\(\int \frac {-15 x+18 x^2+15 x^3-14 x^4+e^3 (12-15 x-16 x^2+15 x^3)+(2 x^3-2 x^4+e^3 (-2 x^2+2 x^3)) \log (-x^5+x^6+e^3 (x^4-x^5))}{100 x^2-100 x^3+e^3 (-100 x+100 x^2)+(20 x^2-20 x^3+e^3 (-20 x+20 x^2)) \log (-x^5+x^6+e^3 (x^4-x^5))+(x^2-x^3+e^3 (-x+x^2)) \log ^2(-x^5+x^6+e^3 (x^4-x^5))} \, dx\) [423]

Optimal result

Integrand size = 209, antiderivative size = 27 \[ \int \frac {-15 x+18 x^2+15 x^3-14 x^4+e^3 \left (12-15 x-16 x^2+15 x^3\right )+\left (2 x^3-2 x^4+e^3 \left (-2 x^2+2 x^3\right )\right ) \log \left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )}{100 x^2-100 x^3+e^3 \left (-100 x+100 x^2\right )+\left (20 x^2-20 x^3+e^3 \left (-20 x+20 x^2\right )\right ) \log \left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )+\left (x^2-x^3+e^3 \left (-x+x^2\right )\right ) \log ^2\left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )} \, dx=\frac {3+x^2}{10+\log \left ((1-x) \left (e^3-x\right ) x^4\right )} \] Output:

(x^2+3)/(10+ln(x^4*(-x+exp(3))*(1-x)))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {-15 x+18 x^2+15 x^3-14 x^4+e^3 \left (12-15 x-16 x^2+15 x^3\right )+\left (2 x^3-2 x^4+e^3 \left (-2 x^2+2 x^3\right )\right ) \log \left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )}{100 x^2-100 x^3+e^3 \left (-100 x+100 x^2\right )+\left (20 x^2-20 x^3+e^3 \left (-20 x+20 x^2\right )\right ) \log \left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )+\left (x^2-x^3+e^3 \left (-x+x^2\right )\right ) \log ^2\left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )} \, dx=\frac {3+x^2}{10+\log \left ((-1+x) x^4 \left (-e^3+x\right )\right )} \] Input:

Integrate[(-15*x + 18*x^2 + 15*x^3 - 14*x^4 + E^3*(12 - 15*x - 16*x^2 + 15 
*x^3) + (2*x^3 - 2*x^4 + E^3*(-2*x^2 + 2*x^3))*Log[-x^5 + x^6 + E^3*(x^4 - 
 x^5)])/(100*x^2 - 100*x^3 + E^3*(-100*x + 100*x^2) + (20*x^2 - 20*x^3 + E 
^3*(-20*x + 20*x^2))*Log[-x^5 + x^6 + E^3*(x^4 - x^5)] + (x^2 - x^3 + E^3* 
(-x + x^2))*Log[-x^5 + x^6 + E^3*(x^4 - x^5)]^2),x]
 

Output:

(3 + x^2)/(10 + Log[(-1 + x)*x^4*(-E^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.

