\(\int \frac {-2 x^2+(-2-8 x^2-10 x^4-4 x^6) \log (\frac {e^{-x^2}}{x})+(4 x^2+4 x^4) \log ^2(\frac {e^{-x^2}}{x})}{x} \, dx\) [765]

Optimal result

Integrand size = 66, antiderivative size = 33 \[ \int \frac {-2 x^2+\left (-2-8 x^2-10 x^4-4 x^6\right ) \log \left (\frac {e^{-x^2}}{x}\right )+\left (4 x^2+4 x^4\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )}{x} \, dx=e^{e^4}-x^2+\left (1+x^2\right )^2 \log ^2\left (\frac {e^{-x^2}}{x}\right ) \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(33)=66\).

Time = 0.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.18 \[ \int \frac {-2 x^2+\left (-2-8 x^2-10 x^4-4 x^6\right ) \log \left (\frac {e^{-x^2}}{x}\right )+\left (4 x^2+4 x^4\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )}{x} \, dx=\log ^2\left (\frac {1}{x}\right )+x^2 \left (2+x^2\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )+2 \log \left (\frac {1}{x}\right ) \log (x)-2 \log \left (\frac {e^{-x^2}}{x}\right ) \left (x^2+\log (x)\right )-x^2 \left (1+x^2+2 \log (x)\right ) \] Input:

Integrate[(-2*x^2 + (-2 - 8*x^2 - 10*x^4 - 4*x^6)*Log[1/(E^x^2*x)] + (4*x^ 
2 + 4*x^4)*Log[1/(E^x^2*x)]^2)/x,x]
 

Output:

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

\(\displaystyle \int \frac {-2 x^2+\left (4 x^4+4 x^2\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )+\left (-4 x^6-10 x^4-8 x^2-2\right ) \log \left (\frac {e^{-x^2}}{x}\right )}{x} \, dx\)

\(\Big \downarrow \) 2010

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

\(\Big \downarrow \) 2009

\(\displaystyle 4 \int x \log ^2\left (\frac {e^{-x^2}}{x}\right )dx-2 \int \frac {\log \left (\frac {e^{-x^2}}{x}\right )}{x}dx+4 \int x^3 \log ^2\left (\frac {e^{-x^2}}{x}\right )dx-\frac {x^8}{6}-\frac {17 x^6}{18}-\frac {21 x^4}{8}-3 x^2-4 x^2 \log \left (\frac {e^{-x^2}}{x}\right )-\frac {2}{3} x^6 \log \left (\frac {e^{-x^2}}{x}\right )-\frac {5}{2} x^4 \log \left (\frac {e^{-x^2}}{x}\right )\)

Input:

Int[(-2*x^2 + (-2 - 8*x^2 - 10*x^4 - 4*x^6)*Log[1/(E^x^2*x)] + (4*x^2 + 4* 
x^4)*Log[1/(E^x^2*x)]^2)/x,x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.67

