\(\int \frac {e^{4 e^x+2 x^2} (6 x-24 e^x x^2+12 x^3)+e^{2 x^2} (6 x^3-12 x^5)-12 e^{2 x^2} x \log (x)+e^{2 x^2} (6 x+12 x^3) \log ^2(x)+e^{2 e^x} (12 e^{2 x^2} x+e^{2 x^2} (-12 x+24 e^x x^2-24 x^3) \log (x))}{e^{12 e^x}-x^6-6 e^{10 e^x} \log (x)+3 x^4 \log ^2(x)-3 x^2 \log ^4(x)+\log ^6(x)+e^{8 e^x} (-3 x^2+15 \log ^2(x))+e^{6 e^x} (12 x^2 \log (x)-20 \log ^3(x))+e^{4 e^x} (3 x^4-18 x^2 \log ^2(x)+15 \log ^4(x))+e^{2 e^x} (-6 x^4 \log (x)+12 x^2 \log ^3(x)-6 \log ^5(x))} \, dx\) [2358]

Optimal result

Integrand size = 281, antiderivative size = 33 \[ \int \frac {e^{4 e^x+2 x^2} \left (6 x-24 e^x x^2+12 x^3\right )+e^{2 x^2} \left (6 x^3-12 x^5\right )-12 e^{2 x^2} x \log (x)+e^{2 x^2} \left (6 x+12 x^3\right ) \log ^2(x)+e^{2 e^x} \left (12 e^{2 x^2} x+e^{2 x^2} \left (-12 x+24 e^x x^2-24 x^3\right ) \log (x)\right )}{e^{12 e^x}-x^6-6 e^{10 e^x} \log (x)+3 x^4 \log ^2(x)-3 x^2 \log ^4(x)+\log ^6(x)+e^{8 e^x} \left (-3 x^2+15 \log ^2(x)\right )+e^{6 e^x} \left (12 x^2 \log (x)-20 \log ^3(x)\right )+e^{4 e^x} \left (3 x^4-18 x^2 \log ^2(x)+15 \log ^4(x)\right )+e^{2 e^x} \left (-6 x^4 \log (x)+12 x^2 \log ^3(x)-6 \log ^5(x)\right )} \, dx=\frac {3 e^{2 x^2}}{\left (-x+\frac {\left (-e^{2 e^x}+\log (x)\right )^2}{x}\right )^2} \]

[Out]

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

Rubi [F]

\[ \int \frac {e^{4 e^x+2 x^2} \left (6 x-24 e^x x^2+12 x^3\right )+e^{2 x^2} \left (6 x^3-12 x^5\right )-12 e^{2 x^2} x \log (x)+e^{2 x^2} \left (6 x+12 x^3\right ) \log ^2(x)+e^{2 e^x} \left (12 e^{2 x^2} x+e^{2 x^2} \left (-12 x+24 e^x x^2-24 x^3\right ) \log (x)\right )}{e^{12 e^x}-x^6-6 e^{10 e^x} \log (x)+3 x^4 \log ^2(x)-3 x^2 \log ^4(x)+\log ^6(x)+e^{8 e^x} \left (-3 x^2+15 \log ^2(x)\right )+e^{6 e^x} \left (12 x^2 \log (x)-20 \log ^3(x)\right )+e^{4 e^x} \left (3 x^4-18 x^2 \log ^2(x)+15 \log ^4(x)\right )+e^{2 e^x} \left (-6 x^4 \log (x)+12 x^2 \log ^3(x)-6 \log ^5(x)\right )} \, dx=\int \frac {e^{4 e^x+2 x^2} \left (6 x-24 e^x x^2+12 x^3\right )+e^{2 x^2} \left (6 x^3-12 x^5\right )-12 e^{2 x^2} x \log (x)+e^{2 x^2} \left (6 x+12 x^3\right ) \log ^2(x)+e^{2 e^x} \left (12 e^{2 x^2} x+e^{2 x^2} \left (-12 x+24 e^x x^2-24 x^3\right ) \log (x)\right )}{e^{12 e^x}-x^6-6 e^{10 e^x} \log (x)+3 x^4 \log ^2(x)-3 x^2 \log ^4(x)+\log ^6(x)+e^{8 e^x} \left (-3 x^2+15 \log ^2(x)\right )+e^{6 e^x} \left (12 x^2 \log (x)-20 \log ^3(x)\right )+e^{4 e^x} \left (3 x^4-18 x^2 \log ^2(x)+15 \log ^4(x)\right )+e^{2 e^x} \left (-6 x^4 \log (x)+12 x^2 \log ^3(x)-6 \log ^5(x)\right )} \, dx \]

[In]

Int[(E^(4*E^x + 2*x^2)*(6*x - 24*E^x*x^2 + 12*x^3) + E^(2*x^2)*(6*x^3 - 12*x^5) - 12*E^(2*x^2)*x*Log[x] + E^(2
*x^2)*(6*x + 12*x^3)*Log[x]^2 + E^(2*E^x)*(12*E^(2*x^2)*x + E^(2*x^2)*(-12*x + 24*E^x*x^2 - 24*x^3)*Log[x]))/(
E^(12*E^x) - x^6 - 6*E^(10*E^x)*Log[x] + 3*x^4*Log[x]^2 - 3*x^2*Log[x]^4 + Log[x]^6 + E^(8*E^x)*(-3*x^2 + 15*L
og[x]^2) + E^(6*E^x)*(12*x^2*Log[x] - 20*Log[x]^3) + E^(4*E^x)*(3*x^4 - 18*x^2*Log[x]^2 + 15*Log[x]^4) + E^(2*
E^x)*(-6*x^4*Log[x] + 12*x^2*Log[x]^3 - 6*Log[x]^5)),x]

