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

Optimal. Leaf size=31 \[ e^{\frac {\log ^2(x)}{x \left (4-x+2 x \left (-\frac {3 e^{4+x}}{4}+x\right )\right )}} \]

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Rubi [F]  time = 46.73, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 31, normalized size = 1.00 \begin {gather*} e^{\frac {2 \log ^2(x)}{x \left (8-2 x-3 e^{4+x} x+4 x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[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)))

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fricas [A]  time = 0.74, size = 32, normalized size = 1.03 \begin {gather*} e^{\left (\frac {2 \, \log \relax (x)^{2}}{4 \, x^{3} - 3 \, x^{2} e^{\left (x + 4\right )} - 2 \, x^{2} + 8 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{208971104256,[0,8,11,40,8]%%%}+%%%{-1950396973056,[0,8,1
1,39,8]%%%}

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maple [A]  time = 0.04, size = 30, normalized size = 0.97

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maxima [B]  time = 0.69, size = 91, normalized size = 2.94 \begin {gather*} e^{\left (-\frac {x \log \relax (x)^{2}}{4 \, x^{2} - 3 \, x e^{\left (x + 4\right )} - 2 \, x + 8} + \frac {3 \, e^{\left (x + 4\right )} \log \relax (x)^{2}}{4 \, {\left (4 \, x^{2} - 3 \, x e^{\left (x + 4\right )} - 2 \, x + 8\right )}} + \frac {\log \relax (x)^{2}}{2 \, {\left (4 \, x^{2} - 3 \, x e^{\left (x + 4\right )} - 2 \, x + 8\right )}} + \frac {\log \relax (x)^{2}}{4 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 4.68, size = 32, normalized size = 1.03 \begin {gather*} {\mathrm {e}}^{\frac {2\,{\ln \relax (x)}^2}{8\,x-2\,x^2+4\,x^3-3\,x^2\,{\mathrm {e}}^4\,{\mathrm {e}}^x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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sympy [A]  time = 2.98, size = 32, normalized size = 1.03 \begin {gather*} e^{- \frac {2 \log {\relax (x )}^{2}}{- 4 x^{3} + 3 x^{2} e^{x + 4} + 2 x^{2} - 8 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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