\(\int \frac {e^{-\frac {2 \log ^2(x)}{-8 x+2 x^2+3 e^{4+x} x^2-4 x^3}} ((32-8 x-12 e^{4+x} x+16 x^2) \log (x)+(-16+8 x-24 x^2+e^{4+x} (12 x+6 x^2)) \log ^2(x))}{64 x^2-32 x^3+68 x^4+9 e^{8+2 x} x^4-16 x^5+16 x^6+e^{4+x} (-48 x^3+12 x^4-24 x^5)} \, dx\) [6028]

Optimal result

Integrand size = 149, antiderivative size = 31 \[ \int \frac {e^{-\frac {2 \log ^2(x)}{-8 x+2 x^2+3 e^{4+x} x^2-4 x^3}} \left (\left (32-8 x-12 e^{4+x} x+16 x^2\right ) \log (x)+\left (-16+8 x-24 x^2+e^{4+x} \left (12 x+6 x^2\right )\right ) \log ^2(x)\right )}{64 x^2-32 x^3+68 x^4+9 e^{8+2 x} x^4-16 x^5+16 x^6+e^{4+x} \left (-48 x^3+12 x^4-24 x^5\right )} \, dx=e^{\frac {\log ^2(x)}{x \left (4-x+2 x \left (-\frac {3 e^{4+x}}{4}+x\right )\right )}} \]

[Out]

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

Rubi [F]

\[ \int \frac {e^{-\frac {2 \log ^2(x)}{-8 x+2 x^2+3 e^{4+x} x^2-4 x^3}} \left (\left (32-8 x-12 e^{4+x} x+16 x^2\right ) \log (x)+\left (-16+8 x-24 x^2+e^{4+x} \left (12 x+6 x^2\right )\right ) \log ^2(x)\right )}{64 x^2-32 x^3+68 x^4+9 e^{8+2 x} x^4-16 x^5+16 x^6+e^{4+x} \left (-48 x^3+12 x^4-24 x^5\right )} \, dx=\int \frac {\exp \left (-\frac {2 \log ^2(x)}{-8 x+2 x^2+3 e^{4+x} x^2-4 x^3}\right ) \left (\left (32-8 x-12 e^{4+x} x+16 x^2\right ) \log (x)+\left (-16+8 x-24 x^2+e^{4+x} \left (12 x+6 x^2\right )\right ) \log ^2(x)\right )}{64 x^2-32 x^3+68 x^4+9 e^{8+2 x} x^4-16 x^5+16 x^6+e^{4+x} \left (-48 x^3+12 x^4-24 x^5\right )} \, dx \]

[In]

Int[((32 - 8*x - 12*E^(4 + x)*x + 16*x^2)*Log[x] + (-16 + 8*x - 24*x^2 + E^(4 + x)*(12*x + 6*x^2))*Log[x]^2)/(
E^((2*Log[x]^2)/(-8*x + 2*x^2 + 3*E^(4 + x)*x^2 - 4*x^3))*(64*x^2 - 32*x^3 + 68*x^4 + 9*E^(8 + 2*x)*x^4 - 16*x
^5 + 16*x^6 + E^(4 + x)*(-48*x^3 + 12*x^4 - 24*x^5))),x]

[Out]

4*Defer[Int][(E^((2*Log[x]^2)/(x*(8 - (2 + 3*E^(4 + x))*x + 4*x^2)))*Log[x])/(x^2*(8 - 2*x - 3*E^(4 + x)*x + 4
*x^2)), x] - 12*Defer[Int][(E^((2*Log[x]^2)/(x*(8 - (2 + 3*E^(4 + x))*x + 4*x^2)))*Log[x]^2)/(8 - 2*x - 3*E^(4
+ x)*x + 4*x^2)^2, x] + 16*Defer[Int][(E^((2*Log[x]^2)/(x*(8 - (2 + 3*E^(4 + x))*x + 4*x^2)))*Log[x]^2)/(x^2*
(8 - 2*x - 3*E^(4 + x)*x + 4*x^2)^2), x] + 16*Defer[Int][(E^((2*Log[x]^2)/(x*(8 - (2 + 3*E^(4 + x))*x + 4*x^2)
))*Log[x]^2)/(x*(8 - 2*x - 3*E^(4 + x)*x + 4*x^2)^2), x] + 8*Defer[Int][(E^((2*Log[x]^2)/(x*(8 - (2 + 3*E^(4 +
x))*x + 4*x^2)))*x*Log[x]^2)/(8 - 2*x - 3*E^(4 + x)*x + 4*x^2)^2, x] - 4*Defer[Int][(E^((2*Log[x]^2)/(x*(8 -
(2 + 3*E^(4 + x))*x + 4*x^2)))*Log[x]^2)/(x^2*(8 - 2*x - 3*E^(4 + x)*x + 4*x^2)), x] - 2*Defer[Int][(E^((2*Log
[x]^2)/(x*(8 - (2 + 3*E^(4 + x))*x + 4*x^2)))*Log[x]^2)/(x*(8 - 2*x - 3*E^(4 + x)*x + 4*x^2)), x]