[Out]

(3*Defer[Int][E^(2*x^2)/(E^(2*E^x) - x - Log[x])^3, x])/4 + (3*Defer[Int][E^(2*(2*E^x + x^2))/(E^(2*E^x) - x -
Log[x])^3, x])/2 - 3*Defer[Int][E^(2*E^x + x + 2*x^2)/(E^(2*E^x) - x - Log[x])^3, x] - (9*Defer[Int][E^(2*x^2
)/(E^(2*E^x) - x - Log[x]), x])/4 - (3*Defer[Int][E^(2*x^2)/(E^(2*E^x) + x - Log[x])^3, x])/4 - (3*Defer[Int][
E^(2*(2*E^x + x^2))/(E^(2*E^x) + x - Log[x])^3, x])/2 - 3*Defer[Int][E^(2*E^x + x + 2*x^2)/(E^(2*E^x) + x - Lo
g[x])^3, x] - (3*Defer[Int][E^(2*(E^x + x^2))/(x^2*(E^(2*E^x) + x - Log[x])^3), x])/2 - (3*Defer[Int][E^(2*(2*
E^x + x^2))/(x^2*(E^(2*E^x) + x - Log[x])^3), x])/4 + (3*Defer[Int][(E^(2*x^2)*x^2)/(E^(2*E^x) + x - Log[x])^3
, x])/2 - (9*Defer[Int][E^(2*(E^x + x^2))/(x^3*(E^(2*E^x) + x - Log[x])^2), x])/4 - (9*Defer[Int][E^(2*(2*E^x
+ x^2))/(x^3*(E^(2*E^x) + x - Log[x])^2), x])/8 - (9*Defer[Int][E^(2*x^2)/(x*(E^(2*E^x) + x - Log[x])^2), x])/
8 - (9*Defer[Int][E^(2*(2*E^x + x^2))/(x*(E^(2*E^x) + x - Log[x])^2), x])/4 - (3*Defer[Int][E^(2*E^x + x + 2*x
^2)/(x*(E^(2*E^x) + x - Log[x])^2), x])/2 + (9*Defer[Int][(E^(2*x^2)*x)/(E^(2*E^x) + x - Log[x])^2, x])/4 + (9
*Defer[Int][E^(2*x^2)/(E^(2*E^x) + x - Log[x]), x])/4 - (9*Defer[Int][E^(2*(E^x + x^2))/(x^4*(E^(2*E^x) + x -
Log[x])), x])/4 - (9*Defer[Int][E^(2*(2*E^x + x^2))/(x^4*(E^(2*E^x) + x - Log[x])), x])/8 - (9*Defer[Int][E^(2
*x^2)/(x^2*(E^(2*E^x) + x - Log[x])), x])/8 - (9*Defer[Int][E^(2*(2*E^x + x^2))/(x^2*(E^(2*E^x) + x - Log[x]))
, x])/4 + 3*Defer[Int][(E^(2*(E^x + x^2))*Log[x])/(E^(2*E^x) + x - Log[x])^3, x] + (3*Defer[Int][(E^(2*x^2)*Lo
g[x])/(x^2*(E^(2*E^x) + x - Log[x])^3), x])/2 + (3*Defer[Int][(E^(2*(E^x + x^2))*Log[x])/(x^2*(E^(2*E^x) + x -
Log[x])^3), x])/2 + (9*Defer[Int][(E^(2*x^2)*Log[x])/(x^3*(E^(2*E^x) + x - Log[x])^2), x])/4 + (9*Defer[Int][
(E^(2*(E^x + x^2))*Log[x])/(x^3*(E^(2*E^x) + x - Log[x])^2), x])/4 + (9*Defer[Int][(E^(2*(E^x + x^2))*Log[x])/
(x*(E^(2*E^x) + x - Log[x])^2), x])/2 + (9*Defer[Int][(E^(2*x^2)*Log[x])/(x^4*(E^(2*E^x) + x - Log[x])), x])/4
+ (9*Defer[Int][(E^(2*(E^x + x^2))*Log[x])/(x^4*(E^(2*E^x) + x - Log[x])), x])/4 + (9*Defer[Int][(E^(2*(E^x +
x^2))*Log[x])/(x^2*(E^(2*E^x) + x - Log[x])), x])/2 - (3*Defer[Int][(E^(2*x^2)*Log[x]^2)/(E^(2*E^x) + x - Log
[x])^3, x])/2 - (3*Defer[Int][(E^(2*x^2)*Log[x]^2)/(x^2*(E^(2*E^x) + x - Log[x])^3), x])/4 - (9*Defer[Int][(E^
(2*x^2)*Log[x]^2)/(x^3*(E^(2*E^x) + x - Log[x])^2), x])/8 - (9*Defer[Int][(E^(2*x^2)*Log[x]^2)/(x*(E^(2*E^x) +
x - Log[x])^2), x])/4 - (9*Defer[Int][(E^(2*x^2)*Log[x]^2)/(x^4*(E^(2*E^x) + x - Log[x])), x])/8 - (9*Defer[I
nt][(E^(2*x^2)*Log[x]^2)/(x^2*(E^(2*E^x) + x - Log[x])), x])/4 - (3*Defer[Int][E^(2*(E^x + x^2))/(x^2*(-E^(2*E
^x) + x + Log[x])^3), x])/2 - (3*Defer[Int][E^(2*(2*E^x + x^2))/(x^2*(-E^(2*E^x) + x + Log[x])^3), x])/4 + (3*
Defer[Int][(E^(2*x^2)*x^2)/(-E^(2*E^x) + x + Log[x])^3, x])/2 + 3*Defer[Int][(E^(2*(E^x + x^2))*Log[x])/(-E^(2
*E^x) + x + Log[x])^3, x] + (3*Defer[Int][(E^(2*x^2)*Log[x])/(x^2*(-E^(2*E^x) + x + Log[x])^3), x])/2 + (3*Def
er[Int][(E^(2*(E^x + x^2))*Log[x])/(x^2*(-E^(2*E^x) + x + Log[x])^3), x])/2 - (3*Defer[Int][(E^(2*x^2)*Log[x]^
2)/(-E^(2*E^x) + x + Log[x])^3, x])/2 - (3*Defer[Int][(E^(2*x^2)*Log[x]^2)/(x^2*(-E^(2*E^x) + x + Log[x])^3),
x])/4 - (9*Defer[Int][E^(2*(E^x + x^2))/(x^3*(-E^(2*E^x) + x + Log[x])^2), x])/4 - (9*Defer[Int][E^(2*(2*E^x +
x^2))/(x^3*(-E^(2*E^x) + x + Log[x])^2), x])/8 - (9*Defer[Int][E^(2*x^2)/(x*(-E^(2*E^x) + x + Log[x])^2), x])
/8 - (9*Defer[Int][E^(2*(2*E^x + x^2))/(x*(-E^(2*E^x) + x + Log[x])^2), x])/4 + (3*Defer[Int][E^(2*E^x + x + 2
*x^2)/(x*(-E^(2*E^x) + x + Log[x])^2), x])/2 + (9*Defer[Int][(E^(2*x^2)*x)/(-E^(2*E^x) + x + Log[x])^2, x])/4
+ (9*Defer[Int][(E^(2*x^2)*Log[x])/(x^3*(-E^(2*E^x) + x + Log[x])^2), x])/4 + (9*Defer[Int][(E^(2*(E^x + x^2))
*Log[x])/(x^3*(-E^(2*E^x) + x + Log[x])^2), x])/4 + (9*Defer[Int][(E^(2*(E^x + x^2))*Log[x])/(x*(-E^(2*E^x) +
x + Log[x])^2), x])/2 - (9*Defer[Int][(E^(2*x^2)*Log[x]^2)/(x^3*(-E^(2*E^x) + x + Log[x])^2), x])/8 - (9*Defer
[Int][(E^(2*x^2)*Log[x]^2)/(x*(-E^(2*E^x) + x + Log[x])^2), x])/4 - (9*Defer[Int][E^(2*(E^x + x^2))/(x^4*(-E^(
2*E^x) + x + Log[x])), x])/4 - (9*Defer[Int][E^(2*(2*E^x + x^2))/(x^4*(-E^(2*E^x) + x + Log[x])), x])/8 - (9*D
efer[Int][E^(2*x^2)/(x^2*(-E^(2*E^x) + x + Log[x])), x])/8 - (9*Defer[Int][E^(2*(2*E^x + x^2))/(x^2*(-E^(2*E^x
) + x + Log[x])), x])/4 + (9*Defer[Int][(E^(2*x^2)*Log[x])/(x^4*(-E^(2*E^x) + x + Log[x])), x])/4 + (9*Defer[I
nt][(E^(2*(E^x + x^2))*Log[x])/(x^4*(-E^(2*E^x) + x + Log[x])), x])/4 + (9*Defer[Int][(E^(2*(E^x + x^2))*Log[x
])/(x^2*(-E^(2*E^x) + x + Log[x])), x])/2 - (9*Defer[Int][(E^(2*x^2)*Log[x]^2)/(x^4*(-E^(2*E^x) + x + Log[x]))
, x])/8 - (9*Defer[Int][(E^(2*x^2)*Log[x]^2)/(x^2*(-E^(2*E^x) + x + Log[x])), x])/4

