{4} - 4 \, {\left (x + 2\right )} \log \left (x\right )^{3} - 2 \, x^{3} + {\left (x^{3} + 9 \, x^{2} + 16 \, x + 4\right )} \log \left (x\right )^{2} + 4 \, x^{2} - 2 \, {\left (x^{4} + 4 \, x^{3} + 3 \, x^{2} + 4 \, x\right )} \log \left (x\right )\right )} e^{\left (\frac {x^{3} + 4 \, x^{2} + {\left (x^{2} + 4 \, x + 4\right )} \log \left (x\right ) + 4 \, x}{x^{2}}\right )}}{3 \, x^{3}} \,d x } \] Input:

integrate(1/3*((-4*x-8)*log(x)^3+(x^3+9*x^2+16*x+4)*log(x)^2+(-2*x^4-8*x^3 
-6*x^2-8*x)*log(x)+x^5+3*x^4-2*x^3+4*x^2)*exp(((x^2+4*x+4)*log(x)+x^3+4*x^ 
2+4*x)/x^2)/x^3,x, algorithm="giac")
 

Output:

integrate(1/3*(x^5 + 3*x^4 - 4*(x + 2)*log(x)^3 - 2*x^3 + (x^3 + 9*x^2 + 1 
6*x + 4)*log(x)^2 + 4*x^2 - 2*(x^4 + 4*x^3 + 3*x^2 + 4*x)*log(x))*e^((x^3 
+ 4*x^2 + (x^2 + 4*x + 4)*log(x) + 4*x)/x^2)/x^3, x)
 

Mupad [B] (verification not implemented)

Time = 3.22 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {e^{\frac {4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log (x)}{x^2}} \left (4 x^2-2 x^3+3 x^4+x^5+\left (-8 x-6 x^2-8 x^3-2 x^4\right ) \log (x)+\left (4+16 x+9 x^2+x^3\right ) \log ^2(x)+(-8-4 x) \log ^3(x)\right )}{3 x^3} \, dx=x\,x^{4/x}\,x^{\frac {4}{x^2}}\,{\mathrm {e}}^{x+\frac {4}{x}+4}\,\left (\frac {x^2}{3}-\frac {2\,x\,\ln \left (x\right )}{3}+\frac {{\ln \left (x\right )}^2}{3}\right ) \] Input:

int((exp((4*x + log(x)*(4*x + x^2 + 4) + 4*x^2 + x^3)/x^2)*(4*x^2 - log(x) 
*(8*x + 6*x^2 + 8*x^3 + 2*x^4) - 2*x^3 + 3*x^4 + x^5 - log(x)^3*(4*x + 8) 
+ log(x)^2*(16*x + 9*x^2 + x^3 + 4)))/(3*x^3),x)
 

Output:

x*x^(4/x)*x^(4/x^2)*exp(x + 4/x + 4)*(log(x)^2/3 - (2*x*log(x))/3 + x^2/3)
 

Reduce [B] (verification not implemented)

Time = 9.54 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {e^{\frac {4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log (x)}{x^2}} \left (4 x^2-2 x^3+3 x^4+x^5+\left (-8 x-6 x^2-8 x^3-2 x^4\right ) \log (x)+\left (4+16 x+9 x^2+x^3\right ) \log ^2(x)+(-8-4 x) \log ^3(x)\right )}{3 x^3} \, dx=\frac {x^{\frac {4 x +4}{x^{2}}} e^{\frac {x^{2}+4}{x}} e^{4} x \left (\mathrm {log}\left (x \right )^{2}-2 \,\mathrm {log}\left (x \right ) x +x^{2}\right )}{3} \] Input:

int(1/3*((-4*x-8)*log(x)^3+(x^3+9*x^2+16*x+4)*log(x)^2+(-2*x^4-8*x^3-6*x^2 
-8*x)*log(x)+x^5+3*x^4-2*x^3+4*x^2)*exp(((x^2+4*x+4)*log(x)+x^3+4*x^2+4*x) 
/x^2)/x^3,x)
 

Output:

(x**((4*x + 4)/x**2)*e**((x**2 + 4)/x)*e**4*x*(log(x)**2 - 2*log(x)*x + x* 
*2))/3