\(\int \frac {e^{\frac {3 x-9 e x-3 x^3+(3 e x+x^3) \log (x^2)}{-9 e+3 e \log (x^2)}} (-20 x+36 x^3+e (36+36 x)+(e (-24-24 x)+4 x-24 x^3) \log (x^2)+(4 x^3+e (4+4 x)) \log ^2(x^2))}{9 e-6 e \log (x^2)+e \log ^2(x^2)} \, dx\) [1107]

Optimal result

Integrand size = 121, antiderivative size = 32 \[ \int \frac {e^{\frac {3 x-9 e x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{-9 e+3 e \log \left (x^2\right )}} \left (-20 x+36 x^3+e (36+36 x)+\left (e (-24-24 x)+4 x-24 x^3\right ) \log \left (x^2\right )+\left (4 x^3+e (4+4 x)\right ) \log ^2\left (x^2\right )\right )}{9 e-6 e \log \left (x^2\right )+e \log ^2\left (x^2\right )} \, dx=4 e^{x-\frac {-\frac {x^3}{3}+\frac {x}{3-\log \left (x^2\right )}}{e}} x \] Output:

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \[ \int \frac {e^{\frac {3 x-9 e x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{-9 e+3 e \log \left (x^2\right )}} \left (-20 x+36 x^3+e (36+36 x)+\left (e (-24-24 x)+4 x-24 x^3\right ) \log \left (x^2\right )+\left (4 x^3+e (4+4 x)\right ) \log ^2\left (x^2\right )\right )}{9 e-6 e \log \left (x^2\right )+e \log ^2\left (x^2\right )} \, dx=4 e^{\frac {x \left (-3 \left (-1+3 e+x^2\right )+\left (3 e+x^2\right ) \log \left (x^2\right )\right )}{3 e \left (-3+\log \left (x^2\right )\right )}} x \] Input:

Integrate[(E^((3*x - 9*E*x - 3*x^3 + (3*E*x + x^3)*Log[x^2])/(-9*E + 3*E*L 
og[x^2]))*(-20*x + 36*x^3 + E*(36 + 36*x) + (E*(-24 - 24*x) + 4*x - 24*x^3 
)*Log[x^2] + (4*x^3 + E*(4 + 4*x))*Log[x^2]^2))/(9*E - 6*E*Log[x^2] + E*Lo 
g[x^2]^2),x]
 

Output:

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

\(\displaystyle \int \frac {\left (36 x^3+\left (4 x^3+e (4 x+4)\right ) \log ^2\left (x^2\right )+\left (-24 x^3+4 x+e (-24 x-24)\right ) \log \left (x^2\right )-20 x+e (36 x+36)\right ) \exp \left (\frac {-3 x^3+\left (x^3+3 e x\right ) \log \left (x^2\right )-9 e x+3 x}{3 e \log \left (x^2\right )-9 e}\right )}{e \log ^2\left (x^2\right )-6 e \log \left (x^2\right )+9 e} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (36 x^3+\left (4 x^3+e (4 x+4)\right ) \log ^2\left (x^2\right )+\left (-24 x^3+4 x+e (-24 x-24)\right ) \log \left (x^2\right )-20 x+e (36 x+36)\right ) \exp \left (\frac {-3 x^3+\left (x^3+3 e x\right ) \log \left (x^2\right )+3 (1-3 e) x}{3 e \left (\log \left (x^2\right )-3\right )}-1\right )}{\left (3-\log \left (x^2\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 x \exp \left (\frac {-3 x^3+\left (x^3+3 e x\right ) \log \left (x^2\right )+3 (1-3 e) x}{3 e \left (\log \left (x^2\right )-3\right )}-1\right )}{\log \left (x^2\right )-3}-\frac {8 x \exp \left (\frac {-3 x^3+\left (x^3+3 e x\right ) \log \left (x^2\right )+3 (1-3 e) x}{3 e \left (\log \left (x^2\right )-3\right )}-1\right )}{\left (\log \left (x^2\right )-3\right )^2}+4 \left (x^3+e x+e\right ) \exp \left (\frac {-3 x^3+\left (x^3+3 e x\right ) \log \left (x^2\right )+3 (1-3 e) x}{3 e \left (\log \left (x^2\right )-3\right )}-1\right )\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 4 \int \exp \left (\frac {-3 x^3+3 (1-3 e) x+\left (x^3+3 e x\right ) \log \left (x^2\right )}{3 e \left (\log \left (x^2\right )-3\right )}\right )dx+4 \int \exp \left (\frac {-3 x^3+3 (1-3 e) x+\left (x^3+3 e x\right ) \log \left (x^2\right )}{3 e \left (\log \left (x^2\right )-3\right )}\right ) xdx+4 \int \exp \left (\frac {-3 x^3+3 (1-3 e) x+\left (x^3+3 e x\right ) \log \left (x^2\right )}{3 e \left (\log \left (x^2\right )-3\right )}-1\right ) x^3dx-8 \int \frac {\exp \left (\frac {-3 x^3+3 (1-3 e) x+\left (x^3+3 e x\right ) \log \left (x^2\right )}{3 e \left (\log \left (x^2\right )-3\right )}-1\right ) x}{\left (\log \left (x^2\right )-3\right )^2}dx+4 \int \frac {\exp \left (\frac {-3 x^3+3 (1-3 e) x+\left (x^3+3 e x\right ) \log \left (x^2\right )}{3 e \left (\log \left (x^2\right )-3\right )}-1\right ) x}{\log \left (x^2\right )-3}dx\)