Rubi steps \begin{align*} \text {integral}& = \int \frac {6 e^{2 x^2} x \left (2 e^{2 e^x}-4 e^{4 e^x+x} x-x^2 \left (-1+2 x^2\right )+e^{4 e^x} \left (1+2 x^2\right )-\left (2-4 e^{2 e^x+x} x+e^{2 e^x} \left (2+4 x^2\right )\right ) \log (x)+\left (1+2 x^2\right ) \log ^2(x)\right )}{\left (e^{4 e^x}-x^2-2 e^{2 e^x} \log (x)+\log ^2(x)\right )^3} \, dx \\ & = 6 \int \frac {e^{2 x^2} x \left (2 e^{2 e^x}-4 e^{4 e^x+x} x-x^2 \left (-1+2 x^2\right )+e^{4 e^x} \left (1+2 x^2\right )-\left (2-4 e^{2 e^x+x} x+e^{2 e^x} \left (2+4 x^2\right )\right ) \log (x)+\left (1+2 x^2\right ) \log ^2(x)\right )}{\left (e^{4 e^x}-x^2-2 e^{2 e^x} \log (x)+\log ^2(x)\right )^3} \, dx \\ & = 6 \int \left (\frac {e^{2 x^2} x^3 \left (-1+2 x^2\right )}{\left (e^{2 e^x}+x-\log (x)\right )^3 \left (-e^{2 e^x}+x+\log (x)\right )^3}+\frac {2 e^{2 x^2} x \log (x)}{\left (e^{2 e^x}+x-\log (x)\right )^3 \left (-e^{2 e^x}+x+\log (x)\right )^3}-\frac {e^{2 x^2} x \left (1+2 x^2\right ) \log ^2(x)}{\left (e^{2 e^x}+x-\log (x)\right )^3 \left (-e^{2 e^x}+x+\log (x)\right )^3}+\frac {2 e^{2 e^x+2 x^2} x}{\left (e^{4 e^x}-x^2-2 e^{2 e^x} \log (x)+\log ^2(x)\right )^3}+\frac {e^{4 e^x+2 x^2} x \left (1+2 x^2\right )}{\left (e^{4 e^x}-x^2-2 e^{2 e^x} \log (x)+\log ^2(x)\right )^3}-\frac {4 e^{2 e^x+x+2 x^2} x^2 \left (e^{2 e^x}-\log (x)\right )}{\left (e^{4 e^x}-x^2-2 e^{2 e^x} \log (x)+\log ^2(x)\right )^3}-\frac {2 e^{2 e^x+2 x^2} x \left (1+2 x^2\right ) \log (x)}{\left (e^{4 e^x}-x^2-2 e^{2 e^x} \log (x)+\log ^2(x)\right )^3}\right ) \, dx \\ & = 6 \int \frac {e^{2 x^2} x^3 \left (-1+2 x^2\right )}{\left (e^{2 e^x}+x-\log (x)\right )^3 \left (-e^{2 e^x}+x+\log (x)\right )^3} \, dx-6 \int \frac {e^{2 x^2} x \left (1+2 x^2\right ) \log ^2(x)}{\left (e^{2 e^x}+x-\log (x)\right )^3 \left (-e^{2 e^x}+x+\log (x)\right )^3} \, dx+6 \int \frac {e^{4 e^x+2 x^2} x \left (1+2 x^2\right )}{\left (e^{4 e^x}-x^2-2 e^{2 e^x} \log (x)+\log ^2(x)\right )^3} \, dx+12 \int \frac {e^{2 x^2} x \log (x)}{\left (e^{2 e^x}+x-\log (x)\right )^3 \left (-e^{2 e^x}+x+\log (x)\right )^3} \, dx+12 \int \frac {e^{2 e^x+2 x^2} x}{\left (e^{4 e^x}-x^2-2 e^{2 e^x} \log (x)+\log ^2(x)\right )^3} \, dx-12 \int \frac {e^{2 e^x+2 x^2} x \left (1+2 x^2\right ) \log (x)}{\left (e^{4 e^x}-x^2-2 e^{2 e^x} \log (x)+\log ^2(x)\right )^3} \, dx-24 \int \frac {e^{2 e^x+x+2 x^2} x^2 \left (e^{2 e^x}-\log (x)\right )}{\left (e^{4 e^x}-x^2-2 e^{2 e^x} \log (x)+\log ^2(x)\right )^3} \, dx \\ & = 6 \int \frac {e^{2 \left (2 e^x+x^2\right )} x \left (1+2 x^2\right )}{\left (e^{4 e^x}-x^2-2 e^{2 e^x} \log (x)+\log ^2(x)\right )^3} \, dx+6 \int \left (\frac {e^{2 x^2} \left (-1+2 x^2\right )}{8 \left (e^{2 e^x}+x-\log (x)\right )^3}+\frac {3 e^{2 x^2} \left (-1+2 x^2\right )}{16 x \left (e^{2 e^x}+x-\log (x)\right )^2}+\frac {3 e^{2 x^2} \left (-1+2 x^2\right )}{16 x^2 \left (e^{2 e^x}+x-\log (x)\right )}+\frac {e^{2 x^2} \left (-1+2 x^2\right )}{8 \left (-e^{2 e^x}+x+\log (x)\right )^3}+\frac {3 e^{2 x^2} \left (-1+2 x^2\right )}{16 x \left (-e^{2 e^x}+x+\log (x)\right )^2}+\frac {3 e^{2 x^2} \left (-1+2 x^2\right )}{16 x^2 \left (-e^{2 e^x}+x+\log (x)\right )}\right ) \, dx-6 \int \left (\frac {e^{2 x^2} \left (1+2 x^2\right ) \log ^2(x)}{8 x^2 \left (e^{2 e^x}+x-\log (x)\right )^3}+\frac {3 e^{2 x^2} \left (1+2 x^2\right ) \log ^2(x)}{16 x^3 \left (e^{2 e^x}+x-\log (x)\right )^2}+\frac {3 e^{2 x^2} \left (1+2 x^2\right ) \log ^2(x)}{16 x^4 \left (e^{2 e^x}+x-\log (x)\right )}+\frac {e^{2 x^2} \left (1+2 x^2\right ) \log ^2(x)}{8 x^2 \left (-e^{2 e^x}+x+\log (x)\right )^3}+\frac {3 e^{2 x^2} \left (1+2 x^2\right ) \log ^2(x)}{16 x^3 \left (-e^{2 e^x}+x+\log (x)\right )^2}+\frac {3 e^{2 x^2} \left (1+2 x^2\right ) \log ^2(x)}{16 x^4 \left (-e^{2 e^x}+x+\log (x)\right )}\right ) \, dx+12 \int \frac {e^{2 \left (e^x+x^2\right )} x}{\left (e^{4 e^x}-x^2-2 e^{2 e^x} \log (x)+\log ^2(x)\right )^3} \, dx-12 \int \frac {e^{2 \left (e^x+x^2\right )} x \left (1+2 x^2\right ) \log (x)}{\left (e^{4 e^x}-x^2-2 e^{2 e^x} \log (x)+\log ^2(x)\right )^3} \, dx+12 \int \left (\frac {e^{2 x^2} \log (x)}{8 x^2 \left (e^{2 e^x}+x-\log (x)\right )^3}+\frac {3 e^{2 x^2} \log (x)}{16 x^3 \left (e^{2 e^x}+x-\log (x)\right )^2}+\frac {3 e^{2 x^2} \log (x)}{16 x^4 \left (e^{2 e^x}+x-\log (x)\right )}+\frac {e^{2 x^2} \log (x)}{8 x^2 \left (-e^{2 e^x}+x+\log (x)\right )^3}+\frac {3 e^{2 x^2} \log (x)}{16 x^3 \left (-e^{2 e^x}+x+\log (x)\right )^2}+\frac {3 e^{2 x^2} \log (x)}{16 x^4 \left (-e^{2 e^x}+x+\log (x)\right )}\right ) \, dx-24 \int \left (\frac {e^{2 e^x+x+2 x^2}}{8 \left (e^{2 e^x}-x-\log (x)\right )^3}+\frac {e^{2 e^x+x+2 x^2}}{8 \left (e^{2 e^x}+x-\log (x)\right )^3}+\frac {e^{2 e^x+x+2 x^2}}{16 x \left (e^{2 e^x}+x-\log (x)\right )^2}-\frac {e^{2 e^x+x+2 x^2}}{16 x \left (-e^{2 e^x}+x+\log (x)\right )^2}\right ) \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \[ \int \frac {e^{4 e^x+2 x^2} \left (6 x-24 e^x x^2+12 x^3\right )+e^{2 x^2} \left (6 x^3-12 x^5\right )-12 e^{2 x^2} x \log (x)+e^{2 x^2} \left (6 x+12 x^3\right ) \log ^2(x)+e^{2 e^x} \left (12 e^{2 x^2} x+e^{2 x^2} \left (-12 x+24 e^x x^2-24 x^3\right ) \log (x)\right )}{e^{12 e^x}-x^6-6 e^{10 e^x} \log (x)+3 x^4 \log ^2(x)-3 x^2 \log ^4(x)+\log ^6(x)+e^{8 e^x} \left (-3 x^2+15 \log ^2(x)\right )+e^{6 e^x} \left (12 x^2 \log (x)-20 \log ^3(x)\right )+e^{4 e^x} \left (3 x^4-18 x^2 \log ^2(x)+15 \log ^4(x)\right )+e^{2 e^x} \left (-6 x^4 \log (x)+12 x^2 \log ^3(x)-6 \log ^5(x)\right )} \, dx=\frac {3 e^{2 x^2} x^2}{\left (e^{4 e^x}-x^2-2 e^{2 e^x} \log (x)+\log ^2(x)\right )^2} \]

