\(\int \frac {180 x^3-120 x^4 \log (3)+e^{2 x} (200 x-600 x^2+200 x^3 \log (3))+e^x (360 x^2-120 x^3-180 x^4+(-200 x^3+60 x^4+60 x^5) \log (3))+(-60 x^2+180 x^3-60 x^4 \log (3)+e^x (-200 x+700 x^2-300 x^3+(-200 x^3+100 x^4) \log (3))) \log (1-3 x+x^2 \log (3))}{9 x^2-27 x^3+9 x^4 \log (3)+e^{2 x} (25-75 x+25 x^2 \log (3))+e^x (30 x-90 x^2+30 x^3 \log (3))} \, dx\) [1387]

Optimal result

Integrand size = 196, antiderivative size = 33 \[ \int \frac {180 x^3-120 x^4 \log (3)+e^{2 x} \left (200 x-600 x^2+200 x^3 \log (3)\right )+e^x \left (360 x^2-120 x^3-180 x^4+\left (-200 x^3+60 x^4+60 x^5\right ) \log (3)\right )+\left (-60 x^2+180 x^3-60 x^4 \log (3)+e^x \left (-200 x+700 x^2-300 x^3+\left (-200 x^3+100 x^4\right ) \log (3)\right )\right ) \log \left (1-3 x+x^2 \log (3)\right )}{9 x^2-27 x^3+9 x^4 \log (3)+e^{2 x} \left (25-75 x+25 x^2 \log (3)\right )+e^x \left (30 x-90 x^2+30 x^3 \log (3)\right )} \, dx=\frac {4 x \left (e^x-\log (1+x (-3+x \log (3)))\right )}{\frac {3}{5}+\frac {e^x}{x}} \] Output:

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

Mathematica [F]

\[ \int \frac {180 x^3-120 x^4 \log (3)+e^{2 x} \left (200 x-600 x^2+200 x^3 \log (3)\right )+e^x \left (360 x^2-120 x^3-180 x^4+\left (-200 x^3+60 x^4+60 x^5\right ) \log (3)\right )+\left (-60 x^2+180 x^3-60 x^4 \log (3)+e^x \left (-200 x+700 x^2-300 x^3+\left (-200 x^3+100 x^4\right ) \log (3)\right )\right ) \log \left (1-3 x+x^2 \log (3)\right )}{9 x^2-27 x^3+9 x^4 \log (3)+e^{2 x} \left (25-75 x+25 x^2 \log (3)\right )+e^x \left (30 x-90 x^2+30 x^3 \log (3)\right )} \, dx=\int \frac {180 x^3-120 x^4 \log (3)+e^{2 x} \left (200 x-600 x^2+200 x^3 \log (3)\right )+e^x \left (360 x^2-120 x^3-180 x^4+\left (-200 x^3+60 x^4+60 x^5\right ) \log (3)\right )+\left (-60 x^2+180 x^3-60 x^4 \log (3)+e^x \left (-200 x+700 x^2-300 x^3+\left (-200 x^3+100 x^4\right ) \log (3)\right )\right ) \log \left (1-3 x+x^2 \log (3)\right )}{9 x^2-27 x^3+9 x^4 \log (3)+e^{2 x} \left (25-75 x+25 x^2 \log (3)\right )+e^x \left (30 x-90 x^2+30 x^3 \log (3)\right )} \, dx \] Input:

Integrate[(180*x^3 - 120*x^4*Log[3] + E^(2*x)*(200*x - 600*x^2 + 200*x^3*L 
og[3]) + E^x*(360*x^2 - 120*x^3 - 180*x^4 + (-200*x^3 + 60*x^4 + 60*x^5)*L 
og[3]) + (-60*x^2 + 180*x^3 - 60*x^4*Log[3] + E^x*(-200*x + 700*x^2 - 300* 
x^3 + (-200*x^3 + 100*x^4)*Log[3]))*Log[1 - 3*x + x^2*Log[3]])/(9*x^2 - 27 
*x^3 + 9*x^4*Log[3] + E^(2*x)*(25 - 75*x + 25*x^2*Log[3]) + E^x*(30*x - 90 
*x^2 + 30*x^3*Log[3])),x]
 

