\(\int \frac {3 e^{3+x} x^4-6 e^{2+2 x} x^4+12 x \log (x^2)-6 x \log ^2(x^2)}{144 x^4-24 e^{3+x} x^4+e^{6+2 x} x^4+e^{4+4 x} x^4+e^{2+2 x} (24 x^4-2 e^{3+x} x^4)+(-24 x^2+2 e^{3+x} x^2-2 e^{2+2 x} x^2) \log ^2(x^2)+\log ^4(x^2)} \, dx\) [511]

Optimal result

Integrand size = 145, antiderivative size = 31 \[ \int \frac {3 e^{3+x} x^4-6 e^{2+2 x} x^4+12 x \log \left (x^2\right )-6 x \log ^2\left (x^2\right )}{144 x^4-24 e^{3+x} x^4+e^{6+2 x} x^4+e^{4+4 x} x^4+e^{2+2 x} \left (24 x^4-2 e^{3+x} x^4\right )+\left (-24 x^2+2 e^{3+x} x^2-2 e^{2+2 x} x^2\right ) \log ^2\left (x^2\right )+\log ^4\left (x^2\right )} \, dx=\frac {3}{12-e^{3+x}+e^{2+2 x}-\frac {\log ^2\left (x^2\right )}{x^2}} \] Output:

3/(12-ln(x^2)^2/x^2-exp(3+x)+exp(2+2*x))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {3 e^{3+x} x^4-6 e^{2+2 x} x^4+12 x \log \left (x^2\right )-6 x \log ^2\left (x^2\right )}{144 x^4-24 e^{3+x} x^4+e^{6+2 x} x^4+e^{4+4 x} x^4+e^{2+2 x} \left (24 x^4-2 e^{3+x} x^4\right )+\left (-24 x^2+2 e^{3+x} x^2-2 e^{2+2 x} x^2\right ) \log ^2\left (x^2\right )+\log ^4\left (x^2\right )} \, dx=\frac {3 x^2}{\left (12-e^{3+x}+e^{2+2 x}\right ) x^2-\log ^2\left (x^2\right )} \] Input:

Integrate[(3*E^(3 + x)*x^4 - 6*E^(2 + 2*x)*x^4 + 12*x*Log[x^2] - 6*x*Log[x 
^2]^2)/(144*x^4 - 24*E^(3 + x)*x^4 + E^(6 + 2*x)*x^4 + E^(4 + 4*x)*x^4 + E 
^(2 + 2*x)*(24*x^4 - 2*E^(3 + x)*x^4) + (-24*x^2 + 2*E^(3 + x)*x^2 - 2*E^( 
2 + 2*x)*x^2)*Log[x^2]^2 + Log[x^2]^4),x]
 

Output:

(3*x^2)/((12 - E^(3 + x) + E^(2 + 2*x))*x^2 - Log[x^2]^2)
 

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.

\(\displaystyle \int \frac {3 e^{x+3} x^4-6 e^{2 x+2} x^4-6 x \log ^2\left (x^2\right )+12 x \log \left (x^2\right )}{-24 e^{x+3} x^4+e^{2 x+6} x^4+e^{4 x+4} x^4+144 x^4+e^{2 x+2} \left (24 x^4-2 e^{x+3} x^4\right )+\log ^4\left (x^2\right )+\left (2 e^{x+3} x^2-2 e^{2 x+2} x^2-24 x^2\right ) \log ^2\left (x^2\right )} \, dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

\(\displaystyle 3 \int \left (-\frac {2 x^2}{-e^{x+3} x^2+e^{2 x+2} x^2+12 x^2-\log ^2\left (x^2\right )}-\frac {\left (e^{x+3} x^3-24 x^3+2 \log ^2\left (x^2\right ) x+2 \log ^2\left (x^2\right )-4 \log \left (x^2\right )\right ) x}{\left (-e^{x+3} x^2+e^{2 x+2} x^2+12 x^2-\log ^2\left (x^2\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 3 \left (4 \int \frac {x \log \left (x^2\right )}{\left (-e^{x+3} x^2+e^{2 x+2} x^2+12 x^2-\log ^2\left (x^2\right )\right )^2}dx-2 \int \frac {x \log ^2\left (x^2\right )}{\left (-e^{x+3} x^2+e^{2 x+2} x^2+12 x^2-\log ^2\left (x^2\right )\right )^2}dx-2 \int \frac {x^2 \log ^2\left (x^2\right )}{\left (-e^{x+3} x^2+e^{2 x+2} x^2+12 x^2-\log ^2\left (x^2\right )\right )^2}dx-2 \int \frac {x^2}{-e^{x+3} x^2+e^{2 x+2} x^2+12 x^2-\log ^2\left (x^2\right )}dx+24 \int \frac {x^4}{\left (-e^{x+3} x^2+e^{2 x+2} x^2+12 x^2-\log ^2\left (x^2\right )\right )^2}dx-\int \frac {e^{x+3} x^4}{\left (-e^{x+3} x^2+e^{2 x+2} x^2+12 x^2-\log ^2\left (x^2\right )\right )^2}dx\right )\)

