3.24.2 \(\int \frac {e^7 (-3 x^2+9 x^3)+(x^3-6 x^4+e^3 (4 x^3-12 x^4)) \log (2)+(e^4 (6 x^2-18 x^3)+(-4 x^3+12 x^4) \log (2)) \log (-x+3 x^2)+e (-3 x^2+9 x^3) \log ^2(-x+3 x^2)}{e^6 (-1+3 x)+e^3 (2-6 x) \log (-x+3 x^2)+(-1+3 x) \log ^2(-x+3 x^2)} \, dx\) [2302]

3.24.2.1 Optimal result

Integrand size = 159, antiderivative size = 28 \[ \int \frac {e^7 \left (-3 x^2+9 x^3\right )+\left (x^3-6 x^4+e^3 \left (4 x^3-12 x^4\right )\right ) \log (2)+\left (e^4 \left (6 x^2-18 x^3\right )+\left (-4 x^3+12 x^4\right ) \log (2)\right ) \log \left (-x+3 x^2\right )+e \left (-3 x^2+9 x^3\right ) \log ^2\left (-x+3 x^2\right )}{e^6 (-1+3 x)+e^3 (2-6 x) \log \left (-x+3 x^2\right )+(-1+3 x) \log ^2\left (-x+3 x^2\right )} \, dx=x^3 \left (e+\frac {x \log (2)}{-e^3+\log \left (-x+3 x^2\right )}\right ) \]

output

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

3.24.2.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(62\) vs. \(2(28)=56\).

Time = 0.21 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.21 \[ \int \frac {e^7 \left (-3 x^2+9 x^3\right )+\left (x^3-6 x^4+e^3 \left (4 x^3-12 x^4\right )\right ) \log (2)+\left (e^4 \left (6 x^2-18 x^3\right )+\left (-4 x^3+12 x^4\right ) \log (2)\right ) \log \left (-x+3 x^2\right )+e \left (-3 x^2+9 x^3\right ) \log ^2\left (-x+3 x^2\right )}{e^6 (-1+3 x)+e^3 (2-6 x) \log \left (-x+3 x^2\right )+(-1+3 x) \log ^2\left (-x+3 x^2\right )} \, dx=\frac {x^3 \left (e^4 (-1+6 x)+x (\log (2)-x \log (64))+(e-6 e x) \log (x (-1+3 x))\right )}{(-1+6 x) \left (e^3-\log (x (-1+3 x))\right )} \]

input

Integrate[(E^7*(-3*x^2 + 9*x^3) + (x^3 - 6*x^4 + E^3*(4*x^3 - 12*x^4))*Log 
[2] + (E^4*(6*x^2 - 18*x^3) + (-4*x^3 + 12*x^4)*Log[2])*Log[-x + 3*x^2] + 
E*(-3*x^2 + 9*x^3)*Log[-x + 3*x^2]^2)/(E^6*(-1 + 3*x) + E^3*(2 - 6*x)*Log[ 
-x + 3*x^2] + (-1 + 3*x)*Log[-x + 3*x^2]^2),x]
 

output

(x^3*(E^4*(-1 + 6*x) + x*(Log[2] - x*Log[64]) + (E - 6*E*x)*Log[x*(-1 + 3* 
x)]))/((-1 + 6*x)*(E^3 - Log[x*(-1 + 3*x)]))
 

3.24.2.3 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.

