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

Optimal. Leaf size=26 \[ e^{\frac {4}{3 e^{2/3}}-2 x+\frac {-2+\log \left (\log \left (x^2\right )\right )}{x}} \]

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Rubi [F]  time = 2.46, 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 {4 x+e^{2/3} \left (-6-6 x^2\right )+3 e^{2/3} \log \left (\log \left (x^2\right )\right )}{3 e^{2/3} x}\right ) \left (2+\left (2-2 x^2\right ) \log \left (x^2\right )-\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right )}{x^2 \log \left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.39, size = 29, normalized size = 1.12 \begin {gather*} e^{\frac {4}{3 e^{2/3}}-\frac {2}{x}-2 x} \sqrt [x]{\log \left (x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.70, size = 30, normalized size = 1.15 \begin {gather*} e^{\left (-\frac {{\left (6 \, {\left (x^{2} + 1\right )} e^{\frac {2}{3}} - 3 \, e^{\frac {2}{3}} \log \left (\log \left (x^{2}\right )\right ) - 4 \, x\right )} e^{\left (-\frac {2}{3}\right )}}{3 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x^2)*log(log(x^2))+(-2*x^2+2)*log(x^2)+2)*exp(1/3*(3*exp(2/3)*log(log(x^2))+(-6*x^2-6)*exp(2/3
)+4*x)/x/exp(2/3))/x^2/log(x^2),x, algorithm="fricas")

[Out]

e^(-1/3*(6*(x^2 + 1)*e^(2/3) - 3*e^(2/3)*log(log(x^2)) - 4*x)*e^(-2/3)/x)

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giac [A]  time = 0.22, size = 23, normalized size = 0.88 \begin {gather*} e^{\left (-2 \, x + \frac {\log \left (\log \left (x^{2}\right )\right )}{x} - \frac {2}{x} + \frac {4}{3} \, e^{\left (-\frac {2}{3}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x^2)*log(log(x^2))+(-2*x^2+2)*log(x^2)+2)*exp(1/3*(3*exp(2/3)*log(log(x^2))+(-6*x^2-6)*exp(2/3
)+4*x)/x/exp(2/3))/x^2/log(x^2),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (-\ln \left (x^{2}\right ) \ln \left (\ln \left (x^{2}\right )\right )+\left (-2 x^{2}+2\right ) \ln \left (x^{2}\right )+2\right ) {\mathrm e}^{\frac {\left (3 \,{\mathrm e}^{\frac {2}{3}} \ln \left (\ln \left (x^{2}\right )\right )+\left (-6 x^{2}-6\right ) {\mathrm e}^{\frac {2}{3}}+4 x \right ) {\mathrm e}^{-\frac {2}{3}}}{3 x}}}{x^{2} \ln \left (x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-ln(x^2)*ln(ln(x^2))+(-2*x^2+2)*ln(x^2)+2)*exp(1/3*(3*exp(2/3)*ln(ln(x^2))+(-6*x^2-6)*exp(2/3)+4*x)/x/exp
(2/3))/x^2/ln(x^2),x)

[Out]

int((-ln(x^2)*ln(ln(x^2))+(-2*x^2+2)*ln(x^2)+2)*exp(1/3*(3*exp(2/3)*ln(ln(x^2))+(-6*x^2-6)*exp(2/3)+4*x)/x/exp
(2/3))/x^2/ln(x^2),x)

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maxima [A]  time = 0.72, size = 27, normalized size = 1.04 \begin {gather*} e^{\left (-2 \, x + \frac {\log \relax (2)}{x} + \frac {\log \left (\log \relax (x)\right )}{x} - \frac {2}{x} + \frac {4}{3} \, e^{\left (-\frac {2}{3}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x^2)*log(log(x^2))+(-2*x^2+2)*log(x^2)+2)*exp(1/3*(3*exp(2/3)*log(log(x^2))+(-6*x^2-6)*exp(2/3
)+4*x)/x/exp(2/3))/x^2/log(x^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.69, size = 24, normalized size = 0.92 \begin {gather*} {\ln \left (x^2\right )}^{1/x}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{-\frac {2}{3}}}{3}}\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-\frac {2}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.49, size = 39, normalized size = 1.50 \begin {gather*} e^{\frac {\frac {4 x}{3} + \frac {\left (- 6 x^{2} - 6\right ) e^{\frac {2}{3}}}{3} + e^{\frac {2}{3}} \log {\left (\log {\left (x^{2} \right )} \right )}}{x e^{\frac {2}{3}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-ln(x**2)*ln(ln(x**2))+(-2*x**2+2)*ln(x**2)+2)*exp(1/3*(3*exp(2/3)*ln(ln(x**2))+(-6*x**2-6)*exp(2/3
)+4*x)/x/exp(2/3))/x**2/ln(x**2),x)

[Out]

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

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