[In]

Integrate[(E^(4*E^x + 2*x^2)*(6*x - 24*E^x*x^2 + 12*x^3) + E^(2*x^2)*(6*x^3 - 12*x^5) - 12*E^(2*x^2)*x*Log[x]
+ E^(2*x^2)*(6*x + 12*x^3)*Log[x]^2 + E^(2*E^x)*(12*E^(2*x^2)*x + E^(2*x^2)*(-12*x + 24*E^x*x^2 - 24*x^3)*Log[
x]))/(E^(12*E^x) - x^6 - 6*E^(10*E^x)*Log[x] + 3*x^4*Log[x]^2 - 3*x^2*Log[x]^4 + Log[x]^6 + E^(8*E^x)*(-3*x^2
+ 15*Log[x]^2) + E^(6*E^x)*(12*x^2*Log[x] - 20*Log[x]^3) + E^(4*E^x)*(3*x^4 - 18*x^2*Log[x]^2 + 15*Log[x]^4) +
E^(2*E^x)*(-6*x^4*Log[x] + 12*x^2*Log[x]^3 - 6*Log[x]^5)),x]

[Out]

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

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21

\[\frac {3 x^{2} {\mathrm e}^{2 x^{2}}}{\left (-{\mathrm e}^{4 \,{\mathrm e}^{x}}+2 \ln \left (x \right ) {\mathrm e}^{2 \,{\mathrm e}^{x}}+x^{2}-\ln \left (x \right )^{2}\right )^{2}}\]

[In]

int(((-24*exp(x)*x^2+12*x^3+6*x)*exp(x^2)^2*exp(exp(x))^4+((24*exp(x)*x^2-24*x^3-12*x)*exp(x^2)^2*ln(x)+12*x*e
xp(x^2)^2)*exp(exp(x))^2+(12*x^3+6*x)*exp(x^2)^2*ln(x)^2-12*x*exp(x^2)^2*ln(x)+(-12*x^5+6*x^3)*exp(x^2)^2)/(ex
p(exp(x))^12-6*ln(x)*exp(exp(x))^10+(15*ln(x)^2-3*x^2)*exp(exp(x))^8+(-20*ln(x)^3+12*x^2*ln(x))*exp(exp(x))^6+
(15*ln(x)^4-18*x^2*ln(x)^2+3*x^4)*exp(exp(x))^4+(-6*ln(x)^5+12*x^2*ln(x)^3-6*x^4*ln(x))*exp(exp(x))^2+ln(x)^6-
3*x^2*ln(x)^4+3*x^4*ln(x)^2-x^6),x)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (29) = 58\).

