\(\int \frac {e^{\frac {16 \log (\frac {1}{x^2})+(-x+x^2) \log (x)}{4 \log (\frac {1}{x^2})}} ((-1+x) \log (\frac {1}{x^2})+(-2+2 x+(-1+2 x) \log (\frac {1}{x^2})) \log (x))}{4 \log ^2(\frac {1}{x^2})} \, dx\) [1554]

Optimal result

Integrand size = 66, antiderivative size = 25 \[ \int \frac {e^{\frac {16 \log \left (\frac {1}{x^2}\right )+\left (-x+x^2\right ) \log (x)}{4 \log \left (\frac {1}{x^2}\right )}} \left ((-1+x) \log \left (\frac {1}{x^2}\right )+\left (-2+2 x+(-1+2 x) \log \left (\frac {1}{x^2}\right )\right ) \log (x)\right )}{4 \log ^2\left (\frac {1}{x^2}\right )} \, dx=36+e^{4+\frac {\left (-x+x^2\right ) \log (x)}{4 \log \left (\frac {1}{x^2}\right )}} \] Output:

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {e^{\frac {16 \log \left (\frac {1}{x^2}\right )+\left (-x+x^2\right ) \log (x)}{4 \log \left (\frac {1}{x^2}\right )}} \left ((-1+x) \log \left (\frac {1}{x^2}\right )+\left (-2+2 x+(-1+2 x) \log \left (\frac {1}{x^2}\right )\right ) \log (x)\right )}{4 \log ^2\left (\frac {1}{x^2}\right )} \, dx=e^4 x^{\frac {(-1+x) x}{4 \log \left (\frac {1}{x^2}\right )}} \] Input:

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

Output:

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

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.

\(\displaystyle \int \frac {\left ((x-1) \log \left (\frac {1}{x^2}\right )+\left ((2 x-1) \log \left (\frac {1}{x^2}\right )+2 x-2\right ) \log (x)\right ) \exp \left (\frac {16 \log \left (\frac {1}{x^2}\right )+\left (x^2-x\right ) \log (x)}{4 \log \left (\frac {1}{x^2}\right )}\right )}{4 \log ^2\left (\frac {1}{x^2}\right )} \, dx\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08

method	result	size
parallelrisch	\({\mathrm e}^{\frac {\ln \left (x \right ) \left (x^{2}-x \right )+16 \ln \left (\frac {1}{x^{2}}\right )}{4 \ln \left (\frac {1}{x^{2}}\right )}}\)	\(27\)
default	\(-\frac {-4 \left (\ln \left (\frac {1}{x^{2}}\right )+2 \ln \left (x \right )\right ) {\mathrm e}^{\frac {\ln \left (x \right ) \left (x^{2}-x \right )+16 \ln \left (\frac {1}{x^{2}}\right )}{4 \ln \left (\frac {1}{x^{2}}\right )}}+8 \ln \left (x \right ) {\mathrm e}^{\frac {\ln \left (x \right ) \left (x^{2}-x \right )+16 \ln \left (\frac {1}{x^{2}}\right )}{4 \ln \left (\frac {1}{x^{2}}\right )}}}{4 \ln \left (\frac {1}{x^{2}}\right )}\)	\(77\)
risch	\({\mathrm e}^{-\frac {8 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-16 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )+8 i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+x^{2} \ln \left (x \right )-x \ln \left (x \right )-32 \ln \left (x \right )}{2 \left (-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+4 \ln \left (x \right )\right )}}\)	\(125\)

Input:

int(1/4*(((-1+2*x)*ln(1/x^2)+2*x-2)*ln(x)+(-1+x)*ln(1/x^2))*exp(1/4*(ln(x) 
*(x^2-x)+16*ln(1/x^2))/ln(1/x^2))/ln(1/x^2)^2,x,method=_RETURNVERBOSE)
 

Output:

exp(1/4*(ln(x)*(x^2-x)+16*ln(1/x^2))/ln(1/x^2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.44 \[ \int \frac {e^{\frac {16 \log \left (\frac {1}{x^2}\right )+\left (-x+x^2\right ) \log (x)}{4 \log \left (\frac {1}{x^2}\right )}} \left ((-1+x) \log \left (\frac {1}{x^2}\right )+\left (-2+2 x+(-1+2 x) \log \left (\frac {1}{x^2}\right )\right ) \log (x)\right )}{4 \log ^2\left (\frac {1}{x^2}\right )} \, dx=e^{\left (-\frac {1}{8} \, x^{2} + \frac {1}{8} \, x + 4\right )} \] Input:

integrate(1/4*(((-1+2*x)*log(1/x^2)+2*x-2)*log(x)+(-1+x)*log(1/x^2))*exp(1 
/4*(log(x)*(x^2-x)+16*log(1/x^2))/log(1/x^2))/log(1/x^2)^2,x, algorithm="f 
ricas")
 

Output:

e^(-1/8*x^2 + 1/8*x + 4)
 

