3.21.19 \(\int \frac {e^{\frac {-2-x}{(-x+2 x^3) \log (x)}} (-2-x+4 x^2+2 x^3+(-2+12 x^2+4 x^3) \log (x))}{(x^2-4 x^4+4 x^6) \log ^2(x)} \, dx\)

Optimal. Leaf size=19 \[ e^{\frac {2+x}{\left (x-2 x^3\right ) \log (x)}} \]

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Rubi [F]  time = 10.94, 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 {e^{\frac {-2-x}{\left (-x+2 x^3\right ) \log (x)}} \left (-2-x+4 x^2+2 x^3+\left (-2+12 x^2+4 x^3\right ) \log (x)\right )}{\left (x^2-4 x^4+4 x^6\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-2 - x)/((-x + 2*x^3)*Log[x]))*(-2 - x + 4*x^2 + 2*x^3 + (-2 + 12*x^2 + 4*x^3)*Log[x]))/((x^2 - 4*x^4
+ 4*x^6)*Log[x]^2),x]

[Out]

-2*Defer[Int][E^((-2 - x)/(x*(-1 + 2*x^2)*Log[x]))/(x^2*Log[x]^2), x] - Defer[Int][E^((-2 - x)/(x*(-1 + 2*x^2)
*Log[x]))/(x*Log[x]^2), x] - 2*Defer[Int][E^((-2 - x)/(x*(-1 + 2*x^2)*Log[x]))/((1 - Sqrt[2]*x)*Log[x]^2), x]
- Defer[Int][E^((-2 - x)/(x*(-1 + 2*x^2)*Log[x]))/((1 - Sqrt[2]*x)*Log[x]^2), x]/Sqrt[2] - 2*Defer[Int][E^((-2
- x)/(x*(-1 + 2*x^2)*Log[x]))/((1 + Sqrt[2]*x)*Log[x]^2), x] + Defer[Int][E^((-2 - x)/(x*(-1 + 2*x^2)*Log[x])
)/((1 + Sqrt[2]*x)*Log[x]^2), x]/Sqrt[2] - 2*Defer[Int][E^((-2 - x)/(x*(-1 + 2*x^2)*Log[x]))/(x^2*Log[x]), x]
- 2*Defer[Int][E^((-2 - x)/(x*(-1 + 2*x^2)*Log[x]))/((1 - Sqrt[2]*x)*Log[x]), x] - 2*Defer[Int][E^((-2 - x)/(x
*(-1 + 2*x^2)*Log[x]))/((1 + Sqrt[2]*x)*Log[x]), x] + 8*Defer[Int][E^((-2 - x)/(x*(-1 + 2*x^2)*Log[x]))/((-1 +
2*x^2)^2*Log[x]), x] + 4*Defer[Int][(E^((-2 - x)/(x*(-1 + 2*x^2)*Log[x]))*x)/((-1 + 2*x^2)^2*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {-2-x}{\left (-x+2 x^3\right ) \log (x)}} \left (-2-x+4 x^2+2 x^3+\left (-2+12 x^2+4 x^3\right ) \log (x)\right )}{x^2 \left (1-4 x^2+4 x^4\right ) \log ^2(x)} \, dx\\ &=4 \int \frac {e^{\frac {-2-x}{\left (-x+2 x^3\right ) \log (x)}} \left (-2-x+4 x^2+2 x^3+\left (-2+12 x^2+4 x^3\right ) \log (x)\right )}{x^2 \left (-2+4 x^2\right )^2 \log ^2(x)} \, dx\\ &=4 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}} \left (-2-x+4 x^2+2 x^3+\left (-2+12 x^2+4 x^3\right ) \log (x)\right )}{x^2 \left (2-4 x^2\right )^2 \log ^2(x)} \, dx\\ &=4 \int \left (\frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}} (2+x)}{4 x^2 \left (-1+2 x^2\right ) \log ^2(x)}+\frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}} \left (-1+6 x^2+2 x^3\right )}{2 x^2 \left (-1+2 x^2\right )^2 \log (x)}\right ) \, dx\\ &=2 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}} \left (-1+6 x^2+2 x^3\right )}{x^2 \left (-1+2 x^2\right )^2 \log (x)} \, dx+\int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}} (2+x)}{x^2 \left (-1+2 x^2\right ) \log ^2(x)} \, dx\\ &=2 \int \left (-\frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{x^2 \log (x)}+\frac {2 e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}} (2+x)}{\left (-1+2 x^2\right )^2 \log (x)}+\frac {2 e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{\left (-1+2 x^2\right ) \log (x)}\right ) \, dx+\int \left (-\frac {2 e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{x^2 \log ^2(x)}-\frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{x \log ^2(x)}+\frac {2 e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}} (2+x)}{\left (-1+2 x^2\right ) \log ^2(x)}\right ) \, dx\\ &=-\left (2 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{x^2 \log ^2(x)} \, dx\right )+2 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}} (2+x)}{\left (-1+2 x^2\right ) \log ^2(x)} \, dx-2 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{x^2 \log (x)} \, dx+4 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}} (2+x)}{\left (-1+2 x^2\right )^2 \log (x)} \, dx+4 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{\left (-1+2 x^2\right ) \log (x)} \, dx-\int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{x \log ^2(x)} \, dx\\ &=2 \int \left (\frac {2 e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{\left (-1+2 x^2\right ) \log ^2(x)}+\frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}} x}{\left (-1+2 x^2\right ) \log ^2(x)}\right ) \, dx-2 