\(\displaystyle \int \frac {-14 x^4+15 x^3+18 x^2+e^3 \left (15 x^3-16 x^2-15 x+12\right )+\left (-2 x^4+2 x^3+e^3 \left (2 x^3-2 x^2\right )\right ) \log \left (x^6-x^5+e^3 \left (x^4-x^5\right )\right )-15 x}{-100 x^3+100 x^2+e^3 \left (100 x^2-100 x\right )+\left (-x^3+x^2+e^3 \left (x^2-x\right )\right ) \log ^2\left (x^6-x^5+e^3 \left (x^4-x^5\right )\right )+\left (-20 x^3+20 x^2+e^3 \left (20 x^2-20 x\right )\right ) \log \left (x^6-x^5+e^3 \left (x^4-x^5\right )\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-2 \left (e^3-x\right ) (x-1) x^2 \log \left ((x-1) x^4 \left (x-e^3\right )\right )-\left (-14 x^3+15 x^2+18 x-15\right ) x-e^3 \left (15 x^3-16 x^2-15 x+12\right )}{(1-x) \left (e^3-x\right ) x \left (\log \left ((x-1) x^4 \left (x-e^3\right )\right )+10\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 x}{\log \left ((x-1) x^4 \left (x-e^3\right )\right )+10}+\frac {\left (-6 x^2+5 \left (1+e^3\right ) x-4 e^3\right ) \left (x^2+3\right )}{(1-x) \left (e^3-x\right ) x \left (\log \left ((x-1) x^4 \left (x-e^3\right )\right )+10\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\left (\left (1+e^3\right ) \int \frac {1}{\left (\log \left ((x-1) x^4 \left (x-e^3\right )\right )+10\right )^2}dx\right )+\left (3+e^6\right ) \int \frac {1}{\left (e^3-x\right ) \left (\log \left ((x-1) x^4 \left (x-e^3\right )\right )+10\right )^2}dx-4 \int \frac {1}{(x-1) \left (\log \left ((x-1) x^4 \left (x-e^3\right )\right )+10\right )^2}dx-12 \int \frac {1}{x \left (\log \left ((x-1) x^4 \left (x-e^3\right )\right )+10\right )^2}dx-6 \int \frac {x}{\left (\log \left ((x-1) x^4 \left (x-e^3\right )\right )+10\right )^2}dx+2 \int \frac {x}{\log \left ((x-1) x^4 \left (x-e^3\right )\right )+10}dx\)

Input:

Int[(-15*x + 18*x^2 + 15*x^3 - 14*x^4 + E^3*(12 - 15*x - 16*x^2 + 15*x^3) 
+ (2*x^3 - 2*x^4 + E^3*(-2*x^2 + 2*x^3))*Log[-x^5 + x^6 + E^3*(x^4 - x^5)] 
)/(100*x^2 - 100*x^3 + E^3*(-100*x + 100*x^2) + (20*x^2 - 20*x^3 + E^3*(-2 
0*x + 20*x^2))*Log[-x^5 + x^6 + E^3*(x^4 - x^5)] + (x^2 - x^3 + E^3*(-x + 
x^2))*Log[-x^5 + x^6 + E^3*(x^4 - x^5)]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 2.38 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22

method	result	size
norman	\(\frac {x^{2}+3}{\ln \left (\left (-x^{5}+x^{4}\right ) {\mathrm e}^{3}+x^{6}-x^{5}\right )+10}\)	\(33\)
risch	\(\frac {x^{2}+3}{\ln \left (\left (-x^{5}+x^{4}\right ) {\mathrm e}^{3}+x^{6}-x^{5}\right )+10}\)	\(33\)
parallelrisch	\(\frac {\left (x^{2} {\mathrm e}^{6}+3 \,{\mathrm e}^{6}\right ) {\mathrm e}^{-6}}{\ln \left (\left (-x^{5}+x^{4}\right ) {\mathrm e}^{3}+x^{6}-x^{5}\right )+10}\)	\(47\)

Input:

int((((2*x^3-2*x^2)*exp(3)-2*x^4+2*x^3)*ln((-x^5+x^4)*exp(3)+x^6-x^5)+(15* 
x^3-16*x^2-15*x+12)*exp(3)-14*x^4+15*x^3+18*x^2-15*x)/(((x^2-x)*exp(3)-x^3 
+x^2)*ln((-x^5+x^4)*exp(3)+x^6-x^5)^2+((20*x^2-20*x)*exp(3)-20*x^3+20*x^2) 
*ln((-x^5+x^4)*exp(3)+x^6-x^5)+(100*x^2-100*x)*exp(3)-100*x^3+100*x^2),x,m 
ethod=_RETURNVERBOSE)
 

Output:

(x^2+3)/(ln((-x^5+x^4)*exp(3)+x^6-x^5)+10)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {-15 x+18 x^2+15 x^3-14 x^4+e^3 \left (12-15 x-16 x^2+15 x^3\right )+\left (2 x^3-2 x^4+e^3 \left (-2 x^2+2 x^3\right )\right ) \log \left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )}{100 x^2-100 x^3+e^3 \left (-100 x+100 x^2\right )+\left (20 x^2-20 x^3+e^3 \left (-20 x+20 x^2\right )\right ) \log \left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )+\left (x^2-x^3+e^3 \left (-x+x^2\right )\right ) \log ^2\left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )} \, dx=\frac {x^{2} + 3}{\log \left (x^{6} - x^{5} - {\left (x^{5} - x^{4}\right )} e^{3}\right ) + 10} \] Input:

integrate((((2*x^3-2*x^2)*exp(3)-2*x^4+2*x^3)*log((-x^5+x^4)*exp(3)+x^6-x^ 
5)+(15*x^3-16*x^2-15*x+12)*exp(3)-14*x^4+15*x^3+18*x^2-15*x)/(((x^2-x)*exp 
(3)-x^3+x^2)*log((-x^5+x^4)*exp(3)+x^6-x^5)^2+((20*x^2-20*x)*exp(3)-20*x^3 
+20*x^2)*log((-x^5+x^4)*exp(3)+x^6-x^5)+(100*x^2-100*x)*exp(3)-100*x^3+100 
*x^2),x, algorithm="fricas")
 

Output:

(x^2 + 3)/(log(x^6 - x^5 - (x^5 - x^4)*e^3) + 10)
 

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {-15 x+18 x^2+15 x^3-14 x^4+e^3 \left (12-15 x-16 x^2+15 x^3\right )+\left (2 x^3-2 x^4+e^3 \left (-2 x^2+2 x^3\right )\right ) \log \left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )}{100 x^2-100 x^3+e^3 \left (-100 x+100 x^2\right )+\left (20 x^2-20 x^3+e^3 \left (-20 x+20 x^2\right )\right ) \log \left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )+\left (x^2-x^3+e^3 \left (-x+x^2\right )\right ) \log ^2\left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )} \, dx=\frac {x^{2} + 3}{\log {\left (x^{6} - x^{5} + \left (- x^{5} + x^{4}\right ) e^{3} \right )} + 10} \] Input:

integrate((((2*x**3-2*x**2)*exp(3)-2*x**4+2*x**3)*ln((-x**5+x**4)*exp(3)+x 
**6-x**5)+(15*x**3-16*x**2-15*x+12)*exp(3)-14*x**4+15*x**3+18*x**2-15*x)/( 
((x**2-x)*exp(3)-x**3+x**2)*ln((-x**5+x**4)*exp(3)+x**6-x**5)**2+((20*x**2 
-20*x)*exp(3)-20*x**3+20*x**2)*ln((-x**5+x**4)*exp(3)+x**6-x**5)+(100*x**2 
-100*x)*exp(3)-100*x**3+100*x**2),x)
 

Output:

(x**2 + 3)/(log(x**6 - x**5 + (-x**5 + x**4)*exp(3)) + 10)
 

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {-15 x+18 x^2+15 x^3-14 x^4+e^3 \left (12-15 x-16 x^2+15 x^3\right )+\left (2 x^3-2 x^4+e^3 \left (-2 x^2+2 x^3\right )\right ) \log \left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )}{100 x^2-100 x^3+e^3 \left (-100 x+100 x^2\right )+\left (20 x^2-20 x^3+e^3 \left (-20 x+20 x^2\right )\right ) \log \left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )+\left (x^2-x^3+e^3 \left (-x+x^2\right )\right ) \log ^2\left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )} \, dx=\frac {x^{2} + 3}{\log \left (x - e^{3}\right ) + \log \left (x - 1\right ) + 4 \, \log \left (x\right ) + 10} \] Input:

integrate((((2*x^3-2*x^2)*exp(3)-2*x^4+2*x^3)*log((-x^5+x^4)*exp(3)+x^6-x^ 
5)+(15*x^3-16*x^2-15*x+12)*exp(3)-14*x^4+15*x^3+18*x^2-15*x)/(((x^2-x)*exp 
(3)-x^3+x^2)*log((-x^5+x^4)*exp(3)+x^6-x^5)^2+((20*x^2-20*x)*exp(3)-20*x^3 
+20*x^2)*log((-x^5+x^4)*exp(3)+x^6-x^5)+(100*x^2-100*x)*exp(3)-100*x^3+100 
*x^2),x, algorithm="maxima")
 

