\(\int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+(-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)) \log (\frac {e^{-2 x} (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x))}{x^4})}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx\) [66]

Optimal result

Integrand size = 125, antiderivative size = 33 \[ \int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx=\frac {\log \left (6-\log (x)+\frac {e^{-2 x} \left (1-\frac {x+\log (x)}{x}\right )^2}{x^2}\right )}{x} \] Output:

ln((1-(x+ln(x))/x)^2/exp(x)^2/x^2-ln(x)+6)/x
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx=2+\frac {\log \left (6-\log (x)+\frac {e^{-2 x} \log ^2(x)}{x^4}\right )}{x} \] Input:

Integrate[(-(E^(2*x)*x^4) + 2*Log[x] + (-4 - 2*x)*Log[x]^2 + (-6*E^(2*x)*x 
^4 + E^(2*x)*x^4*Log[x] - Log[x]^2)*Log[(6*E^(2*x)*x^4 - E^(2*x)*x^4*Log[x 
] + Log[x]^2)/(E^(2*x)*x^4)])/(6*E^(2*x)*x^6 - E^(2*x)*x^6*Log[x] + x^2*Lo 
g[x]^2),x]
 

Output:

2 + Log[6 - Log[x] + Log[x]^2/(E^(2*x)*x^4)]/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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\log (x) \left (2 x \log ^2(x)+4 \log ^2(x)-12 x \log (x)-25 \log (x)+12\right )}{x^2 (\log (x)-6) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )}+\frac {-\log (x) \log \left (\frac {e^{-2 x} \log ^2(x)}{x^4}-\log (x)+6\right )+6 \log \left (\frac {e^{-2 x} \log ^2(x)}{x^4}-\log (x)+6\right )+1}{x^2 (\log (x)-6)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -12 \int \frac {\log ^2(x)}{x (\log (x)-6) \left (-6 e^{2 x} x^4+e^{2 x} \log (x) x^4-\log ^2(x)\right )}dx+2 \int \frac {\log ^3(x)}{x (\log (x)-6) \left (-6 e^{2 x} x^4+e^{2 x} \log (x) x^4-\log ^2(x)\right )}dx+12 \int \frac {\log (x)}{x^2 (\log (x)-6) \left (-6 e^{2 x} x^4+e^{2 x} \log (x) x^4-\log ^2(x)\right )}dx-25 \int \frac {\log ^2(x)}{x^2 (\log (x)-6) \left (-6 e^{2 x} x^4+e^{2 x} \log (x) x^4-\log ^2(x)\right )}dx-\int \frac {\log \left (\frac {e^{-2 x} \log ^2(x)}{x^4}-\log (x)+6\right )}{x^2}dx+4 \int \frac {\log ^3(x)}{x^2 (\log (x)-6) \left (-6 e^{2 x} x^4+e^{2 x} \log (x) x^4-\log ^2(x)\right )}dx+\frac {\operatorname {ExpIntegralEi}(6-\log (x))}{e^6}\)

Input:

Int[(-(E^(2*x)*x^4) + 2*Log[x] + (-4 - 2*x)*Log[x]^2 + (-6*E^(2*x)*x^4 + E 
^(2*x)*x^4*Log[x] - Log[x]^2)*Log[(6*E^(2*x)*x^4 - E^(2*x)*x^4*Log[x] + Lo 
g[x]^2)/(E^(2*x)*x^4)])/(6*E^(2*x)*x^6 - E^(2*x)*x^6*Log[x] + x^2*Log[x]^2 
),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 98.48 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24

method	result	size
parallelrisch	\(\frac {\ln \left (-\frac {\left (-\ln \left (x \right )^{2}+x^{4} {\mathrm e}^{2 x} \ln \left (x \right )-6 \,{\mathrm e}^{2 x} x^{4}\right ) {\mathrm e}^{-2 x}}{x^{4}}\right )}{x}\)	\(41\)
risch	\(\text {Expression too large to display}\)	\(722\)

Input:

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

Output:

ln(-(-ln(x)^2+x^4*exp(x)^2*ln(x)-6*exp(x)^2*x^4)/exp(x)^2/x^4)/x
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21 \[ \int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx=\frac {\log \left (-\frac {{\left (x^{4} e^{\left (2 \, x\right )} \log \left (x\right ) - 6 \, x^{4} e^{\left (2 \, x\right )} - \log \left (x\right )^{2}\right )} e^{\left (-2 \, x\right )}}{x^{4}}\right )}{x} \] Input:

integrate(((-log(x)^2+x^4*exp(x)^2*log(x)-6*exp(x)^2*x^4)*log((log(x)^2-x^ 
4*exp(x)^2*log(x)+6*exp(x)^2*x^4)/exp(x)^2/x^4)+(-2*x-4)*log(x)^2+2*log(x) 
-exp(x)^2*x^4)/(x^2*log(x)^2-x^6*exp(x)^2*log(x)+6*x^6*exp(x)^2),x, algori 
thm="fricas")
 