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{x^2 \log ^2(x)} \, dx-2 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{x^2 \log (x)} \, dx+4 \int \left (-\frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{2 \left (1-\sqrt {2} x\right ) \log (x)}-\frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{2 \left (1+\sqrt {2} x\right ) \log (x)}\right ) \, dx+4 \int \left (\frac {2 e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{\left (-1+2 x^2\right )^2 \log (x)}+\frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}} x}{\left (-1+2 x^2\right )^2 \log (x)}\right ) \, dx-\int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{x \log ^2(x)} \, dx\\ &=-\left (2 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{x^2 \log ^2(x)} \, dx\right )+2 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}} x}{\left (-1+2 x^2\right ) \log ^2(x)} \, dx-2 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{x^2 \log (x)} \, dx-2 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{\left (1-\sqrt {2} x\right ) \log (x)} \, dx-2 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{\left (1+\sqrt {2} x\right ) \log (x)} \, dx+4 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{\left (-1+2 x^2\right ) \log ^2(x)} \, dx+4 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}} x}{\left (-1+2 x^2\right )^2 \log (x)} \, dx+8 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{\left (-1+2 x^2\right )^2 \log (x)} \, dx-\int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{x \log ^2(x)} \, dx\\ &=2 \int \left (-\frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{2 \sqrt {2} \left (1-\sqrt {2} x\right ) \log ^2(x)}+\frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{2 \sqrt {2} \left (1+\sqrt {2} x\right ) \log ^2(x)}\right ) \, dx-2 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{x^2 \log ^2(x)} \, dx-2 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{x^2 \log (x)} \, dx-2 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{\left (1-\sqrt {2} x\right ) \log (x)} \, dx-2 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{\left (1+\sqrt {2} x\right ) \log (x)} \, dx+4 \int \left (-\frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{2 \left (1-\sqrt {2} x\right ) \log ^2(x)}-\frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{2 \left (1+\sqrt {2} x\right ) \log ^2(x)}\right ) \, dx+4 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}} x}{\left (-1+2 x^2\right )^2 \log (x)} \, dx+8 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{\left (-1+2 x^2\right )^2 \log (x)} \, dx-\int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{x \log ^2(x)} \, dx\\ &=-\left (2 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{x^2 \log ^2(x)} \, dx\right )-2 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{\left (1-\sqrt {2} x\right ) \log ^2(x)} \, dx-2 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{\left (1+\sqrt {2} x\right ) \log ^2(x)} \, dx-2 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{x^2 \log (x)} \, dx-2 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{\left (1-\sqrt {2} x\right ) \log (x)} \, dx-2 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{\left (1+\sqrt {2} x\right ) \log (x)} \, dx+4 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}} x}{\left (-1+2 x^2\right )^2 \log (x)} \, dx+8 \int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{\left (-1+2 x^2\right )^2 \log (x)} \, dx-\frac {\int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{\left (1-\sqrt {2} x\right ) \log ^2(x)} \, dx}{\sqrt {2}}+\frac {\int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{\left (1+\sqrt {2} x\right ) \log ^2(x)} \, dx}{\sqrt {2}}-\int \frac {e^{\frac {-2-x}{x \left (-1+2 x^2\right ) \log (x)}}}{x \log ^2(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.24, size = 20, normalized size = 1.05 \begin {gather*} e^{\frac {2+x}{x \log (x)-2 x^3 \log (x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-2 - x)/((-x + 2*x^3)*Log[x]))*(-2 - x + 4*x^2 + 2*x^3 + (-2 + 12*x^2 + 4*x^3)*Log[x]))/((x^2 -
4*x^4 + 4*x^6)*Log[x]^2),x]

