\(\int \frac {e^{\frac {2 e^{x^2}+x^3 \log (e^{x^2}+2 x)}{\log (e^{x^2}+2 x)}} (-4 e^{x^2}-4 e^{2 x^2} x+(4 e^{2 x^2} x+8 e^{x^2} x^2) \log (e^{x^2}+2 x)+(3 e^{x^2} x^2+6 x^3) \log ^2(e^{x^2}+2 x))}{(e^{x^2}+2 x) \log ^2(e^{x^2}+2 x)} \, dx\) [3379]

Optimal result

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

[Out]

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

Rubi [F]

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

[In]

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]

[Out]

3*Defer[Int][E^(x^3 + (2*E^x^2)/Log[E^x^2 + 2*x])*x^2, x] - 4*Defer[Int][E^(x^3 + (2*E^x^2)/Log[E^x^2 + 2*x])/
Log[E^x^2 + 2*x]^2, x] - 4*Defer[Int][(E^(x^2 + x^3 + (2*E^x^2)/Log[E^x^2 + 2*x])*x)/Log[E^x^2 + 2*x]^2, x] +
8*Defer[Int][(E^(x^3 + (2*E^x^2)/Log[E^x^2 + 2*x])*x^2)/Log[E^x^2 + 2*x]^2, x] + 8*Defer[Int][(E^(x^3 + (2*E^x
^2)/Log[E^x^2 + 2*x])*x)/((E^x^2 + 2*x)*Log[E^x^2 + 2*x]^2), x] - 16*Defer[Int][(E^(x^3 + (2*E^x^2)/Log[E^x^2
+ 2*x])*x^3)/((E^x^2 + 2*x)*Log[E^x^2 + 2*x]^2), x] + 4*Defer[Int][(E^(x^2 + x^3 + (2*E^x^2)/Log[E^x^2 + 2*x])
*x)/Log[E^x^2 + 2*x], x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {8 \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 ) x \left (-1+2 x^2\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )}+\frac {4 \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 ) x \left (-1+\log \left (e^{x^2}+2 x\right )\right )}{\log ^2\left (e^{x^2}+2 x\right )}+\frac {\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 (-4+8 x^2+3 x^2 \log ^2\left (e^{x^2}+2 x\right )\right )}{\log ^2\left (e^{x^2}+2 x\right )}\right ) \, dx \\ & = 4 \int \frac {\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 ) x \left (-1+\log \left (e^{x^2}+2 x\right )\right )}{\log ^2\left (e^{x^2}+2 x\right )} \, dx-8 \int \frac {\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 ) x \left (-1+2 x^2\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx+\int \frac {\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 (-4+8 x^2+3 x^2 \log ^2\left (e^{x^2}+2 x\right )\right )}{\log ^2\left (e^{x^2}+2 x\right )} \, dx \\ & = 4 \int \frac {e^{x^2+x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x \left (-1+\log \left (e^{x^2}+2 x\right )\right )}{\log ^2\left (e^{x^2}+2 x\right )} \, dx-8 \int \frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x \left (-1+2 x^2\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx+\int \frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} \left (-4+8 x^2+3 x^2 \log ^2\left (e^{x^2}+2 x\right )\right )}{\log ^2\left (e^{x^2}+2 x\right )} \, dx \\ & = 4 \int \left (-\frac {e^{x^2+x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\log ^2\left (e^{x^2}+2 x\right )}+\frac {e^{x^2+x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\log \left (e^{x^2}+2 x\right )}\right ) \, dx-8 \int \left (-\frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )}+\frac {2 e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x^3}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )}\right ) \, dx+\int \left (3 e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x^2+\frac {4 e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} \left (-1+2 x^2\right )}{\log ^2\left (e^{x^2}+2 x\right )}\right ) \, dx \\ & = 3 \int e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x^2 \, dx-4 \int \frac {e^{x^2+x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\log ^2\left (e^{x^2}+2 x\right )} \, dx+4 \int \frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} \left (-1+2 x^2\right )}{\log ^2\left (e^{x^2}+2 x\right )} \, dx+4 \int \frac {e^{x^2+x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\log \left (e^{x^2}+2 x\right )} \, dx+8 \int \frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx-16 \int \frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x^3}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx \\ & = 3 \int e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x^2 \, dx+4 \int \left (-\frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}}}{\log ^2\left (e^{x^2}+2 x\right )}+\frac {2 e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x^2}{\log ^2\left (e^{x^2}+2 x\right )}\right ) \, dx-4 \int \frac {e^{x^2+x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\log ^2\left (e^{x^2}+2 x\right )} \, dx+4 \int \frac {e^{x^2+x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\log \left (e^{x^2}+2 x\right )} \, dx+8 \int \frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx-16 \int \frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x^3}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx \\ & = 3 \int e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x^2 \, dx-4 \int \frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}}}{\log ^2\left (e^{x^2}+2 x\right )} \, dx-4 \int \frac {e^{x^2+x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\log ^2\left (e^{x^2}+2 x\right )} \, dx+4 \int \frac {e^{x^2+x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\log \left (e^{x^2}+2 x\right )} \, dx+8 \int \frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x^2}{\log ^2\left (e^{x^2}+2 x\right )} \, dx+8 \int \frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx-16 \int \frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x^3}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}} \left (-4 e^{x^2}-4 e^{2 x^2} x+\left (4 e^{2 x^2} x+8 e^{x^2} x^2\right ) \log \left (e^{x^2}+2 x\right )+\left (3 e^{x^2} x^2+6 x^3\right ) \log ^2\left (e^{x^2}+2 x\right )\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx=e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} \]

