\(\int \frac {e^{\frac {2 (2 x^2-x^3-2 x \log (\frac {x}{2})-\log ^2(\frac {x}{2}))}{x^2}} (-4 x^2-2 x^4+e^4 (-4 x-x^2-2 x^3)+(-4 x+4 x^2+e^4 (-4+4 x)) \log (\frac {x}{2})+(4 e^4+4 x) \log ^2(\frac {x}{2}))}{x^4} \, dx\) [1259]

Optimal result

Integrand size = 113, antiderivative size = 31 \[ \int \frac {e^{\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}} \left (-4 x^2-2 x^4+e^4 \left (-4 x-x^2-2 x^3\right )+\left (-4 x+4 x^2+e^4 (-4+4 x)\right ) \log \left (\frac {x}{2}\right )+\left (4 e^4+4 x\right ) \log ^2\left (\frac {x}{2}\right )\right )}{x^4} \, dx=\frac {e^{6-2 x-\frac {2 \left (x+\log \left (\frac {x}{2}\right )\right )^2}{x^2}} \left (e^4+x\right )}{x} \] Output:

exp(3-x-(ln(1/2*x)+x)^2/x^2)^2*(x+exp(4))/x
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {e^{\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}} \left (-4 x^2-2 x^4+e^4 \left (-4 x-x^2-2 x^3\right )+\left (-4 x+4 x^2+e^4 (-4+4 x)\right ) \log \left (\frac {x}{2}\right )+\left (4 e^4+4 x\right ) \log ^2\left (\frac {x}{2}\right )\right )}{x^4} \, dx=16^{\frac {1}{x}} e^{4-2 x-\frac {2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x^{-\frac {4+x}{x}} \left (e^4+x\right ) \] Input:

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

Output:

(16^x^(-1)*E^(4 - 2*x - (2*Log[x/2]^2)/x^2)*(E^4 + x))/x^((4 + x)/x)
 

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.

\(\displaystyle \int \frac {\left (-2 x^4-4 x^2+\left (4 x^2-4 x+e^4 (4 x-4)\right ) \log \left (\frac {x}{2}\right )+e^4 \left (-2 x^3-x^2-4 x\right )+\left (4 x+4 e^4\right ) \log ^2\left (\frac {x}{2}\right )\right ) \exp \left (\frac {2 \left (-x^3+2 x^2-\log ^2\left (\frac {x}{2}\right )-2 x \log \left (\frac {x}{2}\right )\right )}{x^2}\right )}{x^4} \, dx\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 3.46 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32

method	result	size
risch	\(\frac {\left (x +{\mathrm e}^{4}\right ) \left (\frac {x}{2}\right )^{-\frac {4}{x}} {\mathrm e}^{-\frac {2 \left (x^{3}+\ln \left (\frac {x}{2}\right )^{2}-2 x^{2}\right )}{x^{2}}}}{x}\)	\(41\)
parallelrisch	\(\frac {{\mathrm e}^{4} {\mathrm e}^{-\frac {2 \left (x^{3}+\ln \left (\frac {x}{2}\right )^{2}+2 x \ln \left (\frac {x}{2}\right )-2 x^{2}\right )}{x^{2}}}+{\mathrm e}^{-\frac {2 \left (x^{3}+\ln \left (\frac {x}{2}\right )^{2}+2 x \ln \left (\frac {x}{2}\right )-2 x^{2}\right )}{x^{2}}} x}{x}\)	\(71\)

Input:

int(((4*exp(4)+4*x)*ln(1/2*x)^2+((-4+4*x)*exp(4)+4*x^2-4*x)*ln(1/2*x)+(-2* 
x^3-x^2-4*x)*exp(4)-2*x^4-4*x^2)*exp((-ln(1/2*x)^2-2*x*ln(1/2*x)-x^3+2*x^2 
)/x^2)^2/x^4,x,method=_RETURNVERBOSE)
 

Output:

(x+exp(4))/x*((1/2*x)^(-2/x))^2*exp(-2*(x^3+ln(1/2*x)^2-2*x^2)/x^2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}} \left (-4 x^2-2 x^4+e^4 \left (-4 x-x^2-2 x^3\right )+\left (-4 x+4 x^2+e^4 (-4+4 x)\right ) \log \left (\frac {x}{2}\right )+\left (4 e^4+4 x\right ) \log ^2\left (\frac {x}{2}\right )\right )}{x^4} \, dx=\frac {{\left (x + e^{4}\right )} e^{\left (-\frac {2 \, {\left (x^{3} - 2 \, x^{2} + 2 \, x \log \left (\frac {1}{2} \, x\right ) + \log \left (\frac {1}{2} \, x\right )^{2}\right )}}{x^{2}}\right )}}{x} \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}} \left (-4 x^2-2 x^4+e^4 \left (-4 x-x^2-2 x^3\right )+\left (-4 x+4 x^2+e^4 (-4+4 x)\right ) \log \left (\frac {x}{2}\right )+\left (4 e^4+4 x\right ) \log ^2\left (\frac {x}{2}\right )\right )}{x^4} \, dx=\frac {\left (x + e^{4}\right ) e^{\frac {2 \left (- x^{3} + 2 x^{2} - 2 x \log {\left (\frac {x}{2} \right )} - \log {\left (\frac {x}{2} \right )}^{2}\right )}{x^{2}}}}{x} \] Input:

integrate(((4*exp(4)+4*x)*ln(1/2*x)**2+((-4+4*x)*exp(4)+4*x**2-4*x)*ln(1/2 
*x)+(-2*x**3-x**2-4*x)*exp(4)-2*x**4-4*x**2)*exp((-ln(1/2*x)**2-2*x*ln(1/2 
*x)-x**3+2*x**2)/x**2)**2/x**4,x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (27) = 54\).

Time = 0.24 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.84 \[ \int \frac {e^{\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}} \left (-4 x^2-2 x^4+e^4 \left (-4 x-x^2-2 x^3\right )+\left (-4 x+4 x^2+e^4 (-4+4 x)\right ) \log \left (\frac {x}{2}\right )+\left (4 e^4+4 x\right ) \log ^2\left (\frac {x}{2}\right )\right )}{x^4} \, dx=\frac {{\left (x e^{4} + e^{8}\right )} e^{\left (-2 \, x + \frac {4 \, \log \left (2\right )}{x} - \frac {2 \, \log \left (2\right )^{2}}{x^{2}} - \frac {4 \, \log \left (x\right )}{x} + \frac {4 \, \log \left (2\right ) \log \left (x\right )}{x^{2}} - \frac {2 \, \log \left (x\right )^{2}}{x^{2}}\right )}}{x} \] Input:

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

Output:

(x*e^4 + e^8)*e^(-2*x + 4*log(2)/x - 2*log(2)^2/x^2 - 4*log(x)/x + 4*log(2 
)*log(x)/x^2 - 2*log(x)^2/x^2)/x
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (27) = 54\).

Time = 0.31 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.03 \[ \int \frac {e^{\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}} \left (-4 x^2-2 x^4+e^4 \left (-4 x-x^2-2 x^3\right )+\left (-4 x+4 x^2+e^4 (-4+4 x)\right ) \log \left (\frac {x}{2}\right )+\left (4 e^4+4 x\right ) \log ^2\left (\frac {x}{2}\right )\right )}{x^4} \, dx=\frac {x e^{\left (-\frac {2 \, {\left (x^{3} - 2 \, x^{2} + 2 \, x \log \left (\frac {1}{2} \, x\right ) + \log \left (\frac {1}{2} \, x\right )^{2}\right )}}{x^{2}}\right )} + e^{\left (-\frac {2 \, {\left (x^{3} - 4 \, x^{2} + 2 \, x \log \left (\frac {1}{2} \, x\right ) + \log \left (\frac {1}{2} \, x\right )^{2}\right )}}{x^{2}}\right )}}{x} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 3.19 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.84 \[ \int \frac {e^{\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}} \left (-4 x^2-2 x^4+e^4 \left (-4 x-x^2-2 x^3\right )+\left (-4 x+4 x^2+e^4 (-4+4 x)\right ) \log \left (\frac {x}{2}\right )+\left (4 e^4+4 x\right ) \log ^2\left (\frac {x}{2}\right )\right )}{x^4} \, dx=\frac {2^{4/x}\,x^{\frac {4\,\ln \left (2\right )}{x^2}}\,{\mathrm {e}}^{4-\frac {2\,{\ln \left (2\right )}^2}{x^2}-\frac {2\,{\ln \left (x\right )}^2}{x^2}-2\,x}\,\left (x+{\mathrm {e}}^4\right )}{x^{4/x}\,x} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {e^{\frac {2 \left (2 x^2-x^3-2 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x^2}} \left (-4 x^2-2 x^4+e^4 \left (-4 x-x^2-2 x^3\right )+\left (-4 x+4 x^2+e^4 (-4+4 x)\right ) \log \left (\frac {x}{2}\right )+\left (4 e^4+4 x\right ) \log ^2\left (\frac {x}{2}\right )\right )}{x^4} \, dx=\frac {e^{4} \left (e^{4}+x \right )}{e^{\frac {2 \mathrm {log}\left (\frac {x}{2}\right )^{2}+4 \,\mathrm {log}\left (\frac {x}{2}\right ) x +2 x^{3}}{x^{2}}} x} \] Input:

int(((4*exp(4)+4*x)*log(1/2*x)^2+((-4+4*x)*exp(4)+4*x^2-4*x)*log(1/2*x)+(- 
2*x^3-x^2-4*x)*exp(4)-2*x^4-4*x^2)*exp((-log(1/2*x)^2-2*x*log(1/2*x)-x^3+2 
*x^2)/x^2)^2/x^4,x)
 

Output:

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