∫ e−2e2−x2 log(3)+2x log(3) x−2x log(3)−4e2−x2 x2 log(3)+(1+e−2e2−x2 log(3)+2x log(3) x) log(e2e2−x2 log(3)−2x log(3) (−1−e−2e2−x2 log(3)+2x log(3) x))____ (x+e−2e2−x2 log(3)+2x log(3)x2) log(e2e2−x2 log(3)−2x log(3)(−1−e−2e2−x2 log(3)+2x log(3)x)) log(x log(e2e2−x2 log(3)−2x log(3)(−1−e−2e2−x2 log(3)+2x log(3)x))) dx

3.81.43 \(\int \frac {e^{-2 e^{2-x^2} \log (3)+2 x \log (3)} x-2 x \log (3)-4 e^{2-x^2} x^2 \log (3)+(1+e^{-2 e^{2-x^2} \log (3)+2 x \log (3)} x) \log (e^{2 e^{2-x^2} \log (3)-2 x \log (3)} (-1-e^{-2 e^{2-x^2} \log (3)+2 x \log (3)} x))}{(x+e^{-2 e^{2-x^2} \log (3)+2 x \log (3)} x^2) \log (e^{2 e^{2-x^2} \log (3)-2 x \log (3)} (-1-e^{-2 e^{2-x^2} \log (3)+2 x \log (3)} x)) \log (x \log (e^{2 e^{2-x^2} \log (3)-2 x \log (3)} (-1-e^{-2 e^{2-x^2} \log (3)+2 x \log (3)} x)))} \, dx\)

Optimal. Leaf size=28 \[ \log \left (\log \left (x \log \left (-3^{2 e^{2-x^2}-2 x}-x\right )\right )\right ) \]

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Rubi [A] time = 1.77, antiderivative size = 26, normalized size of antiderivative = 0.93, number of steps used = 2, number of rules used = 2, integrand size = 255, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {6688, 6684} \begin {gather*} \log \left (\log \left (x \log \left (-9^{e^{2-x^2}-x}-x\right )\right )\right ) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[(E^(-2*E^(2 - x^2)*Log[3] + 2*x*Log[3])*x - 2*x*Log[3] - 4*E^(2 - x^2)*x^2*Log[3] + (1 + E^(-2*E^(2 - x^2)

*Log[3] + 2*x*Log[3])*x)*Log[E^(2*E^(2 - x^2)*Log[3] - 2*x*Log[3])*(-1 - E^(-2*E^(2 - x^2)*Log[3] + 2*x*Log[3]

)*x)])/((x + E^(-2*E^(2 - x^2)*Log[3] + 2*x*Log[3])*x^2)*Log[E^(2*E^(2 - x^2)*Log[3] - 2*x*Log[3])*(-1 - E^(-2

*E^(2 - x^2)*Log[3] + 2*x*Log[3])*x)]*Log[x*Log[E^(2*E^(2 - x^2)*Log[3] - 2*x*Log[3])*(-1 - E^(-2*E^(2 - x^2)*

Log[3] + 2*x*Log[3])*x)]]),x]

[Out]

Log[Log[x*Log[-9^(E^(2 - x^2) - x) - x]]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /; !Fa

lseQ[q]]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\frac {1}{x}+\frac {e^{-x^2} \left (-4 9^{e^{2-x^2}} e^2 x \log (3)+e^{x^2} \left (9^x-9^{e^{2-x^2}} \log (9)\right )\right )}{\left (9^{e^{2-x^2}}+9^x x\right ) \log \left (-9^{e^{2-x^2}-x}-x\right )}}{\log \left (x \log \left (-9^{e^{2-x^2}-x}-x\right )\right )} \, dx\\ &=\log \left (\log \left (x \log \left (-9^{e^{2-x^2}-x}-x\right )\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [F] time = 0.72, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{-2 e^{2-x^2} \log (3)+2 x \log (3)} x-2 x \log (3)-4 e^{2-x^2} x^2 \log (3)+\left (1+e^{-2 e^{2-x^2} \log (3)+2 x \log (3)} x\right ) \log \left (e^{2 e^{2-x^2} \log (3)-2 x \log (3)} \left (-1-e^{-2 e^{2-x^2} \log (3)+2 x \log (3)} x\right )\right )}{\left (x+e^{-2 e^{2-x^2} \log (3)+2 x \log (3)} x^2\right ) \log \left (e^{2 e^{2-x^2} \log (3)-2 x \log (3)} \left (-1-e^{-2 e^{2-x^2} \log (3)+2 x \log (3)} x\right )\right ) \log \left (x \log \left (e^{2 e^{2-x^2} \log (3)-2 x \log (3)} \left (-1-e^{-2 e^{2-x^2} \log (3)+2 x \log (3)} x\right )\right )\right )} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Integrate[(E^(-2*E^(2 - x^2)*Log[3] + 2*x*Log[3])*x - 2*x*Log[3] - 4*E^(2 - x^2)*x^2*Log[3] + (1 + E^(-2*E^(2

- x^2)*Log[3] + 2*x*Log[3])*x)*Log[E^(2*E^(2 - x^2)*Log[3] - 2*x*Log[3])*(-1 - E^(-2*E^(2 - x^2)*Log[3] + 2*x*

Log[3])*x)])/((x + E^(-2*E^(2 - x^2)*Log[3] + 2*x*Log[3])*x^2)*Log[E^(2*E^(2 - x^2)*Log[3] - 2*x*Log[3])*(-1 -

E^(-2*E^(2 - x^2)*Log[3] + 2*x*Log[3])*x)]*Log[x*Log[E^(2*E^(2 - x^2)*Log[3] - 2*x*Log[3])*(-1 - E^(-2*E^(2 -

x^2)*Log[3] + 2*x*Log[3])*x)]]),x]