Input:

Int[(E^((3*x - 9*E*x - 3*x^3 + (3*E*x + x^3)*Log[x^2])/(-9*E + 3*E*Log[x^2 
]))*(-20*x + 36*x^3 + E*(36 + 36*x) + (E*(-24 - 24*x) + 4*x - 24*x^3)*Log[ 
x^2] + (4*x^3 + E*(4 + 4*x))*Log[x^2]^2))/(9*E - 6*E*Log[x^2] + E*Log[x^2] 
^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 3.85 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.41

method	result	size
risch	\(4 x \,{\mathrm e}^{\frac {x \left (x^{2} \ln \left (x^{2}\right )+3 \,{\mathrm e} \ln \left (x^{2}\right )-3 x^{2}-9 \,{\mathrm e}+3\right ) {\mathrm e}^{-1}}{3 \ln \left (x^{2}\right )-9}}\)	\(45\)
parallelrisch	\(4 \,{\mathrm e}^{\frac {\left (\left (3 x \,{\mathrm e}+x^{3}\right ) \ln \left (x^{2}\right )-9 x \,{\mathrm e}-3 x^{3}+3 x \right ) {\mathrm e}^{-1}}{3 \ln \left (x^{2}\right )-9}} x\)	\(47\)

Input:

int((((4+4*x)*exp(1)+4*x^3)*ln(x^2)^2+((-24*x-24)*exp(1)-24*x^3+4*x)*ln(x^ 
2)+(36*x+36)*exp(1)+36*x^3-20*x)*exp(((3*x*exp(1)+x^3)*ln(x^2)-9*x*exp(1)- 
3*x^3+3*x)/(3*exp(1)*ln(x^2)-9*exp(1)))/(exp(1)*ln(x^2)^2-6*exp(1)*ln(x^2) 
+9*exp(1)),x,method=_RETURNVERBOSE)
 

Output:

4*x*exp(1/3*x*(x^2*ln(x^2)+3*exp(1)*ln(x^2)-3*x^2-9*exp(1)+3)*exp(-1)/(ln( 
x^2)-3))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.53 \[ \int \frac {e^{\frac {3 x-9 e x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{-9 e+3 e \log \left (x^2\right )}} \left (-20 x+36 x^3+e (36+36 x)+\left (e (-24-24 x)+4 x-24 x^3\right ) \log \left (x^2\right )+\left (4 x^3+e (4+4 x)\right ) \log ^2\left (x^2\right )\right )}{9 e-6 e \log \left (x^2\right )+e \log ^2\left (x^2\right )} \, dx=4 \, x e^{\left (-\frac {3 \, x^{3} + 9 \, x e - {\left (x^{3} + 3 \, x e\right )} \log \left (x^{2}\right ) - 3 \, x}{3 \, {\left (e \log \left (x^{2}\right ) - 3 \, e\right )}}\right )} \] Input:

integrate((((4+4*x)*exp(1)+4*x^3)*log(x^2)^2+((-24*x-24)*exp(1)-24*x^3+4*x 
)*log(x^2)+(36*x+36)*exp(1)+36*x^3-20*x)*exp(((3*exp(1)*x+x^3)*log(x^2)-9* 
exp(1)*x-3*x^3+3*x)/(3*exp(1)*log(x^2)-9*exp(1)))/(exp(1)*log(x^2)^2-6*exp 
(1)*log(x^2)+9*exp(1)),x, algorithm="fricas")
 

Output:

4*x*e^(-1/3*(3*x^3 + 9*x*e - (x^3 + 3*x*e)*log(x^2) - 3*x)/(e*log(x^2) - 3 
*e))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (22) = 44\).

