\(\int \frac {3 x^2+e^5 (-x^2-x^3)+(-6 x+2 e^5 x^2) \log (\frac {e^x x}{6})+(-6 x+2 e^5 x^2) \log (\frac {3-e^5 x}{x})}{-3 x^4+e^5 x^5+(-3+e^5 x) \log ^2(\frac {e^x x}{6})+(6 x^2-2 e^5 x^3) \log (\frac {3-e^5 x}{x})+(-3+e^5 x) \log ^2(\frac {3-e^5 x}{x})+\log (\frac {e^x x}{6}) (6 x^2-2 e^5 x^3+(-6+2 e^5 x) \log (\frac {3-e^5 x}{x}))} \, dx\) [1490]

Optimal result

Integrand size = 201, antiderivative size = 33 \[ \int \frac {3 x^2+e^5 \left (-x^2-x^3\right )+\left (-6 x+2 e^5 x^2\right ) \log \left (\frac {e^x x}{6}\right )+\left (-6 x+2 e^5 x^2\right ) \log \left (\frac {3-e^5 x}{x}\right )}{-3 x^4+e^5 x^5+\left (-3+e^5 x\right ) \log ^2\left (\frac {e^x x}{6}\right )+\left (6 x^2-2 e^5 x^3\right ) \log \left (\frac {3-e^5 x}{x}\right )+\left (-3+e^5 x\right ) \log ^2\left (\frac {3-e^5 x}{x}\right )+\log \left (\frac {e^x x}{6}\right ) \left (6 x^2-2 e^5 x^3+\left (-6+2 e^5 x\right ) \log \left (\frac {3-e^5 x}{x}\right )\right )} \, dx=\frac {x^2}{-x^2+\log \left (-e^5+\frac {3}{x}\right )+\log \left (\frac {e^x x}{6}\right )} \] Output:

x^2/(ln(1/6*exp(x)*x)-x^2+ln(3/x-exp(5)))
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {3 x^2+e^5 \left (-x^2-x^3\right )+\left (-6 x+2 e^5 x^2\right ) \log \left (\frac {e^x x}{6}\right )+\left (-6 x+2 e^5 x^2\right ) \log \left (\frac {3-e^5 x}{x}\right )}{-3 x^4+e^5 x^5+\left (-3+e^5 x\right ) \log ^2\left (\frac {e^x x}{6}\right )+\left (6 x^2-2 e^5 x^3\right ) \log \left (\frac {3-e^5 x}{x}\right )+\left (-3+e^5 x\right ) \log ^2\left (\frac {3-e^5 x}{x}\right )+\log \left (\frac {e^x x}{6}\right ) \left (6 x^2-2 e^5 x^3+\left (-6+2 e^5 x\right ) \log \left (\frac {3-e^5 x}{x}\right )\right )} \, dx=\frac {x^2}{-x^2+\log \left (-e^5+\frac {3}{x}\right )+\log \left (\frac {e^x x}{6}\right )} \] Input:

Integrate[(3*x^2 + E^5*(-x^2 - x^3) + (-6*x + 2*E^5*x^2)*Log[(E^x*x)/6] + 
(-6*x + 2*E^5*x^2)*Log[(3 - E^5*x)/x])/(-3*x^4 + E^5*x^5 + (-3 + E^5*x)*Lo 
g[(E^x*x)/6]^2 + (6*x^2 - 2*E^5*x^3)*Log[(3 - E^5*x)/x] + (-3 + E^5*x)*Log 
[(3 - E^5*x)/x]^2 + Log[(E^x*x)/6]*(6*x^2 - 2*E^5*x^3 + (-6 + 2*E^5*x)*Log 
[(3 - E^5*x)/x])),x]
 

Output:

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

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.

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \int \frac {1}{\left (x^2-\log \left (\frac {3}{x}-e^5\right )-\log \left (\frac {e^x x}{6}\right )\right )^2}dx}{e^5}-\int \frac {x}{\left (x^2-\log \left (\frac {3}{x}-e^5\right )-\log \left (\frac {e^x x}{6}\right )\right )^2}dx-\int \frac {x^2}{\left (x^2-\log \left (\frac {3}{x}-e^5\right )-\log \left (\frac {e^x x}{6}\right )\right )^2}dx-\frac {9 \int \frac {1}{\left (e^5 x-3\right ) \left (x^2-\log \left (\frac {3}{x}-e^5\right )-\log \left (\frac {e^x x}{6}\right )\right )^2}dx}{e^5}-2 \int \frac {x}{x^2-\log \left (\frac {3}{x}-e^5\right )-\log \left (\frac {e^x x}{6}\right )}dx+2 \int \frac {x^3}{\left (x^2-\log \left (\frac {3}{x}-e^5\right )-\log \left (\frac {e^x x}{6}\right )\right )^2}dx\)