Time = 0.25 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.45 \[ \int \frac {e^{4 e^x+2 x^2} \left (6 x-24 e^x x^2+12 x^3\right )+e^{2 x^2} \left (6 x^3-12 x^5\right )-12 e^{2 x^2} x \log (x)+e^{2 x^2} \left (6 x+12 x^3\right ) \log ^2(x)+e^{2 e^x} \left (12 e^{2 x^2} x+e^{2 x^2} \left (-12 x+24 e^x x^2-24 x^3\right ) \log (x)\right )}{e^{12 e^x}-x^6-6 e^{10 e^x} \log (x)+3 x^4 \log ^2(x)-3 x^2 \log ^4(x)+\log ^6(x)+e^{8 e^x} \left (-3 x^2+15 \log ^2(x)\right )+e^{6 e^x} \left (12 x^2 \log (x)-20 \log ^3(x)\right )+e^{4 e^x} \left (3 x^4-18 x^2 \log ^2(x)+15 \log ^4(x)\right )+e^{2 e^x} \left (-6 x^4 \log (x)+12 x^2 \log ^3(x)-6 \log ^5(x)\right )} \, dx=\frac {3 \, x^{2} e^{\left (2 \, x^{2}\right )}}{x^{4} - 2 \, x^{2} \log \left (x\right )^{2} + \log \left (x\right )^{4} - 2 \, {\left (x^{2} - 3 \, \log \left (x\right )^{2}\right )} e^{\left (4 \, e^{x}\right )} + 4 \, {\left (x^{2} \log \left (x\right ) - \log \left (x\right )^{3}\right )} e^{\left (2 \, e^{x}\right )} - 4 \, e^{\left (6 \, e^{x}\right )} \log \left (x\right ) + e^{\left (8 \, e^{x}\right )}} \]

[In]

integrate(((-24*exp(x)*x^2+12*x^3+6*x)*exp(x^2)^2*exp(exp(x))^4+((24*exp(x)*x^2-24*x^3-12*x)*exp(x^2)^2*log(x)
+12*x*exp(x^2)^2)*exp(exp(x))^2+(12*x^3+6*x)*exp(x^2)^2*log(x)^2-12*x*exp(x^2)^2*log(x)+(-12*x^5+6*x^3)*exp(x^
2)^2)/(exp(exp(x))^12-6*log(x)*exp(exp(x))^10+(15*log(x)^2-3*x^2)*exp(exp(x))^8+(-20*log(x)^3+12*x^2*log(x))*e
xp(exp(x))^6+(15*log(x)^4-18*x^2*log(x)^2+3*x^4)*exp(exp(x))^4+(-6*log(x)^5+12*x^2*log(x)^3-6*x^4*log(x))*exp(
exp(x))^2+log(x)^6-3*x^2*log(x)^4+3*x^4*log(x)^2-x^6),x, algorithm="fricas")

[Out]

3*x^2*e^(2*x^2)/(x^4 - 2*x^2*log(x)^2 + log(x)^4 - 2*(x^2 - 3*log(x)^2)*e^(4*e^x) + 4*(x^2*log(x) - log(x)^3)*
e^(2*e^x) - 4*e^(6*e^x)*log(x) + e^(8*e^x))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (24) = 48\).

Time = 0.30 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.67 \[ \int \frac {e^{4 e^x+2 x^2} \left (6 x-24 e^x x^2+12 x^3\right )+e^{2 x^2} \left (6 x^3-12 x^5\right )-12 e^{2 x^2} x \log (x)+e^{2 x^2} \left (6 x+12 x^3\right ) \log ^2(x)+e^{2 e^x} \left (12 e^{2 x^2} x+e^{2 x^2} \left (-12 x+24 e^x x^2-24 x^3\right ) \log (x)\right )}{e^{12 e^x}-x^6-6 e^{10 e^x} \log (x)+3 x^4 \log ^2(x)-3 x^2 \log ^4(x)+\log ^6(x)+e^{8 e^x} \left (-3 x^2+15 \log ^2(x)\right )+e^{6 e^x} \left (12 x^2 \log (x)-20 \log ^3(x)\right )+e^{4 e^x} \left (3 x^4-18 x^2 \log ^2(x)+15 \log ^4(x)\right )+e^{2 e^x} \left (-6 x^4 \log (x)+12 x^2 \log ^3(x)-6 \log ^5(x)\right )} \, dx=\frac {3 x^{2} e^{2 x^{2}}}{x^{4} - 2 x^{2} \log {\left (x \right )}^{2} + \left (- 2 x^{2} + 6 \log {\left (x \right )}^{2}\right ) e^{4 e^{x}} + \left (4 x^{2} \log {\left (x \right )} - 4 \log {\left (x \right )}^{3}\right ) e^{2 e^{x}} + e^{8 e^{x}} - 4 e^{6 e^{x}} \log {\left (x \right )} + \log {\left (x \right )}^{4}} \]

[In]

integrate(((-24*exp(x)*x**2+12*x**3+6*x)*exp(x**2)**2*exp(exp(x))**4+((24*exp(x)*x**2-24*x**3-12*x)*exp(x**2)*
*2*ln(x)+12*x*exp(x**2)**2)*exp(exp(x))**2+(12*x**3+6*x)*exp(x**2)**2*ln(x)**2-12*x*exp(x**2)**2*ln(x)+(-12*x*
*5+6*x**3)*exp(x**2)**2)/(exp(exp(x))**12-6*ln(x)*exp(exp(x))**10+(15*ln(x)**2-3*x**2)*exp(exp(x))**8+(-20*ln(
x)**3+12*x**2*ln(x))*exp(exp(x))**6+(15*ln(x)**4-18*x**2*ln(x)**2+3*x**4)*exp(exp(x))**4+(-6*ln(x)**5+12*x**2*
ln(x)**3-6*x**4*ln(x))*exp(exp(x))**2+ln(x)**6-3*x**2*ln(x)**4+3*x**4*ln(x)**2-x**6),x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (29) = 58\).