\(\displaystyle \int \frac {\left (-6 x^4+x^3+e^3 \left (4 x^3-12 x^4\right )\right ) \log (2)+e^7 \left (9 x^3-3 x^2\right )+e \left (9 x^3-3 x^2\right ) \log ^2\left (3 x^2-x\right )+\left (\left (12 x^4-4 x^3\right ) \log (2)+e^4 \left (6 x^2-18 x^3\right )\right ) \log \left (3 x^2-x\right )}{(3 x-1) \log ^2\left (3 x^2-x\right )+e^3 (2-6 x) \log \left (3 x^2-x\right )+e^6 (3 x-1)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-\left (-6 x^4+x^3+e^3 \left (4 x^3-12 x^4\right )\right ) \log (2)-e^7 \left (9 x^3-3 x^2\right )-e \left (9 x^3-3 x^2\right ) \log ^2\left (3 x^2-x\right )-\left (\left (12 x^4-4 x^3\right ) \log (2)+e^4 \left (6 x^2-18 x^3\right )\right ) \log \left (3 x^2-x\right )}{(1-3 x) \left (e^3-\log (x (3 x-1))\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 e x^2 \log ^2(x (3 x-1))}{\left (e^3-\log (x (3 x-1))\right )^2}+\frac {2 x^2 \left (x \log (4)-3 e^4\right ) \log (x (3 x-1))}{\left (e^3-\log (x (3 x-1))\right )^2}+\frac {x^2 \left (6 \left (1+2 e^3\right ) x^2 \log (2)-x \left (9 e^7+\log (2)+e^3 \log (16)\right )+3 e^7\right )}{(1-3 x) \left (e^3-\log (x (3 x-1))\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 e^3 \log (4) \int \frac {x^3}{\left (e^3-\log (x (3 x-1))\right )^2}dx-2 \left (1+2 e^3\right ) \log (2) \int \frac {x^3}{\left (e^3-\log (x (3 x-1))\right )^2}dx-2 \log (4) \int \frac {x^3}{e^3-\log (x (3 x-1))}dx+\frac {1}{3} \left (9 e^7-\log (2)\right ) \int \frac {x^2}{\left (e^3-\log (x (3 x-1))\right )^2}dx-3 e^7 \int \frac {x^2}{\left (e^3-\log (x (3 x-1))\right )^2}dx-\frac {1}{27} \log (2) \int \frac {1}{\left (e^3-\log (x (3 x-1))\right )^2}dx-\frac {1}{9} \log (2) \int \frac {x}{\left (e^3-\log (x (3 x-1))\right )^2}dx-\frac {1}{27} \log (2) \int \frac {1}{(3 x-1) \left (e^3-\log (x (3 x-1))\right )^2}dx+e x^3\)

input

Int[(E^7*(-3*x^2 + 9*x^3) + (x^3 - 6*x^4 + E^3*(4*x^3 - 12*x^4))*Log[2] + 
(E^4*(6*x^2 - 18*x^3) + (-4*x^3 + 12*x^4)*Log[2])*Log[-x + 3*x^2] + E*(-3* 
x^2 + 9*x^3)*Log[-x + 3*x^2]^2)/(E^6*(-1 + 3*x) + E^3*(2 - 6*x)*Log[-x + 3 
*x^2] + (-1 + 3*x)*Log[-x + 3*x^2]^2),x]
 

output

$Aborted
 

3.24.2.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.24.2.4 Maple [A] (verified)

Time = 1.43 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14

method	result	size
risch	\(x^{3} {\mathrm e}-\frac {\ln \left (2\right ) x^{4}}{{\mathrm e}^{3}-\ln \left (3 x^{2}-x \right )}\)	\(32\)
norman	\(\frac {{\mathrm e} \,{\mathrm e}^{3} x^{3}-x^{4} \ln \left (2\right )-x^{3} {\mathrm e} \ln \left (3 x^{2}-x \right )}{{\mathrm e}^{3}-\ln \left (3 x^{2}-x \right )}\)	\(52\)
parallelrisch	\(-\frac {9 x^{4} \ln \left (2\right )-9 \,{\mathrm e} \,{\mathrm e}^{3} x^{3}+9 x^{3} {\mathrm e} \ln \left (3 x^{2}-x \right )}{9 \left ({\mathrm e}^{3}-\ln \left (3 x^{2}-x \right )\right )}\)	\(54\)

input

int(((9*x^3-3*x^2)*exp(1)*ln(3*x^2-x)^2+((12*x^4-4*x^3)*ln(2)+(-18*x^3+6*x 
^2)*exp(1)*exp(3))*ln(3*x^2-x)+((-12*x^4+4*x^3)*exp(3)-6*x^4+x^3)*ln(2)+(9 
*x^3-3*x^2)*exp(1)*exp(3)^2)/((-1+3*x)*ln(3*x^2-x)^2+(-6*x+2)*exp(3)*ln(3* 
x^2-x)+(-1+3*x)*exp(3)^2),x,method=_RETURNVERBOSE)
 

output

x^3*exp(1)-ln(2)*x^4/(exp(3)-ln(3*x^2-x))
 