method	result	size
parallelrisch	\(\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )^{2} x^{4}+2 \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )^{2} x^{2}-x^{2}+\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )^{2}\)	\(55\)
default	\(-2 \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right ) \ln \left (x \right )-8 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right ) \left (\frac {\ln \left ({\mathrm e}^{x^{2}}\right ) x^{4}}{4}+\frac {\ln \left ({\mathrm e}^{x^{2}}\right ) x^{2}}{2}-\frac {x^{6}}{12}-\frac {x^{4}}{4}\right )+\frac {\left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right ) x^{4}}{2}+2 x^{2} \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right )+{\left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right )}^{2} \left (x^{2}+1\right )^{2}+x^{4} \ln \left (x \right )^{2}+\frac {4 x^{6} \ln \left (x \right )}{3}+\frac {3 x^{4} \ln \left (x \right )}{2}+2 x^{2} \ln \left (x \right )^{2}+2 \ln \left ({\mathrm e}^{x^{2}}\right )^{2} x^{2}-\frac {2 \ln \left ({\mathrm e}^{x^{2}}\right ) x^{6}}{3}-2 \ln \left ({\mathrm e}^{x^{2}}\right ) x^{4}+8 \left (\ln \left ({\mathrm e}^{x^{2}}\right )-x^{2}\right ) \left (\frac {x^{4} \ln \left (x \right )}{4}-\frac {x^{4}}{16}\right )+8 \left (\ln \left ({\mathrm e}^{x^{2}}\right )-x^{2}\right ) \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )^{2} x^{4}-4 x^{2} \ln \left (x \right )-\frac {x^{6}}{2}-\ln \left (x \right )^{2}-x^{2}-3 x^{4}-\frac {65}{144}-\frac {2 \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right ) x^{6}}{3}-\frac {5 \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right ) x^{4}}{2}-4 \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right ) x^{2}-2 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right ) x^{4} \ln \left (x \right )-4 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right ) \ln \left (x \right ) x^{2}\)	\(401\)
parts	\(-2 \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right ) \ln \left (x \right )-8 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right ) \left (\frac {\ln \left ({\mathrm e}^{x^{2}}\right ) x^{4}}{4}+\frac {\ln \left ({\mathrm e}^{x^{2}}\right ) x^{2}}{2}-\frac {x^{6}}{12}-\frac {x^{4}}{4}\right )+\frac {\left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right ) x^{4}}{2}+2 x^{2} \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right )+{\left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right )}^{2} \left (x^{2}+1\right )^{2}+x^{4} \ln \left (x \right )^{2}+\frac {4 x^{6} \ln \left (x \right )}{3}+\frac {3 x^{4} \ln \left (x \right )}{2}+2 x^{2} \ln \left (x \right )^{2}+2 \ln \left ({\mathrm e}^{x^{2}}\right )^{2} x^{2}-\frac {2 \ln \left ({\mathrm e}^{x^{2}}\right ) x^{6}}{3}-2 \ln \left ({\mathrm e}^{x^{2}}\right ) x^{4}+8 \left (\ln \left ({\mathrm e}^{x^{2}}\right )-x^{2}\right ) \left (\frac {x^{4} \ln \left (x \right )}{4}-\frac {x^{4}}{16}\right )+8 \left (\ln \left ({\mathrm e}^{x^{2}}\right )-x^{2}\right ) \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )^{2} x^{4}-4 x^{2} \ln \left (x \right )-\frac {x^{6}}{2}-\ln \left (x \right )^{2}-x^{2}-3 x^{4}-\frac {65}{144}-\frac {2 \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right ) x^{6}}{3}-\frac {5 \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right ) x^{4}}{2}-4 \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right ) x^{2}-2 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right ) x^{4} \ln \left (x \right )-4 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right ) \ln \left (x \right ) x^{2}\)	\(401\)
risch	\(\text {Expression too large to display}\)	\(7067\)

Input:

int(((4*x^4+4*x^2)*ln(1/exp(x^2)/x)^2+(-4*x^6-10*x^4-8*x^2-2)*ln(1/exp(x^2 
)/x)-2*x^2)/x,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {-2 x^2+\left (-2-8 x^2-10 x^4-4 x^6\right ) \log \left (\frac {e^{-x^2}}{x}\right )+\left (4 x^2+4 x^4\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )}{x} \, dx={\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right )^{2} - x^{2} \] Input:

integrate(((4*x^4+4*x^2)*log(1/exp(x^2)/x)^2+(-4*x^6-10*x^4-8*x^2-2)*log(1 
/exp(x^2)/x)-2*x^2)/x,x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.73 \[ \int \frac {-2 x^2+\left (-2-8 x^2-10 x^4-4 x^6\right ) \log \left (\frac {e^{-x^2}}{x}\right )+\left (4 x^2+4 x^4\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )}{x} \, dx=- x^{2} + \left (x^{4} + 2 x^{2} + 1\right ) \log {\left (\frac {e^{- x^{2}}}{x} \right )}^{2} \] Input:

integrate(((4*x**4+4*x**2)*ln(1/exp(x**2)/x)**2+(-4*x**6-10*x**4-8*x**2-2) 
*ln(1/exp(x**2)/x)-2*x**2)/x,x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (30) = 60\).

