3.4.21 \(\int \frac {e (1-x) \log (x)+(-x+x^2) \log ^2(x)+(-2 e x+2 x^2 \log (x)) \log (-3+3 x)+(e (1-x)+(-x+x^2) \log (x)) \log ^2(-3+3 x)+(e (-1+x)-x+x^2+(e (1-x)+x-x^2) \log (x)+(e (1-x)+x-x^2) \log ^2(-3+3 x)) \log (-x+x \log (x)+x \log ^2(-3+3 x))}{(x^2-x^3+(-x^2+x^3) \log (x)+(-x^2+x^3) \log ^2(-3+3 x)) \log ^2(-x+x \log (x)+x \log ^2(-3+3 x))} \, dx\) [321]

3.4.21.1 Optimal result

Integrand size = 203, antiderivative size = 31 \begin {dmath*} \int \frac {e (1-x) \log (x)+\left (-x+x^2\right ) \log ^2(x)+\left (-2 e x+2 x^2 \log (x)\right ) \log (-3+3 x)+\left (e (1-x)+\left (-x+x^2\right ) \log (x)\right ) \log ^2(-3+3 x)+\left (e (-1+x)-x+x^2+\left (e (1-x)+x-x^2\right ) \log (x)+\left (e (1-x)+x-x^2\right ) \log ^2(-3+3 x)\right ) \log \left (-x+x \log (x)+x \log ^2(-3+3 x)\right )}{\left (x^2-x^3+\left (-x^2+x^3\right ) \log (x)+\left (-x^2+x^3\right ) \log ^2(-3+3 x)\right ) \log ^2\left (-x+x \log (x)+x \log ^2(-3+3 x)\right )} \, dx=\frac {\frac {e}{x}-\log (x)}{\log \left (-x+x \left (\log (x)+\log ^2(-3+3 x)\right )\right )} \end {dmath*}

output

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

3.4.21.2 Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \begin {dmath*} \int \frac {e (1-x) \log (x)+\left (-x+x^2\right ) \log ^2(x)+\left (-2 e x+2 x^2 \log (x)\right ) \log (-3+3 x)+\left (e (1-x)+\left (-x+x^2\right ) \log (x)\right ) \log ^2(-3+3 x)+\left (e (-1+x)-x+x^2+\left (e (1-x)+x-x^2\right ) \log (x)+\left (e (1-x)+x-x^2\right ) \log ^2(-3+3 x)\right ) \log \left (-x+x \log (x)+x \log ^2(-3+3 x)\right )}{\left (x^2-x^3+\left (-x^2+x^3\right ) \log (x)+\left (-x^2+x^3\right ) \log ^2(-3+3 x)\right ) \log ^2\left (-x+x \log (x)+x \log ^2(-3+3 x)\right )} \, dx=\frac {e-x \log (x)}{x \log \left (x \left (-1+\log ^2(3 (-1+x))+\log (x)\right )\right )} \end {dmath*}

input

Integrate[(E*(1 - x)*Log[x] + (-x + x^2)*Log[x]^2 + (-2*E*x + 2*x^2*Log[x] 
)*Log[-3 + 3*x] + (E*(1 - x) + (-x + x^2)*Log[x])*Log[-3 + 3*x]^2 + (E*(-1 
 + x) - x + x^2 + (E*(1 - x) + x - x^2)*Log[x] + (E*(1 - x) + x - x^2)*Log 
[-3 + 3*x]^2)*Log[-x + x*Log[x] + x*Log[-3 + 3*x]^2])/((x^2 - x^3 + (-x^2 
+ x^3)*Log[x] + (-x^2 + x^3)*Log[-3 + 3*x]^2)*Log[-x + x*Log[x] + x*Log[-3 
 + 3*x]^2]^2),x]
 

output

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

3.4.21.3 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.

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7299

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

input

Int[(E*(1 - x)*Log[x] + (-x + x^2)*Log[x]^2 + (-2*E*x + 2*x^2*Log[x])*Log[ 
-3 + 3*x] + (E*(1 - x) + (-x + x^2)*Log[x])*Log[-3 + 3*x]^2 + (E*(-1 + x) 
- x + x^2 + (E*(1 - x) + x - x^2)*Log[x] + (E*(1 - x) + x - x^2)*Log[-3 + 
3*x]^2)*Log[-x + x*Log[x] + x*Log[-3 + 3*x]^2])/((x^2 - x^3 + (-x^2 + x^3) 
*Log[x] + (-x^2 + x^3)*Log[-3 + 3*x]^2)*Log[-x + x*Log[x] + x*Log[-3 + 3*x 
]^2]^2),x]
 

output

$Aborted
 

3.4.21.3.1 Defintions of rubi rules used

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] 
]
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

3.4.21.4 Maple [F(-1)]

Timed out.