[In]

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]

[Out]

E^(x^3 + (2*E^x^2)/Log[E^x^2 + 2*x])

Maple [A] (verified)

Time = 3.96 (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	\({\mathrm e}^{\frac {x^{3} \ln \left ({\mathrm e}^{x^{2}}+2 x \right )+2 \,{\mathrm e}^{x^{2}}}{\ln \left ({\mathrm e}^{x^{2}}+2 x \right )}}\)	\(34\)

[In]

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,metho
d=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {e^{\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}} \left (-4 e^{x^2}-4 e^{2 x^2} x+\left (4 e^{2 x^2} x+8 e^{x^2} x^2\right ) \log \left (e^{x^2}+2 x\right )+\left (3 e^{x^2} x^2+6 x^3\right ) \log ^2\left (e^{x^2}+2 x\right )\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx=e^{\left (\frac {x^{3} \log \left (2 \, x + e^{\left (x^{2}\right )}\right ) + 2 \, e^{\left (x^{2}\right )}}{\log \left (2 \, x + e^{\left (x^{2}\right )}\right )}\right )} \]

[In]

integrate(((3*x^2*exp(x^2)+6*x^3)*log(exp(x^2)+2*x)^2+(4*x*exp(x^2)^2+8*x^2*exp(x^2))*log(exp(x^2)+2*x)-4*x*ex
p(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")

[Out]

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

Sympy [F(-1)]

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

[In]

integrate(((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)

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(((3*x^2*exp(x^2)+6*x^3)*log(exp(x^2)+2*x)^2+(4*x*exp(x^2)^2+8*x^2*exp(x^2))*log(exp(x^2)+2*x)-4*x*ex
p(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")

[Out]

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)

none

Time = 0.44 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {e^{\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}} \left (-4 e^{x^2}-4 e^{2 x^2} x+\left (4 e^{2 x^2} x+8 e^{x^2} x^2\right ) \log \left (e^{x^2}+2 x\right )+\left (3 e^{x^2} x^2+6 x^3\right ) \log ^2\left (e^{x^2}+2 x\right )\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx=e^{\left (x^{3} + \frac {2 \, e^{\left (x^{2}\right )}}{\log \left (2 \, x + e^{\left (x^{2}\right )}\right )}\right )} \]

[In]

integrate(((3*x^2*exp(x^2)+6*x^3)*log(exp(x^2)+2*x)^2+(4*x*exp(x^2)^2+8*x^2*exp(x^2))*log(exp(x^2)+2*x)-4*x*ex
p(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")

[Out]

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

Mupad [B] (verification not implemented)

Time = 8.93 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}} \left (-4 e^{x^2}-4 e^{2 x^2} x+\left (4 e^{2 x^2} x+8 e^{x^2} x^2\right ) \log \left (e^{x^2}+2 x\right )+\left (3 e^{x^2} x^2+6 x^3\right ) \log ^2\left (e^{x^2}+2 x\right )\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx={\mathrm {e}}^{x^3}\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^{x^2}}{\ln \left (2\,x+{\mathrm {e}}^{x^2}\right )}} \]

[In]

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

[Out]

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