Time = 0.74 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.45 \[ \int \frac {e^{4 e^x+2 x^2} \left (6 x-24 e^x x^2+12 x^3\right )+e^{2 x^2} \left (6 x^3-12 x^5\right )-12 e^{2 x^2} x \log (x)+e^{2 x^2} \left (6 x+12 x^3\right ) \log ^2(x)+e^{2 e^x} \left (12 e^{2 x^2} x+e^{2 x^2} \left (-12 x+24 e^x x^2-24 x^3\right ) \log (x)\right )}{e^{12 e^x}-x^6-6 e^{10 e^x} \log (x)+3 x^4 \log ^2(x)-3 x^2 \log ^4(x)+\log ^6(x)+e^{8 e^x} \left (-3 x^2+15 \log ^2(x)\right )+e^{6 e^x} \left (12 x^2 \log (x)-20 \log ^3(x)\right )+e^{4 e^x} \left (3 x^4-18 x^2 \log ^2(x)+15 \log ^4(x)\right )+e^{2 e^x} \left (-6 x^4 \log (x)+12 x^2 \log ^3(x)-6 \log ^5(x)\right )} \, dx=\frac {3 \, x^{2} e^{\left (2 \, x^{2}\right )}}{x^{4} - 2 \, x^{2} \log \left (x\right )^{2} + \log \left (x\right )^{4} - 2 \, {\left (x^{2} - 3 \, \log \left (x\right )^{2}\right )} e^{\left (4 \, e^{x}\right )} + 4 \, {\left (x^{2} \log \left (x\right ) - \log \left (x\right )^{3}\right )} e^{\left (2 \, e^{x}\right )} - 4 \, e^{\left (6 \, e^{x}\right )} \log \left (x\right ) + e^{\left (8 \, e^{x}\right )}} \]

[In]

integrate(((-24*exp(x)*x^2+12*x^3+6*x)*exp(x^2)^2*exp(exp(x))^4+((24*exp(x)*x^2-24*x^3-12*x)*exp(x^2)^2*log(x)
+12*x*exp(x^2)^2)*exp(exp(x))^2+(12*x^3+6*x)*exp(x^2)^2*log(x)^2-12*x*exp(x^2)^2*log(x)+(-12*x^5+6*x^3)*exp(x^
2)^2)/(exp(exp(x))^12-6*log(x)*exp(exp(x))^10+(15*log(x)^2-3*x^2)*exp(exp(x))^8+(-20*log(x)^3+12*x^2*log(x))*e
xp(exp(x))^6+(15*log(x)^4-18*x^2*log(x)^2+3*x^4)*exp(exp(x))^4+(-6*log(x)^5+12*x^2*log(x)^3-6*x^4*log(x))*exp(
exp(x))^2+log(x)^6-3*x^2*log(x)^4+3*x^4*log(x)^2-x^6),x, algorithm="maxima")

[Out]

3*x^2*e^(2*x^2)/(x^4 - 2*x^2*log(x)^2 + log(x)^4 - 2*(x^2 - 3*log(x)^2)*e^(4*e^x) + 4*(x^2*log(x) - log(x)^3)*
e^(2*e^x) - 4*e^(6*e^x)*log(x) + e^(8*e^x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (29) = 58\).

Time = 0.34 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.67 \[ \int \frac {e^{4 e^x+2 x^2} \left (6 x-24 e^x x^2+12 x^3\right )+e^{2 x^2} \left (6 x^3-12 x^5\right )-12 e^{2 x^2} x \log (x)+e^{2 x^2} \left (6 x+12 x^3\right ) \log ^2(x)+e^{2 e^x} \left (12 e^{2 x^2} x+e^{2 x^2} \left (-12 x+24 e^x x^2-24 x^3\right ) \log (x)\right )}{e^{12 e^x}-x^6-6 e^{10 e^x} \log (x)+3 x^4 \log ^2(x)-3 x^2 \log ^4(x)+\log ^6(x)+e^{8 e^x} \left (-3 x^2+15 \log ^2(x)\right )+e^{6 e^x} \left (12 x^2 \log (x)-20 \log ^3(x)\right )+e^{4 e^x} \left (3 x^4-18 x^2 \log ^2(x)+15 \log ^4(x)\right )+e^{2 e^x} \left (-6 x^4 \log (x)+12 x^2 \log ^3(x)-6 \log ^5(x)\right )} \, dx=\frac {3 \, x^{2} e^{\left (2 \, x^{2}\right )}}{x^{4} + 4 \, x^{2} e^{\left (2 \, e^{x}\right )} \log \left (x\right ) - 2 \, x^{2} \log \left (x\right )^{2} - 4 \, e^{\left (2 \, e^{x}\right )} \log \left (x\right )^{3} + \log \left (x\right )^{4} - 2 \, x^{2} e^{\left (4 \, e^{x}\right )} + 6 \, e^{\left (4 \, e^{x}\right )} \log \left (x\right )^{2} - 4 \, e^{\left (6 \, e^{x}\right )} \log \left (x\right ) + e^{\left (8 \, e^{x}\right )}} \]

[In]

integrate(((-24*exp(x)*x^2+12*x^3+6*x)*exp(x^2)^2*exp(exp(x))^4+((24*exp(x)*x^2-24*x^3-12*x)*exp(x^2)^2*log(x)
+12*x*exp(x^2)^2)*exp(exp(x))^2+(12*x^3+6*x)*exp(x^2)^2*log(x)^2-12*x*exp(x^2)^2*log(x)+(-12*x^5+6*x^3)*exp(x^
2)^2)/(exp(exp(x))^12-6*log(x)*exp(exp(x))^10+(15*log(x)^2-3*x^2)*exp(exp(x))^8+(-20*log(x)^3+12*x^2*log(x))*e
xp(exp(x))^6+(15*log(x)^4-18*x^2*log(x)^2+3*x^4)*exp(exp(x))^4+(-6*log(x)^5+12*x^2*log(x)^3-6*x^4*log(x))*exp(
exp(x))^2+log(x)^6-3*x^2*log(x)^4+3*x^4*log(x)^2-x^6),x, algorithm="giac")