Output:

(x^2 + 3)/(log(x - e^3) + log(x - 1) + 4*log(x) + 10)
 

Giac [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {-15 x+18 x^2+15 x^3-14 x^4+e^3 \left (12-15 x-16 x^2+15 x^3\right )+\left (2 x^3-2 x^4+e^3 \left (-2 x^2+2 x^3\right )\right ) \log \left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )}{100 x^2-100 x^3+e^3 \left (-100 x+100 x^2\right )+\left (20 x^2-20 x^3+e^3 \left (-20 x+20 x^2\right )\right ) \log \left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )+\left (x^2-x^3+e^3 \left (-x+x^2\right )\right ) \log ^2\left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )} \, dx=\frac {x^{2} + 3}{\log \left (x^{6} - x^{5} e^{3} - x^{5} + x^{4} e^{3}\right ) + 10} \] Input:

integrate((((2*x^3-2*x^2)*exp(3)-2*x^4+2*x^3)*log((-x^5+x^4)*exp(3)+x^6-x^ 
5)+(15*x^3-16*x^2-15*x+12)*exp(3)-14*x^4+15*x^3+18*x^2-15*x)/(((x^2-x)*exp 
(3)-x^3+x^2)*log((-x^5+x^4)*exp(3)+x^6-x^5)^2+((20*x^2-20*x)*exp(3)-20*x^3 
+20*x^2)*log((-x^5+x^4)*exp(3)+x^6-x^5)+(100*x^2-100*x)*exp(3)-100*x^3+100 
*x^2),x, algorithm="giac")
 

Output:

(x^2 + 3)/(log(x^6 - x^5*e^3 - x^5 + x^4*e^3) + 10)
 

Mupad [B] (verification not implemented)

Time = 11.13 (sec) , antiderivative size = 213, normalized size of antiderivative = 7.89 \[ \int \frac {-15 x+18 x^2+15 x^3-14 x^4+e^3 \left (12-15 x-16 x^2+15 x^3\right )+\left (2 x^3-2 x^4+e^3 \left (-2 x^2+2 x^3\right )\right ) \log \left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )}{100 x^2-100 x^3+e^3 \left (-100 x+100 x^2\right )+\left (20 x^2-20 x^3+e^3 \left (-20 x+20 x^2\right )\right ) \log \left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )+\left (x^2-x^3+e^3 \left (-x+x^2\right )\right ) \log ^2\left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )} \, dx=\frac {\frac {10\,{\mathrm {e}}^3}{3}-\frac {4\,{\mathrm {e}}^6}{3}+\frac {10\,{\mathrm {e}}^9}{3}+x\,\left (\frac {3\,{\mathrm {e}}^3}{2}+\frac {3\,{\mathrm {e}}^6}{2}-\frac {25\,{\mathrm {e}}^9}{6}-\frac {25}{6}\right )}{108\,x^2+\left (-90\,{\mathrm {e}}^3-90\right )\,x+72\,{\mathrm {e}}^3}+\frac {\frac {15\,x-12\,{\mathrm {e}}^3+15\,x\,{\mathrm {e}}^3+16\,x^2\,{\mathrm {e}}^3-15\,x^3\,{\mathrm {e}}^3-18\,x^2-15\,x^3+14\,x^4}{5\,x-4\,{\mathrm {e}}^3+5\,x\,{\mathrm {e}}^3-6\,x^2}+\frac {2\,x^2\,\ln \left ({\mathrm {e}}^3\,\left (x^4-x^5\right )-x^5+x^6\right )\,\left (x-{\mathrm {e}}^3\right )\,\left (x-1\right )}{5\,x-4\,{\mathrm {e}}^3+5\,x\,{\mathrm {e}}^3-6\,x^2}}{\ln \left ({\mathrm {e}}^3\,\left (x^4-x^5\right )-x^5+x^6\right )+10}+\frac {x^2}{3}-x\,\left (\frac {{\mathrm {e}}^3}{18}+\frac {1}{18}\right ) \] Input:

int((15*x + exp(3)*(15*x + 16*x^2 - 15*x^3 - 12) + log(exp(3)*(x^4 - x^5) 
- x^5 + x^6)*(exp(3)*(2*x^2 - 2*x^3) - 2*x^3 + 2*x^4) - 18*x^2 - 15*x^3 + 
14*x^4)/(log(exp(3)*(x^4 - x^5) - x^5 + x^6)^2*(exp(3)*(x - x^2) - x^2 + x 
^3) + exp(3)*(100*x - 100*x^2) + log(exp(3)*(x^4 - x^5) - x^5 + x^6)*(exp( 
3)*(20*x - 20*x^2) - 20*x^2 + 20*x^3) - 100*x^2 + 100*x^3),x)
 

Output:

((10*exp(3))/3 - (4*exp(6))/3 + (10*exp(9))/3 + x*((3*exp(3))/2 + (3*exp(6 
))/2 - (25*exp(9))/6 - 25/6))/(72*exp(3) + 108*x^2 - x*(90*exp(3) + 90)) + 
 ((15*x - 12*exp(3) + 15*x*exp(3) + 16*x^2*exp(3) - 15*x^3*exp(3) - 18*x^2 
 - 15*x^3 + 14*x^4)/(5*x - 4*exp(3) + 5*x*exp(3) - 6*x^2) + (2*x^2*log(exp 
(3)*(x^4 - x^5) - x^5 + x^6)*(x - exp(3))*(x - 1))/(5*x - 4*exp(3) + 5*x*e 
xp(3) - 6*x^2))/(log(exp(3)*(x^4 - x^5) - x^5 + x^6) + 10) + x^2/3 - x*(ex 
p(3)/18 + 1/18)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.41 \[ \int \frac {-15 x+18 x^2+15 x^3-14 x^4+e^3 \left (12-15 x-16 x^2+15 x^3\right )+\left (2 x^3-2 x^4+e^3 \left (-2 x^2+2 x^3\right )\right ) \log \left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )}{100 x^2-100 x^3+e^3 \left (-100 x+100 x^2\right )+\left (20 x^2-20 x^3+e^3 \left (-20 x+20 x^2\right )\right ) \log \left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )+\left (x^2-x^3+e^3 \left (-x+x^2\right )\right ) \log ^2\left (-x^5+x^6+e^3 \left (x^4-x^5\right )\right )} \, dx=\frac {-3 \,\mathrm {log}\left (-e^{3} x^{5}+e^{3} x^{4}+x^{6}-x^{5}\right )+10 x^{2}}{10 \,\mathrm {log}\left (-e^{3} x^{5}+e^{3} x^{4}+x^{6}-x^{5}\right )+100} \] Input:

int((((2*x^3-2*x^2)*exp(3)-2*x^4+2*x^3)*log((-x^5+x^4)*exp(3)+x^6-x^5)+(15 
*x^3-16*x^2-15*x+12)*exp(3)-14*x^4+15*x^3+18*x^2-15*x)/(((x^2-x)*exp(3)-x^ 
3+x^2)*log((-x^5+x^4)*exp(3)+x^6-x^5)^2+((20*x^2-20*x)*exp(3)-20*x^3+20*x^ 
2)*log((-x^5+x^4)*exp(3)+x^6-x^5)+(100*x^2-100*x)*exp(3)-100*x^3+100*x^2), 
x)
 

Output:

( - 3*log( - e**3*x**5 + e**3*x**4 + x**6 - x**5) + 10*x**2)/(10*(log( - e 
**3*x**5 + e**3*x**4 + x**6 - x**5) + 10))