3.24.2.5 Fricas [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {e^7 \left (-3 x^2+9 x^3\right )+\left (x^3-6 x^4+e^3 \left (4 x^3-12 x^4\right )\right ) \log (2)+\left (e^4 \left (6 x^2-18 x^3\right )+\left (-4 x^3+12 x^4\right ) \log (2)\right ) \log \left (-x+3 x^2\right )+e \left (-3 x^2+9 x^3\right ) \log ^2\left (-x+3 x^2\right )}{e^6 (-1+3 x)+e^3 (2-6 x) \log \left (-x+3 x^2\right )+(-1+3 x) \log ^2\left (-x+3 x^2\right )} \, dx=-\frac {x^{4} \log \left (2\right ) + x^{3} e \log \left (3 \, x^{2} - x\right ) - x^{3} e^{4}}{e^{3} - \log \left (3 \, x^{2} - x\right )} \]

input

integrate(((9*x^3-3*x^2)*exp(1)*log(3*x^2-x)^2+((12*x^4-4*x^3)*log(2)+(-18 
*x^3+6*x^2)*exp(1)*exp(3))*log(3*x^2-x)+((-12*x^4+4*x^3)*exp(3)-6*x^4+x^3) 
*log(2)+(9*x^3-3*x^2)*exp(1)*exp(3)^2)/((-1+3*x)*log(3*x^2-x)^2+(-6*x+2)*e 
xp(3)*log(3*x^2-x)+(-1+3*x)*exp(3)^2),x, algorithm=\
 

output

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

3.24.2.6 Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {e^7 \left (-3 x^2+9 x^3\right )+\left (x^3-6 x^4+e^3 \left (4 x^3-12 x^4\right )\right ) \log (2)+\left (e^4 \left (6 x^2-18 x^3\right )+\left (-4 x^3+12 x^4\right ) \log (2)\right ) \log \left (-x+3 x^2\right )+e \left (-3 x^2+9 x^3\right ) \log ^2\left (-x+3 x^2\right )}{e^6 (-1+3 x)+e^3 (2-6 x) \log \left (-x+3 x^2\right )+(-1+3 x) \log ^2\left (-x+3 x^2\right )} \, dx=\frac {x^{4} \log {\left (2 \right )}}{\log {\left (3 x^{2} - x \right )} - e^{3}} + e x^{3} \]

input

integrate(((9*x**3-3*x**2)*exp(1)*ln(3*x**2-x)**2+((12*x**4-4*x**3)*ln(2)+ 
(-18*x**3+6*x**2)*exp(1)*exp(3))*ln(3*x**2-x)+((-12*x**4+4*x**3)*exp(3)-6* 
x**4+x**3)*ln(2)+(9*x**3-3*x**2)*exp(1)*exp(3)**2)/((-1+3*x)*ln(3*x**2-x)* 
*2+(-6*x+2)*exp(3)*ln(3*x**2-x)+(-1+3*x)*exp(3)**2),x)
 

output

x**4*log(2)/(log(3*x**2 - x) - exp(3)) + E*x**3
 

3.24.2.7 Maxima [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {e^7 \left (-3 x^2+9 x^3\right )+\left (x^3-6 x^4+e^3 \left (4 x^3-12 x^4\right )\right ) \log (2)+\left (e^4 \left (6 x^2-18 x^3\right )+\left (-4 x^3+12 x^4\right ) \log (2)\right ) \log \left (-x+3 x^2\right )+e \left (-3 x^2+9 x^3\right ) \log ^2\left (-x+3 x^2\right )}{e^6 (-1+3 x)+e^3 (2-6 x) \log \left (-x+3 x^2\right )+(-1+3 x) \log ^2\left (-x+3 x^2\right )} \, dx=-\frac {x^{4} \log \left (2\right ) + x^{3} e \log \left (3 \, x - 1\right ) + x^{3} e \log \left (x\right ) - x^{3} e^{4}}{e^{3} - \log \left (3 \, x - 1\right ) - \log \left (x\right )} \]

input

integrate(((9*x^3-3*x^2)*exp(1)*log(3*x^2-x)^2+((12*x^4-4*x^3)*log(2)+(-18 
*x^3+6*x^2)*exp(1)*exp(3))*log(3*x^2-x)+((-12*x^4+4*x^3)*exp(3)-6*x^4+x^3) 
*log(2)+(9*x^3-3*x^2)*exp(1)*exp(3)^2)/((-1+3*x)*log(3*x^2-x)^2+(-6*x+2)*e 
xp(3)*log(3*x^2-x)+(-1+3*x)*exp(3)^2),x, algorithm=\
 