[Out]

3*x^2*e^(2*x^2)/(x^4 + 4*x^2*e^(2*e^x)*log(x) - 2*x^2*log(x)^2 - 4*e^(2*e^x)*log(x)^3 + log(x)^4 - 2*x^2*e^(4*
e^x) + 6*e^(4*e^x)*log(x)^2 - 4*e^(6*e^x)*log(x) + e^(8*e^x))

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{4 e^x+2 x^2} \left (6 x-24 e^x x^2+12 x^3\right )+e^{2 x^2} \left (6 x^3-12 x^5\right )-12 e^{2 x^2} x \log (x)+e^{2 x^2} \left (6 x+12 x^3\right ) \log ^2(x)+e^{2 e^x} \left (12 e^{2 x^2} x+e^{2 x^2} \left (-12 x+24 e^x x^2-24 x^3\right ) \log (x)\right )}{e^{12 e^x}-x^6-6 e^{10 e^x} \log (x)+3 x^4 \log ^2(x)-3 x^2 \log ^4(x)+\log ^6(x)+e^{8 e^x} \left (-3 x^2+15 \log ^2(x)\right )+e^{6 e^x} \left (12 x^2 \log (x)-20 \log ^3(x)\right )+e^{4 e^x} \left (3 x^4-18 x^2 \log ^2(x)+15 \log ^4(x)\right )+e^{2 e^x} \left (-6 x^4 \log (x)+12 x^2 \log ^3(x)-6 \log ^5(x)\right )} \, dx=\int \frac {{\mathrm {e}}^{4\,{\mathrm {e}}^x+2\,x^2}\,\left (6\,x-24\,x^2\,{\mathrm {e}}^x+12\,x^3\right )+{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left (12\,x\,{\mathrm {e}}^{2\,x^2}-{\mathrm {e}}^{2\,x^2}\,\ln \left (x\right )\,\left (12\,x-24\,x^2\,{\mathrm {e}}^x+24\,x^3\right )\right )+{\mathrm {e}}^{2\,x^2}\,\left (6\,x^3-12\,x^5\right )-12\,x\,{\mathrm {e}}^{2\,x^2}\,\ln \left (x\right )+{\mathrm {e}}^{2\,x^2}\,{\ln \left (x\right )}^2\,\left (12\,x^3+6\,x\right )}{{\mathrm {e}}^{12\,{\mathrm {e}}^x}+{\mathrm {e}}^{6\,{\mathrm {e}}^x}\,\left (12\,x^2\,\ln \left (x\right )-20\,{\ln \left (x\right )}^3\right )+{\mathrm {e}}^{8\,{\mathrm {e}}^x}\,\left (15\,{\ln \left (x\right )}^2-3\,x^2\right )+{\ln \left (x\right )}^6-3\,x^2\,{\ln \left (x\right )}^4+3\,x^4\,{\ln \left (x\right )}^2-{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left (6\,x^4\,\ln \left (x\right )-12\,x^2\,{\ln \left (x\right )}^3+6\,{\ln \left (x\right )}^5\right )+{\mathrm {e}}^{4\,{\mathrm {e}}^x}\,\left (3\,x^4-18\,x^2\,{\ln \left (x\right )}^2+15\,{\ln \left (x\right )}^4\right )-x^6-6\,{\mathrm {e}}^{10\,{\mathrm {e}}^x}\,\ln \left (x\right )} \,d x \]

[In]

int((exp(2*exp(x))*(12*x*exp(2*x^2) - exp(2*x^2)*log(x)*(12*x - 24*x^2*exp(x) + 24*x^3)) + exp(2*x^2)*(6*x^3 -
12*x^5) + exp(2*x^2)*exp(4*exp(x))*(6*x - 24*x^2*exp(x) + 12*x^3) - 12*x*exp(2*x^2)*log(x) + exp(2*x^2)*log(x
)^2*(6*x + 12*x^3))/(exp(12*exp(x)) + exp(6*exp(x))*(12*x^2*log(x) - 20*log(x)^3) + exp(8*exp(x))*(15*log(x)^2
- 3*x^2) + log(x)^6 - 3*x^2*log(x)^4 + 3*x^4*log(x)^2 - exp(2*exp(x))*(6*x^4*log(x) + 6*log(x)^5 - 12*x^2*log
(x)^3) + exp(4*exp(x))*(15*log(x)^4 - 18*x^2*log(x)^2 + 3*x^4) - x^6 - 6*exp(10*exp(x))*log(x)),x)

[Out]

int((exp(4*exp(x) + 2*x^2)*(6*x - 24*x^2*exp(x) + 12*x^3) + exp(2*exp(x))*(12*x*exp(2*x^2) - exp(2*x^2)*log(x)
*(12*x - 24*x^2*exp(x) + 24*x^3)) + exp(2*x^2)*(6*x^3 - 12*x^5) - 12*x*exp(2*x^2)*log(x) + exp(2*x^2)*log(x)^2
*(6*x + 12*x^3))/(exp(12*exp(x)) + exp(6*exp(x))*(12*x^2*log(x) - 20*log(x)^3) + exp(8*exp(x))*(15*log(x)^2 -
3*x^2) + log(x)^6 - 3*x^2*log(x)^4 + 3*x^4*log(x)^2 - exp(2*exp(x))*(6*x^4*log(x) + 6*log(x)^5 - 12*x^2*log(x)
^3) + exp(4*exp(x))*(15*log(x)^4 - 18*x^2*log(x)^2 + 3*x^4) - x^6 - 6*exp(10*exp(x))*log(x)), x)