\[\int \frac {\left (\left (\left (1-x \right ) {\mathrm e}-x^{2}+x \right ) \ln \left (-3+3 x \right )^{2}+\left (\left (1-x \right ) {\mathrm e}-x^{2}+x \right ) \ln \left (x \right )+\left (-1+x \right ) {\mathrm e}+x^{2}-x \right ) \ln \left (x \ln \left (-3+3 x \right )^{2}+x \ln \left (x \right )-x \right )+\left (\ln \left (x \right ) \left (x^{2}-x \right )+\left (1-x \right ) {\mathrm e}\right ) \ln \left (-3+3 x \right )^{2}+\left (2 x^{2} \ln \left (x \right )-2 x \,{\mathrm e}\right ) \ln \left (-3+3 x \right )+\left (x^{2}-x \right ) \ln \left (x \right )^{2}+\left (1-x \right ) {\mathrm e} \ln \left (x \right )}{\left (\left (x^{3}-x^{2}\right ) \ln \left (-3+3 x \right )^{2}+\left (x^{3}-x^{2}\right ) \ln \left (x \right )-x^{3}+x^{2}\right ) \ln \left (x \ln \left (-3+3 x \right )^{2}+x \ln \left (x \right )-x \right )^{2}}d x\]

input

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

output

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

3.4.21.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \begin {dmath*} \int \frac {e (1-x) \log (x)+\left (-x+x^2\right ) \log ^2(x)+\left (-2 e x+2 x^2 \log (x)\right ) \log (-3+3 x)+\left (e (1-x)+\left (-x+x^2\right ) \log (x)\right ) \log ^2(-3+3 x)+\left (e (-1+x)-x+x^2+\left (e (1-x)+x-x^2\right ) \log (x)+\left (e (1-x)+x-x^2\right ) \log ^2(-3+3 x)\right ) \log \left (-x+x \log (x)+x \log ^2(-3+3 x)\right )}{\left (x^2-x^3+\left (-x^2+x^3\right ) \log (x)+\left (-x^2+x^3\right ) \log ^2(-3+3 x)\right ) \log ^2\left (-x+x \log (x)+x \log ^2(-3+3 x)\right )} \, dx=-\frac {x \log \left (x\right ) - e}{x \log \left (x \log \left (3 \, x - 3\right )^{2} + x \log \left (x\right ) - x\right )} \end {dmath*}

input

integrate(((((1-x)*exp(1)-x^2+x)*log(-3+3*x)^2+((1-x)*exp(1)-x^2+x)*log(x) 
+(-1+x)*exp(1)+x^2-x)*log(x*log(-3+3*x)^2+x*log(x)-x)+(log(x)*(x^2-x)+(1-x 
)*exp(1))*log(-3+3*x)^2+(2*x^2*log(x)-2*x*exp(1))*log(-3+3*x)+(x^2-x)*log( 
x)^2+(1-x)*exp(1)*log(x))/((x^3-x^2)*log(-3+3*x)^2+(x^3-x^2)*log(x)-x^3+x^ 
2)/log(x*log(-3+3*x)^2+x*log(x)-x)^2,x, algorithm=\
 

output

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

3.4.21.6 Sympy [A] (verification not implemented)

Time = 0.97 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \begin {dmath*} \int \frac {e (1-x) \log (x)+\left (-x+x^2\right ) \log ^2(x)+\left (-2 e x+2 x^2 \log (x)\right ) \log (-3+3 x)+\left (e (1-x)+\left (-x+x^2\right ) \log (x)\right ) \log ^2(-3+3 x)+\left (e (-1+x)-x+x^2+\left (e (1-x)+x-x^2\right ) \log (x)+\left (e (1-x)+x-x^2\right ) \log ^2(-3+3 x)\right ) \log \left (-x+x \log (x)+x \log ^2(-3+3 x)\right )}{\left (x^2-x^3+\left (-x^2+x^3\right ) \log (x)+\left (-x^2+x^3\right ) \log ^2(-3+3 x)\right ) \log ^2\left (-x+x \log (x)+x \log ^2(-3+3 x)\right )} \, dx=\frac {- x \log {\left (x \right )} + e}{x \log {\left (x \log {\left (x \right )} + x \log {\left (3 x - 3 \right )}^{2} - x \right )}} \end {dmath*}

input

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

output

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

3.4.21.7 Maxima [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.39 \begin {dmath*} \int \frac {e (1-x) \log (x)+\left (-x+x^2\right ) \log ^2(x)+\left (-2 e x+2 x^2 \log (x)\right ) \log (-3+3 x)+\left (e (1-x)+\left (-x+x^2\right ) \log (x)\right ) \log ^2(-3+3 x)+\left (e (-1+x)-x+x^2+\left (e (1-x)+x-x^2\right ) \log (x)+\left (e (1-x)+x-x^2\right ) \log ^2(-3+3 x)\right ) \log \left (-x+x \log (x)+x \log ^2(-3+3 x)\right )}{\left (x^2-x^3+\left (-x^2+x^3\right ) \log (x)+\left (-x^2+x^3\right ) \log ^2(-3+3 x)\right ) \log ^2\left (-x+x \log (x)+x \log ^2(-3+3 x)\right )} \, dx=-\frac {x \log \left (x\right ) - e}{x \log \left (\log \left (3\right )^{2} + 2 \, \log \left (3\right ) \log \left (x - 1\right ) + \log \left (x - 1\right )^{2} + \log \left (x\right ) - 1\right ) + x \log \left (x\right )} \end {dmath*}

