\(\int \frac {-e+6 e^{2 x} x \log ^2(5 e^x)+(6-6 e^x x) \log (x)+\log (5 e^x) (-6 e^x+6 e^{2 x} x-6 e^x x \log (x))}{3 e^{2 x} x \log ^2(5 e^x)-e x \log (x)-6 e^x x \log (5 e^x) \log (x)+3 x \log ^2(x)} \, dx\) [2908]

Optimal result

Integrand size = 107, antiderivative size = 25 \[ \int \frac {-e+6 e^{2 x} x \log ^2\left (5 e^x\right )+\left (6-6 e^x x\right ) \log (x)+\log \left (5 e^x\right ) \left (-6 e^x+6 e^{2 x} x-6 e^x x \log (x)\right )}{3 e^{2 x} x \log ^2\left (5 e^x\right )-e x \log (x)-6 e^x x \log \left (5 e^x\right ) \log (x)+3 x \log ^2(x)} \, dx=\log \left (\log (x)-\frac {3 \left (-e^x \log \left (5 e^x\right )+\log (x)\right )^2}{e}\right ) \] Output:

ln(ln(x)-3*(ln(x)-ln(5*exp(x))*exp(x))^2/exp(1))
 

Mathematica [F]

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

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

Output:

Integrate[(-E + 6*E^(2*x)*x*Log[5*E^x]^2 + (6 - 6*E^x*x)*Log[x] + Log[5*E^ 
x]*(-6*E^x + 6*E^(2*x)*x - 6*E^x*x*Log[x]))/(3*E^(2*x)*x*Log[5*E^x]^2 - E* 
x*Log[x] - 6*E^x*x*Log[5*E^x]*Log[x] + 3*x*Log[x]^2), 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 3.13 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44

method	result	size
parallelrisch	\(\ln \left ({\mathrm e}^{2 x} \ln \left (5 \,{\mathrm e}^{x}\right )^{2}-2 \,{\mathrm e}^{x} \ln \left (x \right ) \ln \left (5 \,{\mathrm e}^{x}\right )-\frac {{\mathrm e} \ln \left (x \right )}{3}+\ln \left (x \right )^{2}\right )\)	\(36\)
risch	\(2 x +\ln \left (\ln \left ({\mathrm e}^{x}\right )^{2}+2 \ln \left (5\right ) \ln \left ({\mathrm e}^{x}\right )-2 \ln \left ({\mathrm e}^{x}\right ) \ln \left (x \right ) {\mathrm e}^{-x}-2 \ln \left (5\right ) \ln \left (x \right ) {\mathrm e}^{-x}+\ln \left (5\right )^{2}+\ln \left (x \right )^{2} {\mathrm e}^{-2 x}-\frac {\ln \left (x \right ) {\mathrm e}^{1-2 x}}{3}\right )\)	\(63\)

Input:

int((6*x*exp(x)^2*ln(5*exp(x))^2+(-6*x*exp(x)*ln(x)+6*x*exp(x)^2-6*exp(x)) 
*ln(5*exp(x))+(-6*exp(x)*x+6)*ln(x)-exp(1))/(3*x*exp(x)^2*ln(5*exp(x))^2-6 
*x*exp(x)*ln(x)*ln(5*exp(x))+3*x*ln(x)^2-x*exp(1)*ln(x)),x,method=_RETURNV 
ERBOSE)
 

Output:

ln(exp(x)^2*ln(5*exp(x))^2-2*exp(x)*ln(x)*ln(5*exp(x))-1/3*exp(1)*ln(x)+ln 
(x)^2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {-e+6 e^{2 x} x \log ^2\left (5 e^x\right )+\left (6-6 e^x x\right ) \log (x)+\log \left (5 e^x\right ) \left (-6 e^x+6 e^{2 x} x-6 e^x x \log (x)\right )}{3 e^{2 x} x \log ^2\left (5 e^x\right )-e x \log (x)-6 e^x x \log \left (5 e^x\right ) \log (x)+3 x \log ^2(x)} \, dx=\log \left (3 \, {\left (x^{2} + 2 \, x \log \left (5\right ) + \log \left (5\right )^{2}\right )} e^{\left (2 \, x\right )} - {\left (6 \, {\left (x + \log \left (5\right )\right )} e^{x} + e\right )} \log \left (x\right ) + 3 \, \log \left (x\right )^{2}\right ) \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (24) = 48\).

Time = 1.91 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.40 \[ \int \frac {-e+6 e^{2 x} x \log ^2\left (5 e^x\right )+\left (6-6 e^x x\right ) \log (x)+\log \left (5 e^x\right ) \left (-6 e^x+6 e^{2 x} x-6 e^x x \log (x)\right )}{3 e^{2 x} x \log ^2\left (5 e^x\right )-e x \log (x)-6 e^x x \log \left (5 e^x\right ) \log (x)+3 x \log ^2(x)} \, dx=2 \log {\left (x + \log {\left (5 \right )} \right )} + \log {\left (\frac {3 \log {\left (x \right )}^{2} - e \log {\left (x \right )}}{3 x^{2} + 6 x \log {\left (5 \right )} + 3 \log {\left (5 \right )}^{2}} + e^{2 x} - \frac {2 e^{x} \log {\left (x \right )}}{x + \log {\left (5 \right )}} \right )} \] Input:

integrate((6*x*exp(x)**2*ln(5*exp(x))**2+(-6*x*exp(x)*ln(x)+6*x*exp(x)**2- 
6*exp(x))*ln(5*exp(x))+(-6*exp(x)*x+6)*ln(x)-exp(1))/(3*x*exp(x)**2*ln(5*e 
xp(x))**2-6*x*exp(x)*ln(x)*ln(5*exp(x))+3*x*ln(x)**2-x*exp(1)*ln(x)),x)
 

Output:

2*log(x + log(5)) + log((3*log(x)**2 - E*log(x))/(3*x**2 + 6*x*log(5) + 3* 
log(5)**2) + exp(2*x) - 2*exp(x)*log(x)/(x + log(5)))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (23) = 46\).