Input:

Int[(3*x^2 + E^5*(-x^2 - x^3) + (-6*x + 2*E^5*x^2)*Log[(E^x*x)/6] + (-6*x 
+ 2*E^5*x^2)*Log[(3 - E^5*x)/x])/(-3*x^4 + E^5*x^5 + (-3 + E^5*x)*Log[(E^x 
*x)/6]^2 + (6*x^2 - 2*E^5*x^3)*Log[(3 - E^5*x)/x] + (-3 + E^5*x)*Log[(3 - 
E^5*x)/x]^2 + Log[(E^x*x)/6]*(6*x^2 - 2*E^5*x^3 + (-6 + 2*E^5*x)*Log[(3 - 
E^5*x)/x])),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 10.35 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03

method	result	size
parallelrisch	\(-\frac {x^{2}}{x^{2}-\ln \left (\frac {{\mathrm e}^{x} x}{6}\right )-\ln \left (-\frac {x \,{\mathrm e}^{5}-3}{x}\right )}\)	\(34\)
risch	\(-\frac {2 x^{2}}{i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{x} x \right ) \operatorname {csgn}\left (i {\mathrm e}^{x}\right ) \operatorname {csgn}\left (i x \right )-i \pi \operatorname {csgn}\left (i {\mathrm e}^{x} x \right )^{2} \operatorname {csgn}\left (i x \right )-i \pi {\operatorname {csgn}\left (\frac {i \left (x \,{\mathrm e}^{5}-3\right )}{x}\right )}^{2} \operatorname {csgn}\left (\frac {i}{x}\right )-i \pi \operatorname {csgn}\left (i {\mathrm e}^{x} x \right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{x}\right )+2 i \pi {\operatorname {csgn}\left (\frac {i \left (x \,{\mathrm e}^{5}-3\right )}{x}\right )}^{2}+i \pi \,\operatorname {csgn}\left (i \left (x \,{\mathrm e}^{5}-3\right )\right ) \operatorname {csgn}\left (\frac {i \left (x \,{\mathrm e}^{5}-3\right )}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )-i \pi \,\operatorname {csgn}\left (i \left (x \,{\mathrm e}^{5}-3\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x \,{\mathrm e}^{5}-3\right )}{x}\right )}^{2}+i \pi \operatorname {csgn}\left (i {\mathrm e}^{x} x \right )^{3}-i \pi {\operatorname {csgn}\left (\frac {i \left (x \,{\mathrm e}^{5}-3\right )}{x}\right )}^{3}-2 i \pi +2 x^{2}+2 \ln \left (3\right )+2 \ln \left (2\right )-2 \ln \left (x \,{\mathrm e}^{5}-3\right )-2 \ln \left ({\mathrm e}^{x}\right )}\)	\(239\)

Input:

int(((2*x^2*exp(5)-6*x)*ln(1/6*exp(x)*x)+(2*x^2*exp(5)-6*x)*ln((-x*exp(5)+ 
3)/x)+(-x^3-x^2)*exp(5)+3*x^2)/((x*exp(5)-3)*ln(1/6*exp(x)*x)^2+((2*x*exp( 
5)-6)*ln((-x*exp(5)+3)/x)-2*x^3*exp(5)+6*x^2)*ln(1/6*exp(x)*x)+(x*exp(5)-3 
)*ln((-x*exp(5)+3)/x)^2+(-2*x^3*exp(5)+6*x^2)*ln((-x*exp(5)+3)/x)+x^5*exp( 
5)-3*x^4),x,method=_RETURNVERBOSE)
 

Output:

-x^2/(x^2-ln(1/6*exp(x)*x)-ln(-(x*exp(5)-3)/x))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {3 x^2+e^5 \left (-x^2-x^3\right )+\left (-6 x+2 e^5 x^2\right ) \log \left (\frac {e^x x}{6}\right )+\left (-6 x+2 e^5 x^2\right ) \log \left (\frac {3-e^5 x}{x}\right )}{-3 x^4+e^5 x^5+\left (-3+e^5 x\right ) \log ^2\left (\frac {e^x x}{6}\right )+\left (6 x^2-2 e^5 x^3\right ) \log \left (\frac {3-e^5 x}{x}\right )+\left (-3+e^5 x\right ) \log ^2\left (\frac {3-e^5 x}{x}\right )+\log \left (\frac {e^x x}{6}\right ) \left (6 x^2-2 e^5 x^3+\left (-6+2 e^5 x\right ) \log \left (\frac {3-e^5 x}{x}\right )\right )} \, dx=-\frac {x^{2}}{x^{2} - \log \left (\frac {1}{6} \, x e^{x}\right ) - \log \left (-\frac {x e^{5} - 3}{x}\right )} \] Input:

integrate(((2*x^2*exp(5)-6*x)*log(1/6*exp(x)*x)+(2*x^2*exp(5)-6*x)*log((-x 
*exp(5)+3)/x)+(-x^3-x^2)*exp(5)+3*x^2)/((x*exp(5)-3)*log(1/6*exp(x)*x)^2+( 
(2*x*exp(5)-6)*log((-x*exp(5)+3)/x)-2*x^3*exp(5)+6*x^2)*log(1/6*exp(x)*x)+ 
(x*exp(5)-3)*log((-x*exp(5)+3)/x)^2+(-2*x^3*exp(5)+6*x^2)*log((-x*exp(5)+3 
)/x)+x^5*exp(5)-3*x^4),x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.73 \[ \int \frac {3 x^2+e^5 \left (-x^2-x^3\right )+\left (-6 x+2 e^5 x^2\right ) \log \left (\frac {e^x x}{6}\right )+\left (-6 x+2 e^5 x^2\right ) \log \left (\frac {3-e^5 x}{x}\right )}{-3 x^4+e^5 x^5+\left (-3+e^5 x\right ) \log ^2\left (\frac {e^x x}{6}\right )+\left (6 x^2-2 e^5 x^3\right ) \log \left (\frac {3-e^5 x}{x}\right )+\left (-3+e^5 x\right ) \log ^2\left (\frac {3-e^5 x}{x}\right )+\log \left (\frac {e^x x}{6}\right ) \left (6 x^2-2 e^5 x^3+\left (-6+2 e^5 x\right ) \log \left (\frac {3-e^5 x}{x}\right )\right )} \, dx=\frac {x^{2}}{- x^{2} + \log {\left (\frac {- x e^{5} + 3}{x} \right )} + \log {\left (\frac {x e^{x}}{6} \right )}} \] Input:

integrate(((2*x**2*exp(5)-6*x)*ln(1/6*exp(x)*x)+(2*x**2*exp(5)-6*x)*ln((-x 
*exp(5)+3)/x)+(-x**3-x**2)*exp(5)+3*x**2)/((x*exp(5)-3)*ln(1/6*exp(x)*x)** 
2+((2*x*exp(5)-6)*ln((-x*exp(5)+3)/x)-2*x**3*exp(5)+6*x**2)*ln(1/6*exp(x)* 
x)+(x*exp(5)-3)*ln((-x*exp(5)+3)/x)**2+(-2*x**3*exp(5)+6*x**2)*ln((-x*exp( 
5)+3)/x)+x**5*exp(5)-3*x**4),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {3 x^2+e^5 \left (-x^2-x^3\right )+\left (-6 x+2 e^5 x^2\right ) \log \left (\frac {e^x x}{6}\right )+\left (-6 x+2 e^5 x^2\right ) \log \left (\frac {3-e^5 x}{x}\right )}{-3 x^4+e^5 x^5+\left (-3+e^5 x\right ) \log ^2\left (\frac {e^x x}{6}\right )+\left (6 x^2-2 e^5 x^3\right ) \log \left (\frac {3-e^5 x}{x}\right )+\left (-3+e^5 x\right ) \log ^2\left (\frac {3-e^5 x}{x}\right )+\log \left (\frac {e^x x}{6}\right ) \left (6 x^2-2 e^5 x^3+\left (-6+2 e^5 x\right ) \log \left (\frac {3-e^5 x}{x}\right )\right )} \, dx=-\frac {x^{2}}{x^{2} - x + \log \left (3\right ) + \log \left (2\right ) - \log \left (-x e^{5} + 3\right )} \] Input:

integrate(((2*x^2*exp(5)-6*x)*log(1/6*exp(x)*x)+(2*x^2*exp(5)-6*x)*log((-x 
*exp(5)+3)/x)+(-x^3-x^2)*exp(5)+3*x^2)/((x*exp(5)-3)*log(1/6*exp(x)*x)^2+( 
(2*x*exp(5)-6)*log((-x*exp(5)+3)/x)-2*x^3*exp(5)+6*x^2)*log(1/6*exp(x)*x)+ 
(x*exp(5)-3)*log((-x*exp(5)+3)/x)^2+(-2*x^3*exp(5)+6*x^2)*log((-x*exp(5)+3 
)/x)+x^5*exp(5)-3*x^4),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {3 x^2+e^5 \left (-x^2-x^3\right )+\left (-6 x+2 e^5 x^2\right ) \log \left (\frac {e^x x}{6}\right )+\left (-6 x+2 e^5 x^2\right ) \log \left (\frac {3-e^5 x}{x}\right )}{-3 x^4+e^5 x^5+\left (-3+e^5 x\right ) \log ^2\left (\frac {e^x x}{6}\right )+\left (6 x^2-2 e^5 x^3\right ) \log \left (\frac {3-e^5 x}{x}\right )+\left (-3+e^5 x\right ) \log ^2\left (\frac {3-e^5 x}{x}\right )+\log \left (\frac {e^x x}{6}\right ) \left (6 x^2-2 e^5 x^3+\left (-6+2 e^5 x\right ) \log \left (\frac {3-e^5 x}{x}\right )\right )} \, dx=-\frac {x^{2}}{x^{2} - x + \log \left (6\right ) - \log \left (-x e^{5} + 3\right )} \] Input:

integrate(((2*x^2*exp(5)-6*x)*log(1/6*exp(x)*x)+(2*x^2*exp(5)-6*x)*log((-x 
*exp(5)+3)/x)+(-x^3-x^2)*exp(5)+3*x^2)/((x*exp(5)-3)*log(1/6*exp(x)*x)^2+( 
(2*x*exp(5)-6)*log((-x*exp(5)+3)/x)-2*x^3*exp(5)+6*x^2)*log(1/6*exp(x)*x)+ 
(x*exp(5)-3)*log((-x*exp(5)+3)/x)^2+(-2*x^3*exp(5)+6*x^2)*log((-x*exp(5)+3 
)/x)+x^5*exp(5)-3*x^4),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 3.47 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {3 x^2+e^5 \left (-x^2-x^3\right )+\left (-6 x+2 e^5 x^2\right ) \log \left (\frac {e^x x}{6}\right )+\left (-6 x+2 e^5 x^2\right ) \log \left (\frac {3-e^5 x}{x}\right )}{-3 x^4+e^5 x^5+\left (-3+e^5 x\right ) \log ^2\left (\frac {e^x x}{6}\right )+\left (6 x^2-2 e^5 x^3\right ) \log \left (\frac {3-e^5 x}{x}\right )+\left (-3+e^5 x\right ) \log ^2\left (\frac {3-e^5 x}{x}\right )+\log \left (\frac {e^x x}{6}\right ) \left (6 x^2-2 e^5 x^3+\left (-6+2 e^5 x\right ) \log \left (\frac {3-e^5 x}{x}\right )\right )} \, dx=\frac {x^2}{\ln \left (\frac {x\,{\mathrm {e}}^x}{6}\right )+\ln \left (-\frac {x\,{\mathrm {e}}^5-3}{x}\right )-x^2} \] Input:

int(-(log((x*exp(x))/6)*(6*x - 2*x^2*exp(5)) + log(-(x*exp(5) - 3)/x)*(6*x 
 - 2*x^2*exp(5)) + exp(5)*(x^2 + x^3) - 3*x^2)/(log((x*exp(x))/6)^2*(x*exp 
(5) - 3) + log(-(x*exp(5) - 3)/x)^2*(x*exp(5) - 3) + x^5*exp(5) - log(-(x* 
exp(5) - 3)/x)*(2*x^3*exp(5) - 6*x^2) - 3*x^4 + log((x*exp(x))/6)*(log(-(x 
*exp(5) - 3)/x)*(2*x*exp(5) - 6) - 2*x^3*exp(5) + 6*x^2)),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.52 \[ \int \frac {3 x^2+e^5 \left (-x^2-x^3\right )+\left (-6 x+2 e^5 x^2\right ) \log \left (\frac {e^x x}{6}\right )+\left (-6 x+2 e^5 x^2\right ) \log \left (\frac {3-e^5 x}{x}\right )}{-3 x^4+e^5 x^5+\left (-3+e^5 x\right ) \log ^2\left (\frac {e^x x}{6}\right )+\left (6 x^2-2 e^5 x^3\right ) \log \left (\frac {3-e^5 x}{x}\right )+\left (-3+e^5 x\right ) \log ^2\left (\frac {3-e^5 x}{x}\right )+\log \left (\frac {e^x x}{6}\right ) \left (6 x^2-2 e^5 x^3+\left (-6+2 e^5 x\right ) \log \left (\frac {3-e^5 x}{x}\right )\right )} \, dx=\frac {\mathrm {log}\left (\frac {-e^{5} x +3}{x}\right )+\mathrm {log}\left (\frac {e^{x} x}{6}\right )}{\mathrm {log}\left (\frac {-e^{5} x +3}{x}\right )+\mathrm {log}\left (\frac {e^{x} x}{6}\right )-x^{2}} \] Input:

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

Output:

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