3.9.13 \(\int \frac {e^{\frac {2 (-x^2+(x+e x) \log (x))}{1+e}+e^{\frac {2 (-x^2+(x+e x) \log (x))}{1+e}} x \log ^2(\log (2))} (-1+e (-1-2 x)-2 x+4 x^2+(-2 x-2 e x) \log (x)) \log ^2(\log (2))}{1+e} \, dx\)

Optimal. Leaf size=27 \[ -e^{e^{2 x \left (-\frac {x}{1+e}+\log (x)\right )} x \log ^2(\log (2))} \]

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Rubi [F]  time = 13.43, 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 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))\right ) \left (-1+e (-1-2 x)-2 x+4 x^2+(-2 x-2 e x) \log (x)\right ) \log ^2(\log (2))}{1+e} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 5.16, size = 29, normalized size = 1.07 \begin {gather*} -e^{e^{-\frac {2 x^2}{1+e}} x^{1+2 x} \log ^2(\log (2))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((2*(-x^2 + (x + E*x)*Log[x]))/(1 + E) + E^((2*(-x^2 + (x + E*x)*Log[x]))/(1 + E))*x*Log[Log[2]]^
2)*(-1 + E*(-1 - 2*x) - 2*x + 4*x^2 + (-2*x - 2*E*x)*Log[x])*Log[Log[2]]^2)/(1 + E),x]

[Out]

-E^((x^(1 + 2*x)*Log[Log[2]]^2)/E^((2*x^2)/(1 + E)))

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fricas [B]  time = 0.73, size = 84, normalized size = 3.11 \begin {gather*} -e^{\left (\frac {{\left (x e + x\right )} e^{\left (-\frac {2 \, {\left (x^{2} - {\left (x e + x\right )} \log \relax (x)\right )}}{e + 1}\right )} \log \left (\log \relax (2)\right )^{2} - 2 \, x^{2} + 2 \, {\left (x e + x\right )} \log \relax (x)}{e + 1} + \frac {2 \, {\left (x^{2} - {\left (x e + x\right )} \log \relax (x)\right )}}{e + 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*exp(1)-2*x)*log(x)+(-2*x-1)*exp(1)+4*x^2-2*x-1)*log(log(2))^2*exp(((x*exp(1)+x)*log(x)-x^2)/(
1+exp(1)))^2*exp(x*log(log(2))^2*exp(((x*exp(1)+x)*log(x)-x^2)/(1+exp(1)))^2)/(1+exp(1)),x, algorithm="fricas"
)

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (4 \, x^{2} - {\left (2 \, x + 1\right )} e - 2 \, {\left (x e + x\right )} \log \relax (x) - 2 \, x - 1\right )} e^{\left (x e^{\left (-\frac {2 \, {\left (x^{2} - {\left (x e + x\right )} \log \relax (x)\right )}}{e + 1}\right )} \log \left (\log \relax (2)\right )^{2} - \frac {2 \, {\left (x^{2} - {\left (x e + x\right )} \log \relax (x)\right )}}{e + 1}\right )} \log \left (\log \relax (2)\right )^{2}}{e + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*exp(1)-2*x)*log(x)+(-2*x-1)*exp(1)+4*x^2-2*x-1)*log(log(2))^2*exp(((x*exp(1)+x)*log(x)-x^2)/(
1+exp(1)))^2*exp(x*log(log(2))^2*exp(((x*exp(1)+x)*log(x)-x^2)/(1+exp(1)))^2)/(1+exp(1)),x, algorithm="giac")

[Out]

integrate((4*x^2 - (2*x + 1)*e - 2*(x*e + x)*log(x) - 2*x - 1)*e^(x*e^(-2*(x^2 - (x*e + x)*log(x))/(e + 1))*lo
g(log(2))^2 - 2*(x^2 - (x*e + x)*log(x))/(e + 1))*log(log(2))^2/(e + 1), x)

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maple [A]  time = 0.06, size = 32, normalized size = 1.19

method	result	size
risch	\(-{\mathrm e}^{x \ln \left (\ln \relax (2)\right )^{2} {\mathrm e}^{\frac {2 x \left ({\mathrm e} \ln \relax (x )+\ln \relax (x )-x \right )}{1+{\mathrm e}}}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x*exp(1)-2*x)*ln(x)+(-2*x-1)*exp(1)+4*x^2-2*x-1)*ln(ln(2))^2*exp(((x*exp(1)+x)*ln(x)-x^2)/(1+exp(1)))
^2*exp(x*ln(ln(2))^2*exp(((x*exp(1)+x)*ln(x)-x^2)/(1+exp(1)))^2)/(1+exp(1)),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 1.02, size = 47, normalized size = 1.74 \begin {gather*} -e^{\left (x e^{\left (\frac {2 \, x e \log \relax (x)}{e + 1} - \frac {2 \, x^{2}}{e + 1} + \frac {2 \, x \log \relax (x)}{e + 1}\right )} \log \left (\log \relax (2)\right )^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*exp(1)-2*x)*log(x)+(-2*x-1)*exp(1)+4*x^2-2*x-1)*log(log(2))^2*exp(((x*exp(1)+x)*log(x)-x^2)/(
1+exp(1)))^2*exp(x*log(log(2))^2*exp(((x*exp(1)+x)*log(x)-x^2)/(1+exp(1)))^2)/(1+exp(1)),x, algorithm="maxima"
)

[Out]

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

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mupad [B]  time = 1.24, size = 46, normalized size = 1.70 \begin {gather*} -{\mathrm {e}}^{x\,x^{\frac {2\,x\,\mathrm {e}}{\mathrm {e}+1}}\,x^{\frac {2\,x}{\mathrm {e}+1}}\,{\mathrm {e}}^{-\frac {2\,x^2}{\mathrm {e}+1}}\,{\ln \left (\ln \relax (2)\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x*exp((2*(log(x)*(x + x*exp(1)) - x^2))/(exp(1) + 1))*log(log(2))^2)*exp((2*(log(x)*(x + x*exp(1)) -
x^2))/(exp(1) + 1))*log(log(2))^2*(2*x + log(x)*(2*x + 2*x*exp(1)) - 4*x^2 + exp(1)*(2*x + 1) + 1))/(exp(1) +
1),x)

[Out]

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

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sympy [A]  time = 2.06, size = 34, normalized size = 1.26 \begin {gather*} - e^{x e^{\frac {2 \left (- x^{2} + \left (x + e x\right ) \log {\relax (x )}\right )}{1 + e}} \log {\left (\log {\relax (2 )} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*exp(1)-2*x)*ln(x)+(-2*x-1)*exp(1)+4*x**2-2*x-1)*ln(ln(2))**2*exp(((x*exp(1)+x)*ln(x)-x**2)/(1
+exp(1)))**2*exp(x*ln(ln(2))**2*exp(((x*exp(1)+x)*ln(x)-x**2)/(1+exp(1)))**2)/(1+exp(1)),x)

[Out]

-exp(x*exp(2*(-x**2 + (x + E*x)*log(x))/(1 + E))*log(log(2))**2)

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