Time = 0.16 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.68 \[ \int \frac {-e+6 e^{2 x} x \log ^2\left (5 e^x\right )+\left (6-6 e^x x\right ) \log (x)+\log \left (5 e^x\right ) \left (-6 e^x+6 e^{2 x} x-6 e^x x \log (x)\right )}{3 e^{2 x} x \log ^2\left (5 e^x\right )-e x \log (x)-6 e^x x \log \left (5 e^x\right ) \log (x)+3 x \log ^2(x)} \, dx=2 \, \log \left (x + \log \left (5\right )\right ) + \log \left (-\frac {6 \, {\left (x + \log \left (5\right )\right )} e^{x} \log \left (x\right ) - 3 \, {\left (x^{2} + 2 \, x \log \left (5\right ) + \log \left (5\right )^{2}\right )} e^{\left (2 \, x\right )} + e \log \left (x\right ) - 3 \, \log \left (x\right )^{2}}{3 \, {\left (x^{2} + 2 \, x \log \left (5\right ) + \log \left (5\right )^{2}\right )}}\right ) \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (23) = 46\).

Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int \frac {-e+6 e^{2 x} x \log ^2\left (5 e^x\right )+\left (6-6 e^x x\right ) \log (x)+\log \left (5 e^x\right ) \left (-6 e^x+6 e^{2 x} x-6 e^x x \log (x)\right )}{3 e^{2 x} x \log ^2\left (5 e^x\right )-e x \log (x)-6 e^x x \log \left (5 e^x\right ) \log (x)+3 x \log ^2(x)} \, dx=\log \left (-3 \, x^{2} e^{\left (2 \, x\right )} - 6 \, x e^{\left (2 \, x\right )} \log \left (5\right ) - 3 \, e^{\left (2 \, x\right )} \log \left (5\right )^{2} + 6 \, x e^{x} \log \left (x\right ) + 6 \, e^{x} \log \left (5\right ) \log \left (x\right ) + e \log \left (x\right ) - 3 \, \log \left (x\right )^{2}\right ) \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {-e+6 e^{2 x} x \log ^2\left (5 e^x\right )+\left (6-6 e^x x\right ) \log (x)+\log \left (5 e^x\right ) \left (-6 e^x+6 e^{2 x} x-6 e^x x \log (x)\right )}{3 e^{2 x} x \log ^2\left (5 e^x\right )-e x \log (x)-6 e^x x \log \left (5 e^x\right ) \log (x)+3 x \log ^2(x)} \, dx=\int -\frac {-6\,x\,{\mathrm {e}}^{2\,x}\,{\ln \left (5\,{\mathrm {e}}^x\right )}^2+\left (6\,{\mathrm {e}}^x-6\,x\,{\mathrm {e}}^{2\,x}+6\,x\,{\mathrm {e}}^x\,\ln \left (x\right )\right )\,\ln \left (5\,{\mathrm {e}}^x\right )+\mathrm {e}+\ln \left (x\right )\,\left (6\,x\,{\mathrm {e}}^x-6\right )}{3\,x\,{\mathrm {e}}^{2\,x}\,{\ln \left (5\,{\mathrm {e}}^x\right )}^2-6\,x\,{\mathrm {e}}^x\,\ln \left (5\,{\mathrm {e}}^x\right )\,\ln \left (x\right )+3\,x\,{\ln \left (x\right )}^2-x\,\mathrm {e}\,\ln \left (x\right )} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.08 \[ \int \frac {-e+6 e^{2 x} x \log ^2\left (5 e^x\right )+\left (6-6 e^x x\right ) \log (x)+\log \left (5 e^x\right ) \left (-6 e^x+6 e^{2 x} x-6 e^x x \log (x)\right )}{3 e^{2 x} x \log ^2\left (5 e^x\right )-e x \log (x)-6 e^x x \log \left (5 e^x\right ) \log (x)+3 x \log ^2(x)} \, dx=\mathrm {log}\left (-\sqrt {e}\, \sqrt {\mathrm {log}\left (x \right )}\, \sqrt {3}+3 e^{x} \mathrm {log}\left (5 e^{x}\right )-3 \,\mathrm {log}\left (x \right )\right )+\mathrm {log}\left (\sqrt {e}\, \sqrt {\mathrm {log}\left (x \right )}\, \sqrt {3}+3 e^{x} \mathrm {log}\left (5 e^{x}\right )-3 \,\mathrm {log}\left (x \right )\right ) \] Input:

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

Output:

log( - sqrt(e)*sqrt(log(x))*sqrt(3) + 3*e**x*log(5*e**x) - 3*log(x)) + log 
(sqrt(e)*sqrt(log(x))*sqrt(3) + 3*e**x*log(5*e**x) - 3*log(x))