\(\int \frac {e^{\frac {2 e^{x^2}+x^3 \log (e^{x^2}+2 x)}{\log (e^{x^2}+2 x)}} (-4 e^{x^2}-4 e^{2 x^2} x+(4 e^{2 x^2} x+8 e^{x^2} x^2) \log (e^{x^2}+2 x)+(3 e^{x^2} x^2+6 x^3) \log ^2(e^{x^2}+2 x))}{(e^{x^2}+2 x) \log ^2(e^{x^2}+2 x)} \, dx\) [379]

Optimal result

Integrand size = 140, antiderivative size = 25 \[ \int \frac {e^{\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}} \left (-4 e^{x^2}-4 e^{2 x^2} x+\left (4 e^{2 x^2} x+8 e^{x^2} x^2\right ) \log \left (e^{x^2}+2 x\right )+\left (3 e^{x^2} x^2+6 x^3\right ) \log ^2\left (e^{x^2}+2 x\right )\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx=e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}} \left (-4 e^{x^2}-4 e^{2 x^2} x+\left (4 e^{2 x^2} x+8 e^{x^2} x^2\right ) \log \left (e^{x^2}+2 x\right )+\left (3 e^{x^2} x^2+6 x^3\right ) \log ^2\left (e^{x^2}+2 x\right )\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx=e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} \] Input:

Integrate[(E^((2*E^x^2 + x^3*Log[E^x^2 + 2*x])/Log[E^x^2 + 2*x])*(-4*E^x^2 
 - 4*E^(2*x^2)*x + (4*E^(2*x^2)*x + 8*E^x^2*x^2)*Log[E^x^2 + 2*x] + (3*E^x 
^2*x^2 + 6*x^3)*Log[E^x^2 + 2*x]^2))/((E^x^2 + 2*x)*Log[E^x^2 + 2*x]^2),x]
 

Output:

E^(x^3 + (2*E^x^2)/Log[E^x^2 + 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.

\(\displaystyle \int \frac {\left (-4 e^{x^2}-4 e^{2 x^2} x+\left (8 e^{x^2} x^2+4 e^{2 x^2} x\right ) \log \left (e^{x^2}+2 x\right )+\left (6 x^3+3 e^{x^2} x^2\right ) \log ^2\left (e^{x^2}+2 x\right )\right ) \exp \left (\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -4 \int \frac {\exp \left (\frac {\log \left (2 x+e^{x^2}\right ) x^3+2 e^{x^2}}{\log \left (2 x+e^{x^2}\right )}\right )}{\log ^2\left (2 x+e^{x^2}\right )}dx-4 \int \frac {\exp \left (x^2+\frac {\log \left (2 x+e^{x^2}\right ) x^3+2 e^{x^2}}{\log \left (2 x+e^{x^2}\right )}\right ) x}{\log ^2\left (2 x+e^{x^2}\right )}dx+8 \int \frac {\exp \left (\frac {\log \left (2 x+e^{x^2}\right ) x^3+2 e^{x^2}}{\log \left (2 x+e^{x^2}\right )}\right ) x^2}{\log ^2\left (2 x+e^{x^2}\right )}dx+8 \int \frac {\exp \left (\frac {\log \left (2 x+e^{x^2}\right ) x^3+2 e^{x^2}}{\log \left (2 x+e^{x^2}\right )}\right ) x}{\left (2 x+e^{x^2}\right ) \log ^2\left (2 x+e^{x^2}\right )}dx-16 \int \frac {\exp \left (\frac {\log \left (2 x+e^{x^2}\right ) x^3+2 e^{x^2}}{\log \left (2 x+e^{x^2}\right )}\right ) x^3}{\left (2 x+e^{x^2}\right ) \log ^2\left (2 x+e^{x^2}\right )}dx+3 \int \exp \left (\frac {\log \left (2 x+e^{x^2}\right ) x^3+2 e^{x^2}}{\log \left (2 x+e^{x^2}\right )}\right ) x^2dx+4 \int \frac {\exp \left (x^2+\frac {\log \left (2 x+e^{x^2}\right ) x^3+2 e^{x^2}}{\log \left (2 x+e^{x^2}\right )}\right ) x}{\log \left (2 x+e^{x^2}\right )}dx\)

Input:

Int[(E^((2*E^x^2 + x^3*Log[E^x^2 + 2*x])/Log[E^x^2 + 2*x])*(-4*E^x^2 - 4*E 
^(2*x^2)*x + (4*E^(2*x^2)*x + 8*E^x^2*x^2)*Log[E^x^2 + 2*x] + (3*E^x^2*x^2 
 + 6*x^3)*Log[E^x^2 + 2*x]^2))/((E^x^2 + 2*x)*Log[E^x^2 + 2*x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.62 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36

method	result	size
risch	\({\mathrm e}^{\frac {x^{3} \ln \left ({\mathrm e}^{x^{2}}+2 x \right )+2 \,{\mathrm e}^{x^{2}}}{\ln \left ({\mathrm e}^{x^{2}}+2 x \right )}}\)	\(34\)
parallelrisch	\({\mathrm e}^{\frac {x^{3} \ln \left ({\mathrm e}^{x^{2}}+2 x \right )+2 \,{\mathrm e}^{x^{2}}}{\ln \left ({\mathrm e}^{x^{2}}+2 x \right )}}\)	\(34\)

Input:

int(((3*x^2*exp(x^2)+6*x^3)*ln(exp(x^2)+2*x)^2+(4*x*exp(x^2)^2+8*x^2*exp(x 
^2))*ln(exp(x^2)+2*x)-4*x*exp(x^2)^2-4*exp(x^2))*exp((x^3*ln(exp(x^2)+2*x) 
+2*exp(x^2))/ln(exp(x^2)+2*x))/(exp(x^2)+2*x)/ln(exp(x^2)+2*x)^2,x,method= 
_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {e^{\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}} \left (-4 e^{x^2}-4 e^{2 x^2} x+\left (4 e^{2 x^2} x+8 e^{x^2} x^2\right ) \log \left (e^{x^2}+2 x\right )+\left (3 e^{x^2} x^2+6 x^3\right ) \log ^2\left (e^{x^2}+2 x\right )\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx=e^{\left (\frac {x^{3} \log \left (2 \, x + e^{\left (x^{2}\right )}\right ) + 2 \, e^{\left (x^{2}\right )}}{\log \left (2 \, x + e^{\left (x^{2}\right )}\right )}\right )} \] Input:

integrate(((3*exp(x^2)*x^2+6*x^3)*log(exp(x^2)+2*x)^2+(4*x*exp(x^2)^2+8*ex 
p(x^2)*x^2)*log(exp(x^2)+2*x)-4*x*exp(x^2)^2-4*exp(x^2))*exp((x^3*log(exp( 
x^2)+2*x)+2*exp(x^2))/log(exp(x^2)+2*x))/(exp(x^2)+2*x)/log(exp(x^2)+2*x)^ 
2,x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}} \left (-4 e^{x^2}-4 e^{2 x^2} x+\left (4 e^{2 x^2} x+8 e^{x^2} x^2\right ) \log \left (e^{x^2}+2 x\right )+\left (3 e^{x^2} x^2+6 x^3\right ) \log ^2\left (e^{x^2}+2 x\right )\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx=\text {Timed out} \] Input:

integrate(((3*exp(x**2)*x**2+6*x**3)*ln(exp(x**2)+2*x)**2+(4*x*exp(x**2)** 
2+8*exp(x**2)*x**2)*ln(exp(x**2)+2*x)-4*x*exp(x**2)**2-4*exp(x**2))*exp((x 
**3*ln(exp(x**2)+2*x)+2*exp(x**2))/ln(exp(x**2)+2*x))/(exp(x**2)+2*x)/ln(e 
xp(x**2)+2*x)**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {e^{\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}} \left (-4 e^{x^2}-4 e^{2 x^2} x+\left (4 e^{2 x^2} x+8 e^{x^2} x^2\right ) \log \left (e^{x^2}+2 x\right )+\left (3 e^{x^2} x^2+6 x^3\right ) \log ^2\left (e^{x^2}+2 x\right )\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(((3*exp(x^2)*x^2+6*x^3)*log(exp(x^2)+2*x)^2+(4*x*exp(x^2)^2+8*ex 
p(x^2)*x^2)*log(exp(x^2)+2*x)-4*x*exp(x^2)^2-4*exp(x^2))*exp((x^3*log(exp( 
x^2)+2*x)+2*exp(x^2))/log(exp(x^2)+2*x))/(exp(x^2)+2*x)/log(exp(x^2)+2*x)^ 
2,x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: In function CAR, the value of 
the first argument is  0which is not of the expected type LIST
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {e^{\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}} \left (-4 e^{x^2}-4 e^{2 x^2} x+\left (4 e^{2 x^2} x+8 e^{x^2} x^2\right ) \log \left (e^{x^2}+2 x\right )+\left (3 e^{x^2} x^2+6 x^3\right ) \log ^2\left (e^{x^2}+2 x\right )\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx=e^{\left (x^{3} + \frac {2 \, e^{\left (x^{2}\right )}}{\log \left (2 \, x + e^{\left (x^{2}\right )}\right )}\right )} \] Input:

integrate(((3*exp(x^2)*x^2+6*x^3)*log(exp(x^2)+2*x)^2+(4*x*exp(x^2)^2+8*ex 
p(x^2)*x^2)*log(exp(x^2)+2*x)-4*x*exp(x^2)^2-4*exp(x^2))*exp((x^3*log(exp( 
x^2)+2*x)+2*exp(x^2))/log(exp(x^2)+2*x))/(exp(x^2)+2*x)/log(exp(x^2)+2*x)^ 
2,x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 2.52 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}} \left (-4 e^{x^2}-4 e^{2 x^2} x+\left (4 e^{2 x^2} x+8 e^{x^2} x^2\right ) \log \left (e^{x^2}+2 x\right )+\left (3 e^{x^2} x^2+6 x^3\right ) \log ^2\left (e^{x^2}+2 x\right )\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx={\mathrm {e}}^{x^3}\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^{x^2}}{\ln \left (2\,x+{\mathrm {e}}^{x^2}\right )}} \] Input:

int(-(exp((2*exp(x^2) + x^3*log(2*x + exp(x^2)))/log(2*x + exp(x^2)))*(4*e 
xp(x^2) + 4*x*exp(2*x^2) - log(2*x + exp(x^2))*(4*x*exp(2*x^2) + 8*x^2*exp 
(x^2)) - log(2*x + exp(x^2))^2*(3*x^2*exp(x^2) + 6*x^3)))/(log(2*x + exp(x 
^2))^2*(2*x + exp(x^2))),x)
 

Output:

exp(x^3)*exp((2*exp(x^2))/log(2*x + exp(x^2)))
 

Reduce [F]

\[ \int \frac {e^{\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}} \left (-4 e^{x^2}-4 e^{2 x^2} x+\left (4 e^{2 x^2} x+8 e^{x^2} x^2\right ) \log \left (e^{x^2}+2 x\right )+\left (3 e^{x^2} x^2+6 x^3\right ) \log ^2\left (e^{x^2}+2 x\right )\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx=\int \frac {\left (\left (3 \,{\mathrm e}^{x^{2}} x^{2}+6 x^{3}\right ) \mathrm {log}\left ({\mathrm e}^{x^{2}}+2 x \right )^{2}+\left (4 x \left ({\mathrm e}^{x^{2}}\right )^{2}+8 \,{\mathrm e}^{x^{2}} x^{2}\right ) \mathrm {log}\left ({\mathrm e}^{x^{2}}+2 x \right )-4 x \left ({\mathrm e}^{x^{2}}\right )^{2}-4 \,{\mathrm e}^{x^{2}}\right ) {\mathrm e}^{\frac {x^{3} \mathrm {log}\left ({\mathrm e}^{x^{2}}+2 x \right )+2 \,{\mathrm e}^{x^{2}}}{\mathrm {log}\left ({\mathrm e}^{x^{2}}+2 x \right )}}}{\left ({\mathrm e}^{x^{2}}+2 x \right ) \mathrm {log}\left ({\mathrm e}^{x^{2}}+2 x \right )^{2}}d x \] Input:

int(((3*exp(x^2)*x^2+6*x^3)*log(exp(x^2)+2*x)^2+(4*x*exp(x^2)^2+8*exp(x^2) 
*x^2)*log(exp(x^2)+2*x)-4*x*exp(x^2)^2-4*exp(x^2))*exp((x^3*log(exp(x^2)+2 
*x)+2*exp(x^2))/log(exp(x^2)+2*x))/(exp(x^2)+2*x)/log(exp(x^2)+2*x)^2,x)
 

Output:

int(((3*exp(x^2)*x^2+6*x^3)*log(exp(x^2)+2*x)^2+(4*x*exp(x^2)^2+8*exp(x^2) 
*x^2)*log(exp(x^2)+2*x)-4*x*exp(x^2)^2-4*exp(x^2))*exp((x^3*log(exp(x^2)+2 
*x)+2*exp(x^2))/log(exp(x^2)+2*x))/(exp(x^2)+2*x)/log(exp(x^2)+2*x)^2,x)