[Out]

Integrate[(E^(-2*E^(2 - x^2)*Log[3] + 2*x*Log[3])*x - 2*x*Log[3] - 4*E^(2 - x^2)*x^2*Log[3] + (1 + E^(-2*E^(2

- x^2)*Log[3] + 2*x*Log[3])*x)*Log[E^(2*E^(2 - x^2)*Log[3] - 2*x*Log[3])*(-1 - E^(-2*E^(2 - x^2)*Log[3] + 2*x*

Log[3])*x)])/((x + E^(-2*E^(2 - x^2)*Log[3] + 2*x*Log[3])*x^2)*Log[E^(2*E^(2 - x^2)*Log[3] - 2*x*Log[3])*(-1 -

E^(-2*E^(2 - x^2)*Log[3] + 2*x*Log[3])*x)]*Log[x*Log[E^(2*E^(2 - x^2)*Log[3] - 2*x*Log[3])*(-1 - E^(-2*E^(2 -

x^2)*Log[3] + 2*x*Log[3])*x)]]), x]

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fricas [A] time = 1.02, size = 30, normalized size = 1.07 \begin {gather*} \log \left (\log \left (x \log \left (-x - e^{\left (-2 \, x \log \relax (3) + 2 \, e^{\left (-x^{2} + 2\right )} \log \relax (3)\right )}\right )\right )\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

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

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

og(3))^2)/log(x*log((-x*exp(-log(3)*exp(-x^2+2)+x*log(3))^2-1)/exp(-log(3)*exp(-x^2+2)+x*log(3))^2)),x, algori

thm="fricas")

[Out]

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

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giac [A] time = 1.23, size = 49, normalized size = 1.75 \begin {gather*} \log \left (\log \left (x \log \left (-{\left (x e^{\left (2 \, x \log \relax (3) - 2 \, e^{\left (-x^{2} + 2\right )} \log \relax (3)\right )} + 1\right )} e^{\left (-2 \, x \log \relax (3) + 2 \, e^{\left (-x^{2} + 2\right )} \log \relax (3)\right )}\right )\right )\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

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

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

og(3))^2)/log(x*log((-x*exp(-log(3)*exp(-x^2+2)+x*log(3))^2-1)/exp(-log(3)*exp(-x^2+2)+x*log(3))^2)),x, algori

thm="giac")

[Out]

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

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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {\left (x \,{\mathrm e}^{{\mathrm e}^{-x^{2}+2} \ln \left (\frac {1}{9}\right )+x \ln \relax (9)}+1\right ) \ln \left (\left (-x \,{\mathrm e}^{{\mathrm e}^{-x^{2}+2} \ln \left (\frac {1}{9}\right )+x \ln \relax (9)}-1\right ) {\mathrm e}^{{\mathrm e}^{-x^{2}+2} \ln \relax (9)+x \ln \left (\frac {1}{9}\right )}\right )+x \,{\mathrm e}^{{\mathrm e}^{-x^{2}+2} \ln \left (\frac {1}{9}\right )+x \ln \relax (9)}-4 x^{2} \ln \relax (3) {\mathrm e}^{-x^{2}+2}-2 x \ln \relax (3)}{\left (x^{2} {\mathrm e}^{{\mathrm e}^{-x^{2}+2} \ln \left (\frac {1}{9}\right )+x \ln \relax (9)}+x \right ) \ln \left (\left (-x \,{\mathrm e}^{{\mathrm e}^{-x^{2}+2} \ln \left (\frac {1}{9}\right )+x \ln \relax (9)}-1\right ) {\mathrm e}^{{\mathrm e}^{-x^{2}+2} \ln \relax (9)+x \ln \left (\frac {1}{9}\right )}\right ) \ln \left (x \ln \left (\left (-x \,{\mathrm e}^{{\mathrm e}^{-x^{2}+2} \ln \left (\frac {1}{9}\right )+x \ln \relax (9)}-1\right ) {\mathrm e}^{{\mathrm e}^{-x^{2}+2} \ln \relax (9)+x \ln \left (\frac {1}{9}\right )}\right )\right )}\, dx\]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

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

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

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

[Out]

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

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

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

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

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maxima [A] time = 0.66, size = 35, normalized size = 1.25 \begin {gather*} \log \left (\log \left (-2 \, x \log \relax (3) + \log \left (-3^{2 \, x} x - 3^{2 \, e^{\left (-x^{2} + 2\right )}}\right )\right ) + \log \relax (x)\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

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

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

og(3))^2)/log(x*log((-x*exp(-log(3)*exp(-x^2+2)+x*log(3))^2-1)/exp(-log(3)*exp(-x^2+2)+x*log(3))^2)),x, algori

thm="maxima")

[Out]

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

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mupad [B] time = 8.09, size = 36, normalized size = 1.29 \begin {gather*} \ln \left (\ln \left (x\,\ln \left (-3^{2\,x}\,x-3^{2\,{\mathrm {e}}^2\,{\mathrm {e}}^{-x^2}}\right )-2\,x^2\,\ln \relax (3)\right )\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((log(-exp(2*exp(2 - x^2)*log(3) - 2*x*log(3))*(x*exp(2*x*log(3) - 2*exp(2 - x^2)*log(3)) + 1))*(x*exp(2*x*

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

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

g(x*log(-exp(2*exp(2 - x^2)*log(3) - 2*x*log(3))*(x*exp(2*x*log(3) - 2*exp(2 - x^2)*log(3)) + 1)))*(x + x^2*ex

p(2*x*log(3) - 2*exp(2 - x^2)*log(3)))),x)

[Out]

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

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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

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

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

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

[Out]

Timed out

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