Output:

Integrate[(180*x^3 - 120*x^4*Log[3] + E^(2*x)*(200*x - 600*x^2 + 200*x^3*L 
og[3]) + E^x*(360*x^2 - 120*x^3 - 180*x^4 + (-200*x^3 + 60*x^4 + 60*x^5)*L 
og[3]) + (-60*x^2 + 180*x^3 - 60*x^4*Log[3] + E^x*(-200*x + 700*x^2 - 300* 
x^3 + (-200*x^3 + 100*x^4)*Log[3]))*Log[1 - 3*x + x^2*Log[3]])/(9*x^2 - 27 
*x^3 + 9*x^4*Log[3] + E^(2*x)*(25 - 75*x + 25*x^2*Log[3]) + E^x*(30*x - 90 
*x^2 + 30*x^3*Log[3])), 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.

\(\displaystyle \int \frac {-120 x^4 \log (3)+180 x^3+e^{2 x} \left (200 x^3 \log (3)-600 x^2+200 x\right )+\left (-60 x^4 \log (3)+180 x^3-60 x^2+e^x \left (-300 x^3+700 x^2+\left (100 x^4-200 x^3\right ) \log (3)-200 x\right )\right ) \log \left (x^2 \log (3)-3 x+1\right )+e^x \left (-180 x^4-120 x^3+360 x^2+\left (60 x^5+60 x^4-200 x^3\right ) \log (3)\right )}{9 x^4 \log (3)-27 x^3+9 x^2+e^{2 x} \left (25 x^2 \log (3)-75 x+25\right )+e^x \left (30 x^3 \log (3)-90 x^2+30 x\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-120 x^4 \log (3)+180 x^3+e^{2 x} \left (200 x^3 \log (3)-600 x^2+200 x\right )+\left (-60 x^4 \log (3)+180 x^3-60 x^2+e^x \left (-300 x^3+700 x^2+\left (100 x^4-200 x^3\right ) \log (3)-200 x\right )\right ) \log \left (x^2 \log (3)-3 x+1\right )+e^x \left (-180 x^4-120 x^3+360 x^2+\left (60 x^5+60 x^4-200 x^3\right ) \log (3)\right )}{\left (3 x+5 e^x\right )^2 \left (x^2 \log (3)-3 x+1\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (-\frac {12 (x-1) x^2 \left (5 \log \left (x^2 \log (3)-3 x+1\right )+3 x\right )}{\left (3 x+5 e^x\right )^2}+\frac {4 x \left (x^4 \log (27)-9 x^3 (1+\log (3))-15 x^2 \left (1+\frac {2 \log (3)}{3}\right ) \log \left (x^2 \log (3)-3 x+1\right )+30 x^2 \left (1-\frac {\log (3)}{3}\right )+35 x \log \left (x^2 \log (3)-3 x+1\right )-10 \log \left (x^2 \log (3)-3 x+1\right )+5 x^3 \log (3) \log \left (x^2 \log (3)-3 x+1\right )+6 x\right )}{\left (3 x+5 e^x\right ) \left (x^2 \log (3)-3 x+1\right )}+8 x\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (-\frac {12 (x-1) x^2 \left (5 \log \left (x^2 \log (3)-3 x+1\right )+3 x\right )}{\left (3 x+5 e^x\right )^2}+\frac {4 x \left (x^4 \log (27)-9 x^3 (1+\log (3))-15 x^2 \left (1+\frac {2 \log (3)}{3}\right ) \log \left (x^2 \log (3)-3 x+1\right )+30 x^2 \left (1-\frac {\log (3)}{3}\right )+35 x \log \left (x^2 \log (3)-3 x+1\right )-10 \log \left (x^2 \log (3)-3 x+1\right )+5 x^3 \log (3) \log \left (x^2 \log (3)-3 x+1\right )+6 x\right )}{\left (3 x+5 e^x\right ) \left (x^2 \log (3)-3 x+1\right )}+8 x\right )dx\)

Input:

Int[(180*x^3 - 120*x^4*Log[3] + E^(2*x)*(200*x - 600*x^2 + 200*x^3*Log[3]) 
 + E^x*(360*x^2 - 120*x^3 - 180*x^4 + (-200*x^3 + 60*x^4 + 60*x^5)*Log[3]) 
 + (-60*x^2 + 180*x^3 - 60*x^4*Log[3] + E^x*(-200*x + 700*x^2 - 300*x^3 + 
(-200*x^3 + 100*x^4)*Log[3]))*Log[1 - 3*x + x^2*Log[3]])/(9*x^2 - 27*x^3 + 
 9*x^4*Log[3] + E^(2*x)*(25 - 75*x + 25*x^2*Log[3]) + E^x*(30*x - 90*x^2 + 
 30*x^3*Log[3])),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39

\[-\frac {20 x^{2} \ln \left (x^{2} \ln \left (3\right )-3 x +1\right )}{3 x +5 \,{\mathrm e}^{x}}+\frac {20 \,{\mathrm e}^{x} x^{2}}{3 x +5 \,{\mathrm e}^{x}}\]

Input:

int(((((100*x^4-200*x^3)*ln(3)-300*x^3+700*x^2-200*x)*exp(x)-60*x^4*ln(3)+ 
180*x^3-60*x^2)*ln(x^2*ln(3)-3*x+1)+(200*x^3*ln(3)-600*x^2+200*x)*exp(x)^2 
+((60*x^5+60*x^4-200*x^3)*ln(3)-180*x^4-120*x^3+360*x^2)*exp(x)-120*x^4*ln 
(3)+180*x^3)/((25*x^2*ln(3)-75*x+25)*exp(x)^2+(30*x^3*ln(3)-90*x^2+30*x)*e 
xp(x)+9*x^4*ln(3)-27*x^3+9*x^2),x)
 

Output:

-20*x^2/(3*x+5*exp(x))*ln(x^2*ln(3)-3*x+1)+20*exp(x)*x^2/(3*x+5*exp(x))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {180 x^3-120 x^4 \log (3)+e^{2 x} \left (200 x-600 x^2+200 x^3 \log (3)\right )+e^x \left (360 x^2-120 x^3-180 x^4+\left (-200 x^3+60 x^4+60 x^5\right ) \log (3)\right )+\left (-60 x^2+180 x^3-60 x^4 \log (3)+e^x \left (-200 x+700 x^2-300 x^3+\left (-200 x^3+100 x^4\right ) \log (3)\right )\right ) \log \left (1-3 x+x^2 \log (3)\right )}{9 x^2-27 x^3+9 x^4 \log (3)+e^{2 x} \left (25-75 x+25 x^2 \log (3)\right )+e^x \left (30 x-90 x^2+30 x^3 \log (3)\right )} \, dx=\frac {20 \, {\left (x^{2} e^{x} - x^{2} \log \left (x^{2} \log \left (3\right ) - 3 \, x + 1\right )\right )}}{3 \, x + 5 \, e^{x}} \] Input:

integrate(((((100*x^4-200*x^3)*log(3)-300*x^3+700*x^2-200*x)*exp(x)-60*x^4 
*log(3)+180*x^3-60*x^2)*log(x^2*log(3)-3*x+1)+(200*x^3*log(3)-600*x^2+200* 
x)*exp(x)^2+((60*x^5+60*x^4-200*x^3)*log(3)-180*x^4-120*x^3+360*x^2)*exp(x 
)-120*x^4*log(3)+180*x^3)/((25*x^2*log(3)-75*x+25)*exp(x)^2+(30*x^3*log(3) 
-90*x^2+30*x)*exp(x)+9*x^4*log(3)-27*x^3+9*x^2),x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {180 x^3-120 x^4 \log (3)+e^{2 x} \left (200 x-600 x^2+200 x^3 \log (3)\right )+e^x \left (360 x^2-120 x^3-180 x^4+\left (-200 x^3+60 x^4+60 x^5\right ) \log (3)\right )+\left (-60 x^2+180 x^3-60 x^4 \log (3)+e^x \left (-200 x+700 x^2-300 x^3+\left (-200 x^3+100 x^4\right ) \log (3)\right )\right ) \log \left (1-3 x+x^2 \log (3)\right )}{9 x^2-27 x^3+9 x^4 \log (3)+e^{2 x} \left (25-75 x+25 x^2 \log (3)\right )+e^x \left (30 x-90 x^2+30 x^3 \log (3)\right )} \, dx=4 x^{2} + \frac {- 12 x^{3} - 20 x^{2} \log {\left (x^{2} \log {\left (3 \right )} - 3 x + 1 \right )}}{3 x + 5 e^{x}} \] Input:

integrate(((((100*x**4-200*x**3)*ln(3)-300*x**3+700*x**2-200*x)*exp(x)-60* 
x**4*ln(3)+180*x**3-60*x**2)*ln(x**2*ln(3)-3*x+1)+(200*x**3*ln(3)-600*x**2 
+200*x)*exp(x)**2+((60*x**5+60*x**4-200*x**3)*ln(3)-180*x**4-120*x**3+360* 
x**2)*exp(x)-120*x**4*ln(3)+180*x**3)/((25*x**2*ln(3)-75*x+25)*exp(x)**2+( 
30*x**3*ln(3)-90*x**2+30*x)*exp(x)+9*x**4*ln(3)-27*x**3+9*x**2),x)
 

Output:

4*x**2 + (-12*x**3 - 20*x**2*log(x**2*log(3) - 3*x + 1))/(3*x + 5*exp(x))
 

Maxima [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {180 x^3-120 x^4 \log (3)+e^{2 x} \left (200 x-600 x^2+200 x^3 \log (3)\right )+e^x \left (360 x^2-120 x^3-180 x^4+\left (-200 x^3+60 x^4+60 x^5\right ) \log (3)\right )+\left (-60 x^2+180 x^3-60 x^4 \log (3)+e^x \left (-200 x+700 x^2-300 x^3+\left (-200 x^3+100 x^4\right ) \log (3)\right )\right ) \log \left (1-3 x+x^2 \log (3)\right )}{9 x^2-27 x^3+9 x^4 \log (3)+e^{2 x} \left (25-75 x+25 x^2 \log (3)\right )+e^x \left (30 x-90 x^2+30 x^3 \log (3)\right )} \, dx=\frac {20 \, {\left (x^{2} e^{x} - x^{2} \log \left (x^{2} \log \left (3\right ) - 3 \, x + 1\right )\right )}}{3 \, x + 5 \, e^{x}} \] Input:

integrate(((((100*x^4-200*x^3)*log(3)-300*x^3+700*x^2-200*x)*exp(x)-60*x^4 
*log(3)+180*x^3-60*x^2)*log(x^2*log(3)-3*x+1)+(200*x^3*log(3)-600*x^2+200* 
x)*exp(x)^2+((60*x^5+60*x^4-200*x^3)*log(3)-180*x^4-120*x^3+360*x^2)*exp(x 
)-120*x^4*log(3)+180*x^3)/((25*x^2*log(3)-75*x+25)*exp(x)^2+(30*x^3*log(3) 
-90*x^2+30*x)*exp(x)+9*x^4*log(3)-27*x^3+9*x^2),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {180 x^3-120 x^4 \log (3)+e^{2 x} \left (200 x-600 x^2+200 x^3 \log (3)\right )+e^x \left (360 x^2-120 x^3-180 x^4+\left (-200 x^3+60 x^4+60 x^5\right ) \log (3)\right )+\left (-60 x^2+180 x^3-60 x^4 \log (3)+e^x \left (-200 x+700 x^2-300 x^3+\left (-200 x^3+100 x^4\right ) \log (3)\right )\right ) \log \left (1-3 x+x^2 \log (3)\right )}{9 x^2-27 x^3+9 x^4 \log (3)+e^{2 x} \left (25-75 x+25 x^2 \log (3)\right )+e^x \left (30 x-90 x^2+30 x^3 \log (3)\right )} \, dx=\frac {20 \, {\left (x^{2} e^{x} - x^{2} \log \left (x^{2} \log \left (3\right ) - 3 \, x + 1\right )\right )}}{3 \, x + 5 \, e^{x}} \] Input:

integrate(((((100*x^4-200*x^3)*log(3)-300*x^3+700*x^2-200*x)*exp(x)-60*x^4 
*log(3)+180*x^3-60*x^2)*log(x^2*log(3)-3*x+1)+(200*x^3*log(3)-600*x^2+200* 
x)*exp(x)^2+((60*x^5+60*x^4-200*x^3)*log(3)-180*x^4-120*x^3+360*x^2)*exp(x 
)-120*x^4*log(3)+180*x^3)/((25*x^2*log(3)-75*x+25)*exp(x)^2+(30*x^3*log(3) 
-90*x^2+30*x)*exp(x)+9*x^4*log(3)-27*x^3+9*x^2),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {180 x^3-120 x^4 \log (3)+e^{2 x} \left (200 x-600 x^2+200 x^3 \log (3)\right )+e^x \left (360 x^2-120 x^3-180 x^4+\left (-200 x^3+60 x^4+60 x^5\right ) \log (3)\right )+\left (-60 x^2+180 x^3-60 x^4 \log (3)+e^x \left (-200 x+700 x^2-300 x^3+\left (-200 x^3+100 x^4\right ) \log (3)\right )\right ) \log \left (1-3 x+x^2 \log (3)\right )}{9 x^2-27 x^3+9 x^4 \log (3)+e^{2 x} \left (25-75 x+25 x^2 \log (3)\right )+e^x \left (30 x-90 x^2+30 x^3 \log (3)\right )} \, dx=-\frac {20\,x^2\,\left (\ln \left (\ln \left (3\right )\,x^2-3\,x+1\right )-{\mathrm {e}}^x\right )}{3\,x+5\,{\mathrm {e}}^x} \] Input:

int((exp(x)*(log(3)*(60*x^4 - 200*x^3 + 60*x^5) + 360*x^2 - 120*x^3 - 180* 
x^4) - log(x^2*log(3) - 3*x + 1)*(exp(x)*(200*x + log(3)*(200*x^3 - 100*x^ 
4) - 700*x^2 + 300*x^3) + 60*x^4*log(3) + 60*x^2 - 180*x^3) - 120*x^4*log( 
3) + exp(2*x)*(200*x + 200*x^3*log(3) - 600*x^2) + 180*x^3)/(exp(x)*(30*x 
+ 30*x^3*log(3) - 90*x^2) + exp(2*x)*(25*x^2*log(3) - 75*x + 25) + 9*x^4*l 
og(3) + 9*x^2 - 27*x^3),x)
                                                                                    
                                                                                    
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {180 x^3-120 x^4 \log (3)+e^{2 x} \left (200 x-600 x^2+200 x^3 \log (3)\right )+e^x \left (360 x^2-120 x^3-180 x^4+\left (-200 x^3+60 x^4+60 x^5\right ) \log (3)\right )+\left (-60 x^2+180 x^3-60 x^4 \log (3)+e^x \left (-200 x+700 x^2-300 x^3+\left (-200 x^3+100 x^4\right ) \log (3)\right )\right ) \log \left (1-3 x+x^2 \log (3)\right )}{9 x^2-27 x^3+9 x^4 \log (3)+e^{2 x} \left (25-75 x+25 x^2 \log (3)\right )+e^x \left (30 x-90 x^2+30 x^3 \log (3)\right )} \, dx=\frac {20 x^{2} \left (e^{x}-\mathrm {log}\left (\mathrm {log}\left (3\right ) x^{2}-3 x +1\right )\right )}{5 e^{x}+3 x} \] Input:

int(((((100*x^4-200*x^3)*log(3)-300*x^3+700*x^2-200*x)*exp(x)-60*x^4*log(3 
)+180*x^3-60*x^2)*log(x^2*log(3)-3*x+1)+(200*x^3*log(3)-600*x^2+200*x)*exp 
(x)^2+((60*x^5+60*x^4-200*x^3)*log(3)-180*x^4-120*x^3+360*x^2)*exp(x)-120* 
x^4*log(3)+180*x^3)/((25*x^2*log(3)-75*x+25)*exp(x)^2+(30*x^3*log(3)-90*x^ 
2+30*x)*exp(x)+9*x^4*log(3)-27*x^3+9*x^2),x)
 

Output:

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