Time = 4.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.53 \[ \int \frac {e^{\frac {3 x-9 e x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{-9 e+3 e \log \left (x^2\right )}} \left (-20 x+36 x^3+e (36+36 x)+\left (e (-24-24 x)+4 x-24 x^3\right ) \log \left (x^2\right )+\left (4 x^3+e (4+4 x)\right ) \log ^2\left (x^2\right )\right )}{9 e-6 e \log \left (x^2\right )+e \log ^2\left (x^2\right )} \, dx=4 x e^{\frac {- 3 x^{3} - 9 e x + 3 x + \left (x^{3} + 3 e x\right ) \log {\left (x^{2} \right )}}{3 e \log {\left (x^{2} \right )} - 9 e}} \] Input:

integrate((((4+4*x)*exp(1)+4*x**3)*ln(x**2)**2+((-24*x-24)*exp(1)-24*x**3+ 
4*x)*ln(x**2)+(36*x+36)*exp(1)+36*x**3-20*x)*exp(((3*exp(1)*x+x**3)*ln(x** 
2)-9*exp(1)*x-3*x**3+3*x)/(3*exp(1)*ln(x**2)-9*exp(1)))/(exp(1)*ln(x**2)** 
2-6*exp(1)*ln(x**2)+9*exp(1)),x)
 

Output:

4*x*exp((-3*x**3 - 9*E*x + 3*x + (x**3 + 3*E*x)*log(x**2))/(3*E*log(x**2) 
- 9*E))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (25) = 50\).

Time = 0.22 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.56 \[ \int \frac {e^{\frac {3 x-9 e x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{-9 e+3 e \log \left (x^2\right )}} \left (-20 x+36 x^3+e (36+36 x)+\left (e (-24-24 x)+4 x-24 x^3\right ) \log \left (x^2\right )+\left (4 x^3+e (4+4 x)\right ) \log ^2\left (x^2\right )\right )}{9 e-6 e \log \left (x^2\right )+e \log ^2\left (x^2\right )} \, dx=4 \, x e^{\left (\frac {2 \, x^{3} \log \left (x\right )}{3 \, {\left (2 \, e \log \left (x\right ) - 3 \, e\right )}} - \frac {x^{3}}{2 \, e \log \left (x\right ) - 3 \, e} + \frac {2 \, x \log \left (x\right )}{2 \, \log \left (x\right ) - 3} + \frac {x}{2 \, e \log \left (x\right ) - 3 \, e} - \frac {3 \, x}{2 \, \log \left (x\right ) - 3}\right )} \] Input:

integrate((((4+4*x)*exp(1)+4*x^3)*log(x^2)^2+((-24*x-24)*exp(1)-24*x^3+4*x 
)*log(x^2)+(36*x+36)*exp(1)+36*x^3-20*x)*exp(((3*exp(1)*x+x^3)*log(x^2)-9* 
exp(1)*x-3*x^3+3*x)/(3*exp(1)*log(x^2)-9*exp(1)))/(exp(1)*log(x^2)^2-6*exp 
(1)*log(x^2)+9*exp(1)),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (25) = 50\).

Time = 1.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.59 \[ \int \frac {e^{\frac {3 x-9 e x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{-9 e+3 e \log \left (x^2\right )}} \left (-20 x+36 x^3+e (36+36 x)+\left (e (-24-24 x)+4 x-24 x^3\right ) \log \left (x^2\right )+\left (4 x^3+e (4+4 x)\right ) \log ^2\left (x^2\right )\right )}{9 e-6 e \log \left (x^2\right )+e \log ^2\left (x^2\right )} \, dx=4 \, x e^{\left (\frac {x^{3} \log \left (x^{2}\right ) - 3 \, x^{3} + 3 \, x e \log \left (x^{2}\right ) - 9 \, x e + 3 \, x}{3 \, {\left (e \log \left (x^{2}\right ) - 3 \, e\right )}}\right )} \] Input:

integrate((((4+4*x)*exp(1)+4*x^3)*log(x^2)^2+((-24*x-24)*exp(1)-24*x^3+4*x 
)*log(x^2)+(36*x+36)*exp(1)+36*x^3-20*x)*exp(((3*exp(1)*x+x^3)*log(x^2)-9* 
exp(1)*x-3*x^3+3*x)/(3*exp(1)*log(x^2)-9*exp(1)))/(exp(1)*log(x^2)^2-6*exp 
(1)*log(x^2)+9*exp(1)),x, algorithm="giac")
 