Output:

log(-(x^4*e^(2*x)*log(x) - 6*x^4*e^(2*x) - log(x)^2)*e^(-2*x)/x^4)/x
 

Sympy [A] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx=\frac {\log {\left (\frac {\left (- x^{4} e^{2 x} \log {\left (x \right )} + 6 x^{4} e^{2 x} + \log {\left (x \right )}^{2}\right ) e^{- 2 x}}{x^{4}} \right )}}{x} \] Input:

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

Output:

log((-x**4*exp(2*x)*log(x) + 6*x**4*exp(2*x) + log(x)**2)*exp(-2*x)/x**4)/ 
x
 

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx=\frac {\log \left (-x^{4} e^{\left (2 \, x\right )} \log \left (x\right ) + 6 \, x^{4} e^{\left (2 \, x\right )} + \log \left (x\right )^{2}\right ) - 4 \, \log \left (x\right )}{x} \] Input:

integrate(((-log(x)^2+x^4*exp(x)^2*log(x)-6*exp(x)^2*x^4)*log((log(x)^2-x^ 
4*exp(x)^2*log(x)+6*exp(x)^2*x^4)/exp(x)^2/x^4)+(-2*x-4)*log(x)^2+2*log(x) 
-exp(x)^2*x^4)/(x^2*log(x)^2-x^6*exp(x)^2*log(x)+6*x^6*exp(x)^2),x, algori 
thm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \[ \int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx=\frac {\log \left (-{\left (x^{4} e^{\left (2 \, x\right )} \log \left (x\right ) - 6 \, x^{4} e^{\left (2 \, x\right )} - \log \left (x\right )^{2}\right )} e^{\left (-2 \, x\right )}\right ) - 4 \, \log \left (x\right )}{x} \] Input:

integrate(((-log(x)^2+x^4*exp(x)^2*log(x)-6*exp(x)^2*x^4)*log((log(x)^2-x^ 
4*exp(x)^2*log(x)+6*exp(x)^2*x^4)/exp(x)^2/x^4)+(-2*x-4)*log(x)^2+2*log(x) 
-exp(x)^2*x^4)/(x^2*log(x)^2-x^6*exp(x)^2*log(x)+6*x^6*exp(x)^2),x, algori 
thm="giac")
 

Output:

(log(-(x^4*e^(2*x)*log(x) - 6*x^4*e^(2*x) - log(x)^2)*e^(-2*x)) - 4*log(x) 
)/x
 

Mupad [B] (verification not implemented)

Time = 4.53 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx=\frac {\ln \left (\frac {1}{x^4}\right )+\ln \left (6\,x^4-x^4\,\ln \left (x\right )+{\mathrm {e}}^{-2\,x}\,{\ln \left (x\right )}^2\right )}{x} \] Input:

int(-(log((exp(-2*x)*(log(x)^2 + 6*x^4*exp(2*x) - x^4*exp(2*x)*log(x)))/x^ 
4)*(log(x)^2 + 6*x^4*exp(2*x) - x^4*exp(2*x)*log(x)) - 2*log(x) + x^4*exp( 
2*x) + log(x)^2*(2*x + 4))/(6*x^6*exp(2*x) + x^2*log(x)^2 - x^6*exp(2*x)*l 
og(x)),x)
 

Output:

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

Reduce [F]

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

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

Output:

4*int(log(x)**2/(e**(2*x)*log(x)*x**6 - 6*e**(2*x)*x**6 - log(x)**2*x**2), 
x) + 2*int(log(x)**2/(e**(2*x)*log(x)*x**5 - 6*e**(2*x)*x**5 - log(x)**2*x 
),x) - 2*int(log(x)/(e**(2*x)*log(x)*x**6 - 6*e**(2*x)*x**6 - log(x)**2*x* 
*2),x) + int((e**(2*x)*x**2)/(e**(2*x)*log(x)*x**4 - 6*e**(2*x)*x**4 - log 
(x)**2),x) + 6*int((e**(2*x)*log(( - e**(2*x)*log(x)*x**4 + 6*e**(2*x)*x** 
4 + log(x)**2)/(e**(2*x)*x**4))*x**2)/(e**(2*x)*log(x)*x**4 - 6*e**(2*x)*x 
**4 - log(x)**2),x) - int((e**(2*x)*log(( - e**(2*x)*log(x)*x**4 + 6*e**(2 
*x)*x**4 + log(x)**2)/(e**(2*x)*x**4))*log(x)*x**2)/(e**(2*x)*log(x)*x**4 
- 6*e**(2*x)*x**4 - log(x)**2),x) + int((log(( - e**(2*x)*log(x)*x**4 + 6* 
e**(2*x)*x**4 + log(x)**2)/(e**(2*x)*x**4))*log(x)**2)/(e**(2*x)*log(x)*x* 
*6 - 6*e**(2*x)*x**6 - log(x)**2*x**2),x)