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {e^{\frac {16 \log \left (\frac {1}{x^2}\right )+\left (-x+x^2\right ) \log (x)}{4 \log \left (\frac {1}{x^2}\right )}} \left ((-1+x) \log \left (\frac {1}{x^2}\right )+\left (-2+2 x+(-1+2 x) \log \left (\frac {1}{x^2}\right )\right ) \log (x)\right )}{4 \log ^2\left (\frac {1}{x^2}\right )} \, dx=e^{- \frac {\frac {\left (x^{2} - x\right ) \log {\left (x \right )}}{4} - 8 \log {\left (x \right )}}{2 \log {\left (x \right )}}} \] Input:

integrate(1/4*(((-1+2*x)*ln(1/x**2)+2*x-2)*ln(x)+(-1+x)*ln(1/x**2))*exp(1/ 
4*(ln(x)*(x**2-x)+16*ln(1/x**2))/ln(1/x**2))/ln(1/x**2)**2,x)
 

Output:

exp(-((x**2 - x)*log(x)/4 - 8*log(x))/(2*log(x)))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.44 \[ \int \frac {e^{\frac {16 \log \left (\frac {1}{x^2}\right )+\left (-x+x^2\right ) \log (x)}{4 \log \left (\frac {1}{x^2}\right )}} \left ((-1+x) \log \left (\frac {1}{x^2}\right )+\left (-2+2 x+(-1+2 x) \log \left (\frac {1}{x^2}\right )\right ) \log (x)\right )}{4 \log ^2\left (\frac {1}{x^2}\right )} \, dx=e^{\left (-\frac {1}{8} \, x^{2} + \frac {1}{8} \, x + 4\right )} \] Input:

integrate(1/4*(((-1+2*x)*log(1/x^2)+2*x-2)*log(x)+(-1+x)*log(1/x^2))*exp(1 
/4*(log(x)*(x^2-x)+16*log(1/x^2))/log(1/x^2))/log(1/x^2)^2,x, algorithm="m 
axima")
 

Output:

e^(-1/8*x^2 + 1/8*x + 4)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {e^{\frac {16 \log \left (\frac {1}{x^2}\right )+\left (-x+x^2\right ) \log (x)}{4 \log \left (\frac {1}{x^2}\right )}} \left ((-1+x) \log \left (\frac {1}{x^2}\right )+\left (-2+2 x+(-1+2 x) \log \left (\frac {1}{x^2}\right )\right ) \log (x)\right )}{4 \log ^2\left (\frac {1}{x^2}\right )} \, dx=e^{\left (-\frac {x^{2} \log \left (x\right )}{4 \, \log \left (x^{2}\right )} + \frac {x \log \left (x\right )}{4 \, \log \left (x^{2}\right )} + 4\right )} \] Input:

integrate(1/4*(((-1+2*x)*log(1/x^2)+2*x-2)*log(x)+(-1+x)*log(1/x^2))*exp(1 
/4*(log(x)*(x^2-x)+16*log(1/x^2))/log(1/x^2))/log(1/x^2)^2,x, algorithm="g 
iac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 3.14 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\frac {16 \log \left (\frac {1}{x^2}\right )+\left (-x+x^2\right ) \log (x)}{4 \log \left (\frac {1}{x^2}\right )}} \left ((-1+x) \log \left (\frac {1}{x^2}\right )+\left (-2+2 x+(-1+2 x) \log \left (\frac {1}{x^2}\right )\right ) \log (x)\right )}{4 \log ^2\left (\frac {1}{x^2}\right )} \, dx={\mathrm {e}}^{\frac {x^2\,\ln \left (x\right )}{4\,\ln \left (\frac {1}{x^2}\right )}}\,{\mathrm {e}}^4\,{\mathrm {e}}^{-\frac {x\,\ln \left (x\right )}{4\,\ln \left (\frac {1}{x^2}\right )}} \] Input:

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

Output:

exp((x^2*log(x))/(4*log(1/x^2)))*exp(4)*exp(-(x*log(x))/(4*log(1/x^2)))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {e^{\frac {16 \log \left (\frac {1}{x^2}\right )+\left (-x+x^2\right ) \log (x)}{4 \log \left (\frac {1}{x^2}\right )}} \left ((-1+x) \log \left (\frac {1}{x^2}\right )+\left (-2+2 x+(-1+2 x) \log \left (\frac {1}{x^2}\right )\right ) \log (x)\right )}{4 \log ^2\left (\frac {1}{x^2}\right )} \, dx=\frac {e^{\frac {\mathrm {log}\left (x \right ) x}{4 \,\mathrm {log}\left (x^{2}\right )}} e^{4}}{e^{\frac {\mathrm {log}\left (x \right ) x^{2}}{4 \,\mathrm {log}\left (x^{2}\right )}}} \] Input:

int(1/4*(((-1+2*x)*log(1/x^2)+2*x-2)*log(x)+(-1+x)*log(1/x^2))*exp(1/4*(lo 
g(x)*(x^2-x)+16*log(1/x^2))/log(1/x^2))/log(1/x^2)^2,x)
 

Output:

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