Output:

4*x*e^(1/3*(x^3*log(x^2) - 3*x^3 + 3*x*e*log(x^2) - 9*x*e + 3*x)/(e*log(x^ 
2) - 3*e))
 

Mupad [B] (verification not implemented)

Time = 7.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.97 \[ \int \frac {e^{\frac {3 x-9 e x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{-9 e+3 e \log \left (x^2\right )}} \left (-20 x+36 x^3+e (36+36 x)+\left (e (-24-24 x)+4 x-24 x^3\right ) \log \left (x^2\right )+\left (4 x^3+e (4+4 x)\right ) \log ^2\left (x^2\right )\right )}{9 e-6 e \log \left (x^2\right )+e \log ^2\left (x^2\right )} \, dx=\frac {4\,x\,{\mathrm {e}}^{\frac {3\,x^3}{9\,\mathrm {e}-3\,\ln \left (x^2\right )\,\mathrm {e}}}\,{\mathrm {e}}^{\frac {9\,x\,\mathrm {e}}{9\,\mathrm {e}-3\,\ln \left (x^2\right )\,\mathrm {e}}}\,{\mathrm {e}}^{-\frac {3\,x}{9\,\mathrm {e}-3\,\ln \left (x^2\right )\,\mathrm {e}}}}{{\left (x^2\right )}^{\frac {x^3+3\,\mathrm {e}\,x}{3\,\left (3\,\mathrm {e}-\ln \left (x^2\right )\,\mathrm {e}\right )}}} \] Input:

int((exp(-(3*x + log(x^2)*(3*x*exp(1) + x^3) - 9*x*exp(1) - 3*x^3)/(9*exp( 
1) - 3*log(x^2)*exp(1)))*(log(x^2)^2*(4*x^3 + exp(1)*(4*x + 4)) - log(x^2) 
*(24*x^3 - 4*x + exp(1)*(24*x + 24)) - 20*x + 36*x^3 + exp(1)*(36*x + 36)) 
)/(9*exp(1) - 6*log(x^2)*exp(1) + log(x^2)^2*exp(1)),x)
 

Output:

(4*x*exp((3*x^3)/(9*exp(1) - 3*log(x^2)*exp(1)))*exp((9*x*exp(1))/(9*exp(1 
) - 3*log(x^2)*exp(1)))*exp(-(3*x)/(9*exp(1) - 3*log(x^2)*exp(1))))/(x^2)^ 
((3*x*exp(1) + x^3)/(3*(3*exp(1) - log(x^2)*exp(1))))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.97 \[ \int \frac {e^{\frac {3 x-9 e x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{-9 e+3 e \log \left (x^2\right )}} \left (-20 x+36 x^3+e (36+36 x)+\left (e (-24-24 x)+4 x-24 x^3\right ) \log \left (x^2\right )+\left (4 x^3+e (4+4 x)\right ) \log ^2\left (x^2\right )\right )}{9 e-6 e \log \left (x^2\right )+e \log ^2\left (x^2\right )} \, dx=\frac {4 e^{\frac {3 \,\mathrm {log}\left (x^{2}\right ) e x +\mathrm {log}\left (x^{2}\right ) x^{3}-9 e x +3 x}{3 \,\mathrm {log}\left (x^{2}\right ) e -9 e}} x}{e^{\frac {x^{3}}{\mathrm {log}\left (x^{2}\right ) e -3 e}}} \] Input:

int((((4+4*x)*exp(1)+4*x^3)*log(x^2)^2+((-24*x-24)*exp(1)-24*x^3+4*x)*log( 
x^2)+(36*x+36)*exp(1)+36*x^3-20*x)*exp(((3*exp(1)*x+x^3)*log(x^2)-9*exp(1) 
*x-3*x^3+3*x)/(3*exp(1)*log(x^2)-9*exp(1)))/(exp(1)*log(x^2)^2-6*exp(1)*lo 
g(x^2)+9*exp(1)),x)
 

Output:

(4*e**((3*log(x**2)*e*x + log(x**2)*x**3 - 9*e*x + 3*x)/(3*log(x**2)*e - 9 
*e))*x)/e**(x**3/(log(x**2)*e - 3*e))