Input:

Int[(3*E^(3 + x)*x^4 - 6*E^(2 + 2*x)*x^4 + 12*x*Log[x^2] - 6*x*Log[x^2]^2) 
/(144*x^4 - 24*E^(3 + x)*x^4 + E^(6 + 2*x)*x^4 + E^(4 + 4*x)*x^4 + E^(2 + 
2*x)*(24*x^4 - 2*E^(3 + x)*x^4) + (-24*x^2 + 2*E^(3 + x)*x^2 - 2*E^(2 + 2* 
x)*x^2)*Log[x^2]^2 + Log[x^2]^4),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 3.89 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32

method	result	size
parallelrisch	\(\frac {3 x^{2}}{x^{2} {\mathrm e}^{2+2 x}-x^{2} {\mathrm e}^{3+x}+12 x^{2}-\ln \left (x^{2}\right )^{2}}\)	\(41\)
risch	\(-\frac {12 x^{2}}{-\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6}+4 \pi ^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5}-6 \pi ^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4}+4 \pi ^{2} \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3}-\pi ^{2} \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2}-8 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+16 i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-8 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+4 x^{2} {\mathrm e}^{3+x}-4 x^{2} {\mathrm e}^{2+2 x}-48 x^{2}+16 \ln \left (x \right )^{2}}\)	\(191\)

Input:

int((-6*x*ln(x^2)^2+12*x*ln(x^2)-6*x^4*exp(2+2*x)+3*x^4*exp(3+x))/(ln(x^2) 
^4+(-2*x^2*exp(2+2*x)+2*x^2*exp(3+x)-24*x^2)*ln(x^2)^2+x^4*exp(2+2*x)^2+(- 
2*x^4*exp(3+x)+24*x^4)*exp(2+2*x)+x^4*exp(3+x)^2-24*x^4*exp(3+x)+144*x^4), 
x,method=_RETURNVERBOSE)
 

Output:

3*x^2/(x^2*exp(2+2*x)-x^2*exp(3+x)+12*x^2-ln(x^2)^2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {3 e^{3+x} x^4-6 e^{2+2 x} x^4+12 x \log \left (x^2\right )-6 x \log ^2\left (x^2\right )}{144 x^4-24 e^{3+x} x^4+e^{6+2 x} x^4+e^{4+4 x} x^4+e^{2+2 x} \left (24 x^4-2 e^{3+x} x^4\right )+\left (-24 x^2+2 e^{3+x} x^2-2 e^{2+2 x} x^2\right ) \log ^2\left (x^2\right )+\log ^4\left (x^2\right )} \, dx=\frac {3 \, x^{2} e^{4}}{12 \, x^{2} e^{4} + x^{2} e^{\left (2 \, x + 6\right )} - x^{2} e^{\left (x + 7\right )} - e^{4} \log \left (x^{2}\right )^{2}} \] Input:

integrate((-6*x*log(x^2)^2+12*x*log(x^2)-6*x^4*exp(2+2*x)+3*x^4*exp(3+x))/ 
(log(x^2)^4+(-2*x^2*exp(2+2*x)+2*x^2*exp(3+x)-24*x^2)*log(x^2)^2+x^4*exp(2 
+2*x)^2+(-2*x^4*exp(3+x)+24*x^4)*exp(2+2*x)+x^4*exp(3+x)^2-24*x^4*exp(3+x) 
+144*x^4),x, algorithm="fricas")
 

Output:

3*x^2*e^4/(12*x^2*e^4 + x^2*e^(2*x + 6) - x^2*e^(x + 7) - e^4*log(x^2)^2)
 

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.55 \[ \int \frac {3 e^{3+x} x^4-6 e^{2+2 x} x^4+12 x \log \left (x^2\right )-6 x \log ^2\left (x^2\right )}{144 x^4-24 e^{3+x} x^4+e^{6+2 x} x^4+e^{4+4 x} x^4+e^{2+2 x} \left (24 x^4-2 e^{3+x} x^4\right )+\left (-24 x^2+2 e^{3+x} x^2-2 e^{2+2 x} x^2\right ) \log ^2\left (x^2\right )+\log ^4\left (x^2\right )} \, dx=\frac {3 x^{2} e^{4}}{- x^{2} e^{4} e^{x + 3} + x^{2} e^{2 x + 6} + 12 x^{2} e^{4} - e^{4} \log {\left (x^{2} \right )}^{2}} \] Input:

integrate((-6*x*ln(x**2)**2+12*x*ln(x**2)-6*x**4*exp(2+2*x)+3*x**4*exp(3+x 
))/(ln(x**2)**4+(-2*x**2*exp(2+2*x)+2*x**2*exp(3+x)-24*x**2)*ln(x**2)**2+x 
**4*exp(2+2*x)**2+(-2*x**4*exp(3+x)+24*x**4)*exp(2+2*x)+x**4*exp(3+x)**2-2 
4*x**4*exp(3+x)+144*x**4),x)
 

Output:

3*x**2*exp(4)/(-x**2*exp(4)*exp(x + 3) + x**2*exp(2*x + 6) + 12*x**2*exp(4 
) - exp(4)*log(x**2)**2)
 

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23 \[ \int \frac {3 e^{3+x} x^4-6 e^{2+2 x} x^4+12 x \log \left (x^2\right )-6 x \log ^2\left (x^2\right )}{144 x^4-24 e^{3+x} x^4+e^{6+2 x} x^4+e^{4+4 x} x^4+e^{2+2 x} \left (24 x^4-2 e^{3+x} x^4\right )+\left (-24 x^2+2 e^{3+x} x^2-2 e^{2+2 x} x^2\right ) \log ^2\left (x^2\right )+\log ^4\left (x^2\right )} \, dx=\frac {3 \, x^{2}}{x^{2} e^{\left (2 \, x + 2\right )} - x^{2} e^{\left (x + 3\right )} + 12 \, x^{2} - 4 \, \log \left (x\right )^{2}} \] Input:

integrate((-6*x*log(x^2)^2+12*x*log(x^2)-6*x^4*exp(2+2*x)+3*x^4*exp(3+x))/ 
(log(x^2)^4+(-2*x^2*exp(2+2*x)+2*x^2*exp(3+x)-24*x^2)*log(x^2)^2+x^4*exp(2 
+2*x)^2+(-2*x^4*exp(3+x)+24*x^4)*exp(2+2*x)+x^4*exp(3+x)^2-24*x^4*exp(3+x) 
+144*x^4),x, algorithm="maxima")
 

Output:

3*x^2/(x^2*e^(2*x + 2) - x^2*e^(x + 3) + 12*x^2 - 4*log(x)^2)
 

Giac [F(-1)]