[Out]

E^((2 + x)/(x*Log[x] - 2*x^3*Log[x]))

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fricas [A]  time = 0.58, size = 21, normalized size = 1.11 \begin {gather*} e^{\left (-\frac {x + 2}{{\left (2 \, x^{3} - x\right )} \log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+12*x^2-2)*log(x)+2*x^3+4*x^2-x-2)*exp((-x-2)/(2*x^3-x)/log(x))/(4*x^6-4*x^4+x^2)/log(x)^2,x,
algorithm="fricas")

[Out]

e^(-(x + 2)/((2*x^3 - x)*log(x)))

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giac [A]  time = 0.42, size = 37, normalized size = 1.95 \begin {gather*} e^{\left (-\frac {x}{2 \, x^{3} \log \relax (x) - x \log \relax (x)} - \frac {2}{2 \, x^{3} \log \relax (x) - x \log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+12*x^2-2)*log(x)+2*x^3+4*x^2-x-2)*exp((-x-2)/(2*x^3-x)/log(x))/(4*x^6-4*x^4+x^2)/log(x)^2,x,
algorithm="giac")

[Out]

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

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maple [A]  time = 0.03, size = 23, normalized size = 1.21

method	result	size
risch	\({\mathrm e}^{-\frac {2+x}{x \left (2 x^{2}-1\right ) \ln \relax (x )}}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^3+12*x^2-2)*ln(x)+2*x^3+4*x^2-x-2)*exp((-x-2)/(2*x^3-x)/ln(x))/(4*x^6-4*x^4+x^2)/ln(x)^2,x,method=_R
ETURNVERBOSE)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+12*x^2-2)*log(x)+2*x^3+4*x^2-x-2)*exp((-x-2)/(2*x^3-x)/log(x))/(4*x^6-4*x^4+x^2)/log(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

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mupad [B]  time = 1.35, size = 32, normalized size = 1.68 \begin {gather*} {\mathrm {e}}^{\frac {1}{\ln \relax (x)-2\,x^2\,\ln \relax (x)}}\,{\mathrm {e}}^{-\frac {2}{2\,x^3\,\ln \relax (x)-x\,\ln \relax (x)}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((x + 2)/(log(x)*(x - 2*x^3)))*(log(x)*(12*x^2 + 4*x^3 - 2) - x + 4*x^2 + 2*x^3 - 2))/(log(x)^2*(x^2 -
4*x^4 + 4*x^6)),x)

[Out]

exp(1/(log(x) - 2*x^2*log(x)))*exp(-2/(2*x^3*log(x) - x*log(x)))

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sympy [A]  time = 0.56, size = 15, normalized size = 0.79 \begin {gather*} e^{\frac {- x - 2}{\left (2 x^{3} - x\right ) \log {\relax (x )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**3+12*x**2-2)*ln(x)+2*x**3+4*x**2-x-2)*exp((-x-2)/(2*x**3-x)/ln(x))/(4*x**6-4*x**4+x**2)/ln(x)
**2,x)

[Out]

exp((-x - 2)/((2*x**3 - x)*log(x)))

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