Time = 0.06 (sec) , antiderivative size = 171, normalized size of antiderivative = 5.18 \[ \int \frac {-2 x^2+\left (-2-8 x^2-10 x^4-4 x^6\right ) \log \left (\frac {e^{-x^2}}{x}\right )+\left (4 x^2+4 x^4\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )}{x} \, dx=-\frac {2}{3} \, x^{6} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) + x^{4} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right )^{2} - \frac {5}{2} \, x^{4} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) - x^{4} + 2 \, x^{2} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right )^{2} - 4 \, x^{2} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) - x^{2} - {\left (2 \, x^{2} + \log \left (x^{2}\right )\right )} \log \left (x\right ) + \log \left (x\right )^{2} + \frac {1}{6} \, {\left (4 \, x^{6} + 3 \, x^{4}\right )} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) + 2 \, {\left (x^{4} + x^{2}\right )} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) - 2 \, \log \left (x\right ) \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) \] Input:

integrate(((4*x^4+4*x^2)*log(1/exp(x^2)/x)^2+(-4*x^6-10*x^4-8*x^2-2)*log(1 
/exp(x^2)/x)-2*x^2)/x,x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int \frac {-2 x^2+\left (-2-8 x^2-10 x^4-4 x^6\right ) \log \left (\frac {e^{-x^2}}{x}\right )+\left (4 x^2+4 x^4\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )}{x} \, dx=x^{8} + 2 \, x^{6} + x^{4} + {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (x\right )^{2} - x^{2} + 2 \, {\left (x^{6} + 2 \, x^{4} + x^{2}\right )} \log \left (x\right ) \] Input:

integrate(((4*x^4+4*x^2)*log(1/exp(x^2)/x)^2+(-4*x^6-10*x^4-8*x^2-2)*log(1 
/exp(x^2)/x)-2*x^2)/x,x, algorithm="giac")
 

Output:

x^8 + 2*x^6 + x^4 + (x^4 + 2*x^2 + 1)*log(x)^2 - x^2 + 2*(x^6 + 2*x^4 + x^ 
2)*log(x)
 

Mupad [B] (verification not implemented)

Time = 2.53 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {-2 x^2+\left (-2-8 x^2-10 x^4-4 x^6\right ) \log \left (\frac {e^{-x^2}}{x}\right )+\left (4 x^2+4 x^4\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )}{x} \, dx={\ln \left (\frac {{\mathrm {e}}^{-x^2}}{x}\right )}^2\,\left (x^4+2\,x^2+1\right )-x^2 \] Input:

int(-(log(exp(-x^2)/x)*(8*x^2 + 10*x^4 + 4*x^6 + 2) - log(exp(-x^2)/x)^2*( 
4*x^2 + 4*x^4) + 2*x^2)/x,x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36 \[ \int \frac {-2 x^2+\left (-2-8 x^2-10 x^4-4 x^6\right ) \log \left (\frac {e^{-x^2}}{x}\right )+\left (4 x^2+4 x^4\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )}{x} \, dx=\mathrm {log}\left (e^{x^{2}} x \right )^{2} x^{4}+2 \mathrm {log}\left (e^{x^{2}} x \right )^{2} x^{2}+\mathrm {log}\left (e^{x^{2}} x \right )^{2}-x^{2} \] Input:

int(((4*x^4+4*x^2)*log(1/exp(x^2)/x)^2+(-4*x^6-10*x^4-8*x^2-2)*log(1/exp(x 
^2)/x)-2*x^2)/x,x)
 

Output:

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