Timed out. \[ \int \frac {3 e^{3+x} x^4-6 e^{2+2 x} x^4+12 x \log \left (x^2\right )-6 x \log ^2\left (x^2\right )}{144 x^4-24 e^{3+x} x^4+e^{6+2 x} x^4+e^{4+4 x} x^4+e^{2+2 x} \left (24 x^4-2 e^{3+x} x^4\right )+\left (-24 x^2+2 e^{3+x} x^2-2 e^{2+2 x} x^2\right ) \log ^2\left (x^2\right )+\log ^4\left (x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((-6*x*log(x^2)^2+12*x*log(x^2)-6*x^4*exp(2+2*x)+3*x^4*exp(3+x))/ 
(log(x^2)^4+(-2*x^2*exp(2+2*x)+2*x^2*exp(3+x)-24*x^2)*log(x^2)^2+x^4*exp(2 
+2*x)^2+(-2*x^4*exp(3+x)+24*x^4)*exp(2+2*x)+x^4*exp(3+x)^2-24*x^4*exp(3+x) 
+144*x^4),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {3 e^{3+x} x^4-6 e^{2+2 x} x^4+12 x \log \left (x^2\right )-6 x \log ^2\left (x^2\right )}{144 x^4-24 e^{3+x} x^4+e^{6+2 x} x^4+e^{4+4 x} x^4+e^{2+2 x} \left (24 x^4-2 e^{3+x} x^4\right )+\left (-24 x^2+2 e^{3+x} x^2-2 e^{2+2 x} x^2\right ) \log ^2\left (x^2\right )+\log ^4\left (x^2\right )} \, dx=\int \frac {12\,x\,\ln \left (x^2\right )+3\,x^4\,{\mathrm {e}}^{x+3}-6\,x\,{\ln \left (x^2\right )}^2-6\,x^4\,{\mathrm {e}}^{2\,x+2}}{{\ln \left (x^2\right )}^4-24\,x^4\,{\mathrm {e}}^{x+3}-{\mathrm {e}}^{2\,x+2}\,\left (2\,x^4\,{\mathrm {e}}^{x+3}-24\,x^4\right )+x^4\,{\mathrm {e}}^{2\,x+6}+x^4\,{\mathrm {e}}^{4\,x+4}+144\,x^4-{\ln \left (x^2\right )}^2\,\left (2\,x^2\,{\mathrm {e}}^{2\,x+2}-2\,x^2\,{\mathrm {e}}^{x+3}+24\,x^2\right )} \,d x \] Input:

int((12*x*log(x^2) + 3*x^4*exp(x + 3) - 6*x*log(x^2)^2 - 6*x^4*exp(2*x + 2 
))/(log(x^2)^4 - 24*x^4*exp(x + 3) - exp(2*x + 2)*(2*x^4*exp(x + 3) - 24*x 
^4) + x^4*exp(2*x + 6) + x^4*exp(4*x + 4) + 144*x^4 - log(x^2)^2*(2*x^2*ex 
p(2*x + 2) - 2*x^2*exp(x + 3) + 24*x^2)),x)
 

Output:

int((12*x*log(x^2) + 3*x^4*exp(x + 3) - 6*x*log(x^2)^2 - 6*x^4*exp(2*x + 2 
))/(log(x^2)^4 - 24*x^4*exp(x + 3) - exp(2*x + 2)*(2*x^4*exp(x + 3) - 24*x 
^4) + x^4*exp(2*x + 6) + x^4*exp(4*x + 4) + 144*x^4 - log(x^2)^2*(2*x^2*ex 
p(2*x + 2) - 2*x^2*exp(x + 3) + 24*x^2)), x)
 

Reduce [F]

\[ \int \frac {3 e^{3+x} x^4-6 e^{2+2 x} x^4+12 x \log \left (x^2\right )-6 x \log ^2\left (x^2\right )}{144 x^4-24 e^{3+x} x^4+e^{6+2 x} x^4+e^{4+4 x} x^4+e^{2+2 x} \left (24 x^4-2 e^{3+x} x^4\right )+\left (-24 x^2+2 e^{3+x} x^2-2 e^{2+2 x} x^2\right ) \log ^2\left (x^2\right )+\log ^4\left (x^2\right )} \, dx=-6 \left (\int \frac {\mathrm {log}\left (x^{2}\right )^{2} x}{e^{4 x} e^{4} x^{4}-2 e^{3 x} e^{5} x^{4}-2 e^{2 x} \mathrm {log}\left (x^{2}\right )^{2} e^{2} x^{2}+e^{2 x} e^{6} x^{4}+24 e^{2 x} e^{2} x^{4}+2 e^{x} \mathrm {log}\left (x^{2}\right )^{2} e^{3} x^{2}-24 e^{x} e^{3} x^{4}+\mathrm {log}\left (x^{2}\right )^{4}-24 \mathrm {log}\left (x^{2}\right )^{2} x^{2}+144 x^{4}}d x \right )-6 \left (\int \frac {e^{2 x} x^{4}}{e^{4 x} e^{4} x^{4}-2 e^{3 x} e^{5} x^{4}-2 e^{2 x} \mathrm {log}\left (x^{2}\right )^{2} e^{2} x^{2}+e^{2 x} e^{6} x^{4}+24 e^{2 x} e^{2} x^{4}+2 e^{x} \mathrm {log}\left (x^{2}\right )^{2} e^{3} x^{2}-24 e^{x} e^{3} x^{4}+\mathrm {log}\left (x^{2}\right )^{4}-24 \mathrm {log}\left (x^{2}\right )^{2} x^{2}+144 x^{4}}d x \right ) e^{2}+3 \left (\int \frac {e^{x} x^{4}}{e^{4 x} e^{4} x^{4}-2 e^{3 x} e^{5} x^{4}-2 e^{2 x} \mathrm {log}\left (x^{2}\right )^{2} e^{2} x^{2}+e^{2 x} e^{6} x^{4}+24 e^{2 x} e^{2} x^{4}+2 e^{x} \mathrm {log}\left (x^{2}\right )^{2} e^{3} x^{2}-24 e^{x} e^{3} x^{4}+\mathrm {log}\left (x^{2}\right )^{4}-24 \mathrm {log}\left (x^{2}\right )^{2} x^{2}+144 x^{4}}d x \right ) e^{3}+12 \left (\int \frac {\mathrm {log}\left (x^{2}\right ) x}{e^{4 x} e^{4} x^{4}-2 e^{3 x} e^{5} x^{4}-2 e^{2 x} \mathrm {log}\left (x^{2}\right )^{2} e^{2} x^{2}+e^{2 x} e^{6} x^{4}+24 e^{2 x} e^{2} x^{4}+2 e^{x} \mathrm {log}\left (x^{2}\right )^{2} e^{3} x^{2}-24 e^{x} e^{3} x^{4}+\mathrm {log}\left (x^{2}\right )^{4}-24 \mathrm {log}\left (x^{2}\right )^{2} x^{2}+144 x^{4}}d x \right ) \] Input:

int((-6*x*log(x^2)^2+12*x*log(x^2)-6*x^4*exp(2+2*x)+3*x^4*exp(3+x))/(log(x 
^2)^4+(-2*x^2*exp(2+2*x)+2*x^2*exp(3+x)-24*x^2)*log(x^2)^2+x^4*exp(2+2*x)^ 
2+(-2*x^4*exp(3+x)+24*x^4)*exp(2+2*x)+x^4*exp(3+x)^2-24*x^4*exp(3+x)+144*x 
^4),x)
 

Output:

3*( - 2*int((log(x**2)**2*x)/(e**(4*x)*e**4*x**4 - 2*e**(3*x)*e**5*x**4 - 
2*e**(2*x)*log(x**2)**2*e**2*x**2 + e**(2*x)*e**6*x**4 + 24*e**(2*x)*e**2* 
x**4 + 2*e**x*log(x**2)**2*e**3*x**2 - 24*e**x*e**3*x**4 + log(x**2)**4 - 
24*log(x**2)**2*x**2 + 144*x**4),x) - 2*int((e**(2*x)*x**4)/(e**(4*x)*e**4 
*x**4 - 2*e**(3*x)*e**5*x**4 - 2*e**(2*x)*log(x**2)**2*e**2*x**2 + e**(2*x 
)*e**6*x**4 + 24*e**(2*x)*e**2*x**4 + 2*e**x*log(x**2)**2*e**3*x**2 - 24*e 
**x*e**3*x**4 + log(x**2)**4 - 24*log(x**2)**2*x**2 + 144*x**4),x)*e**2 + 
int((e**x*x**4)/(e**(4*x)*e**4*x**4 - 2*e**(3*x)*e**5*x**4 - 2*e**(2*x)*lo 
g(x**2)**2*e**2*x**2 + e**(2*x)*e**6*x**4 + 24*e**(2*x)*e**2*x**4 + 2*e**x 
*log(x**2)**2*e**3*x**2 - 24*e**x*e**3*x**4 + log(x**2)**4 - 24*log(x**2)* 
*2*x**2 + 144*x**4),x)*e**3 + 4*int((log(x**2)*x)/(e**(4*x)*e**4*x**4 - 2* 
e**(3*x)*e**5*x**4 - 2*e**(2*x)*log(x**2)**2*e**2*x**2 + e**(2*x)*e**6*x** 
4 + 24*e**(2*x)*e**2*x**4 + 2*e**x*log(x**2)**2*e**3*x**2 - 24*e**x*e**3*x 
**4 + log(x**2)**4 - 24*log(x**2)**2*x**2 + 144*x**4),x))