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-\frac {2 \log ^2(x)}{-8 x+2 x^2+3 e^{4+x} x^2-4 x^3}} \left (\left (32-8 x-12 e^{4+x} x+16 x^2\right ) \log (x)+\left (-16+8 x-24 x^2+e^{4+x} \left (12 x+6 x^2\right )\right ) \log ^2(x)\right )}{64 x^2-32 x^3+68 x^4+9 e^{8+2 x} x^4-16 x^5+16 x^6+e^{4+x} \left (-48 x^3+12 x^4-24 x^5\right )} \, dx=e^{\frac {2 \log ^2(x)}{x \left (8-2 x-3 e^{4+x} x+4 x^2\right )}} \]

[In]

Integrate[((32 - 8*x - 12*E^(4 + x)*x + 16*x^2)*Log[x] + (-16 + 8*x - 24*x^2 + E^(4 + x)*(12*x + 6*x^2))*Log[x
]^2)/(E^((2*Log[x]^2)/(-8*x + 2*x^2 + 3*E^(4 + x)*x^2 - 4*x^3))*(64*x^2 - 32*x^3 + 68*x^4 + 9*E^(8 + 2*x)*x^4
- 16*x^5 + 16*x^6 + E^(4 + x)*(-48*x^3 + 12*x^4 - 24*x^5))),x]

[Out]

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

Maple [A] (verified)

Time = 133.61 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97

method	result	size
risch	\({\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{x \left (-3 x \,{\mathrm e}^{4+x}+4 x^{2}-2 x +8\right )}}\)	\(30\)
parallelrisch	\({\mathrm e}^{-\frac {2 \ln \left (x \right )^{2}}{x \left (3 x \,{\mathrm e}^{4+x}-4 x^{2}+2 x -8\right )}}\)	\(30\)

[In]

int((((6*x^2+12*x)*exp(4+x)-24*x^2+8*x-16)*ln(x)^2+(-12*x*exp(4+x)+16*x^2-8*x+32)*ln(x))*exp(-2*ln(x)^2/(3*x^2
*exp(4+x)-4*x^3+2*x^2-8*x))/(9*x^4*exp(4+x)^2+(-24*x^5+12*x^4-48*x^3)*exp(4+x)+16*x^6-16*x^5+68*x^4-32*x^3+64*
x^2),x,method=_RETURNVERBOSE)

[Out]

exp(2*ln(x)^2/x/(-3*x*exp(4+x)+4*x^2-2*x+8))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-\frac {2 \log ^2(x)}{-8 x+2 x^2+3 e^{4+x} x^2-4 x^3}} \left (\left (32-8 x-12 e^{4+x} x+16 x^2\right ) \log (x)+\left (-16+8 x-24 x^2+e^{4+x} \left (12 x+6 x^2\right )\right ) \log ^2(x)\right )}{64 x^2-32 x^3+68 x^4+9 e^{8+2 x} x^4-16 x^5+16 x^6+e^{4+x} \left (-48 x^3+12 x^4-24 x^5\right )} \, dx=e^{\left (\frac {2 \, \log \left (x\right )^{2}}{4 \, x^{3} - 3 \, x^{2} e^{\left (x + 4\right )} - 2 \, x^{2} + 8 \, x}\right )} \]

[In]

integrate((((6*x^2+12*x)*exp(4+x)-24*x^2+8*x-16)*log(x)^2+(-12*x*exp(4+x)+16*x^2-8*x+32)*log(x))*exp(-2*log(x)
^2/(3*x^2*exp(4+x)-4*x^3+2*x^2-8*x))/(9*x^4*exp(4+x)^2+(-24*x^5+12*x^4-48*x^3)*exp(4+x)+16*x^6-16*x^5+68*x^4-3
2*x^3+64*x^2),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 1.47 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-\frac {2 \log ^2(x)}{-8 x+2 x^2+3 e^{4+x} x^2-4 x^3}} \left (\left (32-8 x-12 e^{4+x} x+16 x^2\right ) \log (x)+\left (-16+8 x-24 x^2+e^{4+x} \left (12 x+6 x^2\right )\right ) \log ^2(x)\right )}{64 x^2-32 x^3+68 x^4+9 e^{8+2 x} x^4-16 x^5+16 x^6+e^{4+x} \left (-48 x^3+12 x^4-24 x^5\right )} \, dx=e^{- \frac {2 \log {\left (x \right )}^{2}}{- 4 x^{3} + 3 x^{2} e^{x + 4} + 2 x^{2} - 8 x}} \]

[In]

integrate((((6*x**2+12*x)*exp(4+x)-24*x**2+8*x-16)*ln(x)**2+(-12*x*exp(4+x)+16*x**2-8*x+32)*ln(x))*exp(-2*ln(x
)**2/(3*x**2*exp(4+x)-4*x**3+2*x**2-8*x))/(9*x**4*exp(4+x)**2+(-24*x**5+12*x**4-48*x**3)*exp(4+x)+16*x**6-16*x
**5+68*x**4-32*x**3+64*x**2),x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (29) = 58\).