input

integrate(((((1-x)*exp(1)-x^2+x)*log(-3+3*x)^2+((1-x)*exp(1)-x^2+x)*log(x) 
+(-1+x)*exp(1)+x^2-x)*log(x*log(-3+3*x)^2+x*log(x)-x)+(log(x)*(x^2-x)+(1-x 
)*exp(1))*log(-3+3*x)^2+(2*x^2*log(x)-2*x*exp(1))*log(-3+3*x)+(x^2-x)*log( 
x)^2+(1-x)*exp(1)*log(x))/((x^3-x^2)*log(-3+3*x)^2+(x^3-x^2)*log(x)-x^3+x^ 
2)/log(x*log(-3+3*x)^2+x*log(x)-x)^2,x, algorithm=\
 

output

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

3.4.21.8 Giac [A] (verification not implemented)

Time = 0.63 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \begin {dmath*} \int \frac {e (1-x) \log (x)+\left (-x+x^2\right ) \log ^2(x)+\left (-2 e x+2 x^2 \log (x)\right ) \log (-3+3 x)+\left (e (1-x)+\left (-x+x^2\right ) \log (x)\right ) \log ^2(-3+3 x)+\left (e (-1+x)-x+x^2+\left (e (1-x)+x-x^2\right ) \log (x)+\left (e (1-x)+x-x^2\right ) \log ^2(-3+3 x)\right ) \log \left (-x+x \log (x)+x \log ^2(-3+3 x)\right )}{\left (x^2-x^3+\left (-x^2+x^3\right ) \log (x)+\left (-x^2+x^3\right ) \log ^2(-3+3 x)\right ) \log ^2\left (-x+x \log (x)+x \log ^2(-3+3 x)\right )} \, dx=-\frac {x \log \left (x\right ) - e}{x \log \left (\log \left (3 \, x - 3\right )^{2} + \log \left (x\right ) - 1\right ) + x \log \left (x\right )} \end {dmath*}

input

integrate(((((1-x)*exp(1)-x^2+x)*log(-3+3*x)^2+((1-x)*exp(1)-x^2+x)*log(x) 
+(-1+x)*exp(1)+x^2-x)*log(x*log(-3+3*x)^2+x*log(x)-x)+(log(x)*(x^2-x)+(1-x 
)*exp(1))*log(-3+3*x)^2+(2*x^2*log(x)-2*x*exp(1))*log(-3+3*x)+(x^2-x)*log( 
x)^2+(1-x)*exp(1)*log(x))/((x^3-x^2)*log(-3+3*x)^2+(x^3-x^2)*log(x)-x^3+x^ 
2)/log(x*log(-3+3*x)^2+x*log(x)-x)^2,x, algorithm=\
 

output

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

3.4.21.9 Mupad [F(-1)]

Timed out. \begin {dmath*} \int \frac {e (1-x) \log (x)+\left (-x+x^2\right ) \log ^2(x)+\left (-2 e x+2 x^2 \log (x)\right ) \log (-3+3 x)+\left (e (1-x)+\left (-x+x^2\right ) \log (x)\right ) \log ^2(-3+3 x)+\left (e (-1+x)-x+x^2+\left (e (1-x)+x-x^2\right ) \log (x)+\left (e (1-x)+x-x^2\right ) \log ^2(-3+3 x)\right ) \log \left (-x+x \log (x)+x \log ^2(-3+3 x)\right )}{\left (x^2-x^3+\left (-x^2+x^3\right ) \log (x)+\left (-x^2+x^3\right ) \log ^2(-3+3 x)\right ) \log ^2\left (-x+x \log (x)+x \log ^2(-3+3 x)\right )} \, dx=\int \frac {\ln \left (x\,{\ln \left (3\,x-3\right )}^2-x+x\,\ln \left (x\right )\right )\,\left (x+{\ln \left (3\,x-3\right )}^2\,\left (\mathrm {e}\,\left (x-1\right )-x+x^2\right )-\mathrm {e}\,\left (x-1\right )-x^2+\ln \left (x\right )\,\left (\mathrm {e}\,\left (x-1\right )-x+x^2\right )\right )+{\ln \left (3\,x-3\right )}^2\,\left (\ln \left (x\right )\,\left (x-x^2\right )+\mathrm {e}\,\left (x-1\right )\right )+{\ln \left (x\right )}^2\,\left (x-x^2\right )-\ln \left (3\,x-3\right )\,\left (2\,x^2\,\ln \left (x\right )-2\,x\,\mathrm {e}\right )+\mathrm {e}\,\ln \left (x\right )\,\left (x-1\right )}{{\ln \left (x\,{\ln \left (3\,x-3\right )}^2-x+x\,\ln \left (x\right )\right )}^2\,\left (\ln \left (x\right )\,\left (x^2-x^3\right )+{\ln \left (3\,x-3\right )}^2\,\left (x^2-x^3\right )-x^2+x^3\right )} \,d x \end {dmath*}

input

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

output

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