output

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

3.24.2.8 Giac [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {e^7 \left (-3 x^2+9 x^3\right )+\left (x^3-6 x^4+e^3 \left (4 x^3-12 x^4\right )\right ) \log (2)+\left (e^4 \left (6 x^2-18 x^3\right )+\left (-4 x^3+12 x^4\right ) \log (2)\right ) \log \left (-x+3 x^2\right )+e \left (-3 x^2+9 x^3\right ) \log ^2\left (-x+3 x^2\right )}{e^6 (-1+3 x)+e^3 (2-6 x) \log \left (-x+3 x^2\right )+(-1+3 x) \log ^2\left (-x+3 x^2\right )} \, dx=-\frac {x^{4} \log \left (2\right ) + x^{3} e \log \left (3 \, x^{2} - x\right ) - x^{3} e^{4}}{e^{3} - \log \left (3 \, x^{2} - x\right )} \]

input

integrate(((9*x^3-3*x^2)*exp(1)*log(3*x^2-x)^2+((12*x^4-4*x^3)*log(2)+(-18 
*x^3+6*x^2)*exp(1)*exp(3))*log(3*x^2-x)+((-12*x^4+4*x^3)*exp(3)-6*x^4+x^3) 
*log(2)+(9*x^3-3*x^2)*exp(1)*exp(3)^2)/((-1+3*x)*log(3*x^2-x)^2+(-6*x+2)*e 
xp(3)*log(3*x^2-x)+(-1+3*x)*exp(3)^2),x, algorithm=\
 

output

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

3.24.2.9 Mupad [B] (verification not implemented)

Time = 16.24 (sec) , antiderivative size = 129, normalized size of antiderivative = 4.61 \[ \int \frac {e^7 \left (-3 x^2+9 x^3\right )+\left (x^3-6 x^4+e^3 \left (4 x^3-12 x^4\right )\right ) \log (2)+\left (e^4 \left (6 x^2-18 x^3\right )+\left (-4 x^3+12 x^4\right ) \log (2)\right ) \log \left (-x+3 x^2\right )+e \left (-3 x^2+9 x^3\right ) \log ^2\left (-x+3 x^2\right )}{e^6 (-1+3 x)+e^3 (2-6 x) \log \left (-x+3 x^2\right )+(-1+3 x) \log ^2\left (-x+3 x^2\right )} \, dx=\frac {\frac {x^3\,\left (x\,\ln \left (2\right )-6\,x^2\,\ln \left (2\right )-12\,x^2\,{\mathrm {e}}^3\,\ln \left (2\right )+4\,x\,{\mathrm {e}}^3\,\ln \left (2\right )\right )}{6\,x-1}+\frac {4\,x^4\,\ln \left (2\right )\,\ln \left (3\,x^2-x\right )\,\left (3\,x-1\right )}{6\,x-1}}{{\mathrm {e}}^3-\ln \left (3\,x^2-x\right )}-\frac {x\,\ln \left (2\right )}{108}-\frac {\ln \left (2\right )}{6\,\left (648\,x-108\right )}-\frac {x^2\,\ln \left (2\right )}{18}+2\,x^4\,\ln \left (2\right )+x^3\,\left (\mathrm {e}-\frac {\ln \left (2\right )}{3}\right ) \]

input

int(-(exp(7)*(3*x^2 - 9*x^3) - log(3*x^2 - x)*(exp(4)*(6*x^2 - 18*x^3) - l 
og(2)*(4*x^3 - 12*x^4)) - log(2)*(exp(3)*(4*x^3 - 12*x^4) + x^3 - 6*x^4) + 
 exp(1)*log(3*x^2 - x)^2*(3*x^2 - 9*x^3))/(log(3*x^2 - x)^2*(3*x - 1) + ex 
p(6)*(3*x - 1) - exp(3)*log(3*x^2 - x)*(6*x - 2)),x)
 

output

((x^3*(x*log(2) - 6*x^2*log(2) - 12*x^2*exp(3)*log(2) + 4*x*exp(3)*log(2)) 
)/(6*x - 1) + (4*x^4*log(2)*log(3*x^2 - x)*(3*x - 1))/(6*x - 1))/(exp(3) - 
 log(3*x^2 - x)) - (x*log(2))/108 - log(2)/(6*(648*x - 108)) - (x^2*log(2) 
)/18 + 2*x^4*log(2) + x^3*(exp(1) - log(2)/3)