Time = 0.50 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.94 \[ \int \frac {e^{-\frac {2 \log ^2(x)}{-8 x+2 x^2+3 e^{4+x} x^2-4 x^3}} \left (\left (32-8 x-12 e^{4+x} x+16 x^2\right ) \log (x)+\left (-16+8 x-24 x^2+e^{4+x} \left (12 x+6 x^2\right )\right ) \log ^2(x)\right )}{64 x^2-32 x^3+68 x^4+9 e^{8+2 x} x^4-16 x^5+16 x^6+e^{4+x} \left (-48 x^3+12 x^4-24 x^5\right )} \, dx=e^{\left (-\frac {x \log \left (x\right )^{2}}{4 \, x^{2} - 3 \, x e^{\left (x + 4\right )} - 2 \, x + 8} + \frac {3 \, e^{\left (x + 4\right )} \log \left (x\right )^{2}}{4 \, {\left (4 \, x^{2} - 3 \, x e^{\left (x + 4\right )} - 2 \, x + 8\right )}} + \frac {\log \left (x\right )^{2}}{2 \, {\left (4 \, x^{2} - 3 \, x e^{\left (x + 4\right )} - 2 \, x + 8\right )}} + \frac {\log \left (x\right )^{2}}{4 \, x}\right )} \]

[In]

integrate((((6*x^2+12*x)*exp(4+x)-24*x^2+8*x-16)*log(x)^2+(-12*x*exp(4+x)+16*x^2-8*x+32)*log(x))*exp(-2*log(x)
^2/(3*x^2*exp(4+x)-4*x^3+2*x^2-8*x))/(9*x^4*exp(4+x)^2+(-24*x^5+12*x^4-48*x^3)*exp(4+x)+16*x^6-16*x^5+68*x^4-3
2*x^3+64*x^2),x, algorithm="maxima")

[Out]

e^(-x*log(x)^2/(4*x^2 - 3*x*e^(x + 4) - 2*x + 8) + 3/4*e^(x + 4)*log(x)^2/(4*x^2 - 3*x*e^(x + 4) - 2*x + 8) +
1/2*log(x)^2/(4*x^2 - 3*x*e^(x + 4) - 2*x + 8) + 1/4*log(x)^2/x)

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{-\frac {2 \log ^2(x)}{-8 x+2 x^2+3 e^{4+x} x^2-4 x^3}} \left (\left (32-8 x-12 e^{4+x} x+16 x^2\right ) \log (x)+\left (-16+8 x-24 x^2+e^{4+x} \left (12 x+6 x^2\right )\right ) \log ^2(x)\right )}{64 x^2-32 x^3+68 x^4+9 e^{8+2 x} x^4-16 x^5+16 x^6+e^{4+x} \left (-48 x^3+12 x^4-24 x^5\right )} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((((6*x^2+12*x)*exp(4+x)-24*x^2+8*x-16)*log(x)^2+(-12*x*exp(4+x)+16*x^2-8*x+32)*log(x))*exp(-2*log(x)
^2/(3*x^2*exp(4+x)-4*x^3+2*x^2-8*x))/(9*x^4*exp(4+x)^2+(-24*x^5+12*x^4-48*x^3)*exp(4+x)+16*x^6-16*x^5+68*x^4-3
2*x^3+64*x^2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:Unable to divide,
perhaps due to rounding error%%%{17414258688,[0,8,9,38,8]%%%}+%%%{-156728328192,[0,8,9,37,8]%%%}+%%%{79670233
4976,[0,8,9,36

Mupad [B] (verification not implemented)

Time = 12.36 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-\frac {2 \log ^2(x)}{-8 x+2 x^2+3 e^{4+x} x^2-4 x^3}} \left (\left (32-8 x-12 e^{4+x} x+16 x^2\right ) \log (x)+\left (-16+8 x-24 x^2+e^{4+x} \left (12 x+6 x^2\right )\right ) \log ^2(x)\right )}{64 x^2-32 x^3+68 x^4+9 e^{8+2 x} x^4-16 x^5+16 x^6+e^{4+x} \left (-48 x^3+12 x^4-24 x^5\right )} \, dx={\mathrm {e}}^{\frac {2\,{\ln \left (x\right )}^2}{8\,x-2\,x^2+4\,x^3-3\,x^2\,{\mathrm {e}}^4\,{\mathrm {e}}^x}} \]

[In]

int((exp((2*log(x)^2)/(8*x - 3*x^2*exp(x + 4) - 2*x^2 + 4*x^3))*(log(x)^2*(8*x + exp(x + 4)*(12*x + 6*x^2) - 2
4*x^2 - 16) - log(x)*(8*x + 12*x*exp(x + 4) - 16*x^2 - 32)))/(9*x^4*exp(2*x + 8) - exp(x + 4)*(48*x^3 - 12*x^4
+ 24*x^5) + 64*x^2 - 32*x^3 + 68*x^4 - 16*x^5 + 16*x^6),x)

[Out]

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