3.28.76 \(\int \frac {(-4+e^{e^{x^2}} (4 x+(4 x+8 e^{x^2} x^3) \log (x))) \log ^3(\frac {1}{4} (25 \log (x)-25 e^{e^{x^2}} x \log (x)))}{-x \log (x)+e^{e^{x^2}} x^2 \log (x)} \, dx\)

Optimal. Leaf size=21 \[ \log ^4\left (\frac {5}{4} \left (5-5 e^{e^{x^2}} x\right ) \log (x)\right ) \]

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Rubi [F]  time = 5.97, 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 {\left (-4+e^{e^{x^2}} \left (4 x+\left (4 x+8 e^{x^2} x^3\right ) \log (x)\right )\right ) \log ^3\left (\frac {1}{4} \left (25 \log (x)-25 e^{e^{x^2}} x \log (x)\right )\right )}{-x \log (x)+e^{e^{x^2}} x^2 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-4 + E^E^x^2*(4*x + (4*x + 8*E^x^2*x^3)*Log[x]))*Log[(25*Log[x] - 25*E^E^x^2*x*Log[x])/4]^3)/(-(x*Log[x]
) + E^E^x^2*x^2*Log[x]),x]

[Out]

4*Defer[Int][Log[(-25*(-1 + E^E^x^2*x)*Log[x])/4]^3/x, x] + 4*Defer[Int][Log[(-25*(-1 + E^E^x^2*x)*Log[x])/4]^
3/(x*(-1 + E^E^x^2*x)), x] + 8*Defer[Int][(E^(E^x^2 + x^2)*x^2*Log[(-25*(-1 + E^E^x^2*x)*Log[x])/4]^3)/(-1 + E
^E^x^2*x), x] + 4*Defer[Int][Log[(-25*(-1 + E^E^x^2*x)*Log[x])/4]^3/(x*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (4-e^{e^{x^2}} \left (4 x+\left (4 x+8 e^{x^2} x^3\right ) \log (x)\right )\right ) \log ^3\left (-\frac {25}{4} \left (-1+e^{e^{x^2}} x\right ) \log (x)\right )}{x \left (1-e^{e^{x^2}} x\right ) \log (x)} \, dx\\ &=\int \left (\frac {8 e^{e^{x^2}+x^2} x^2 \log ^3\left (-\frac {25}{4} \left (-1+e^{e^{x^2}} x\right ) \log (x)\right )}{-1+e^{e^{x^2}} x}+\frac {4 \left (-1+e^{e^{x^2}} x+e^{e^{x^2}} x \log (x)\right ) \log ^3\left (-\frac {25}{4} \left (-1+e^{e^{x^2}} x\right ) \log (x)\right )}{x \left (-1+e^{e^{x^2}} x\right ) \log (x)}\right ) \, dx\\ &=4 \int \frac {\left (-1+e^{e^{x^2}} x+e^{e^{x^2}} x \log (x)\right ) \log ^3\left (-\frac {25}{4} \left (-1+e^{e^{x^2}} x\right ) \log (x)\right )}{x \left (-1+e^{e^{x^2}} x\right ) \log (x)} \, dx+8 \int \frac {e^{e^{x^2}+x^2} x^2 \log ^3\left (-\frac {25}{4} \left (-1+e^{e^{x^2}} x\right ) \log (x)\right )}{-1+e^{e^{x^2}} x} \, dx\\ &=4 \int \left (\frac {\log ^3\left (-\frac {25}{4} \left (-1+e^{e^{x^2}} x\right ) \log (x)\right )}{x \left (-1+e^{e^{x^2}} x\right )}+\frac {(1+\log (x)) \log ^3\left (-\frac {25}{4} \left (-1+e^{e^{x^2}} x\right ) \log (x)\right )}{x \log (x)}\right ) \, dx+8 \int \frac {e^{e^{x^2}+x^2} x^2 \log ^3\left (-\frac {25}{4} \left (-1+e^{e^{x^2}} x\right ) \log (x)\right )}{-1+e^{e^{x^2}} x} \, dx\\ &=4 \int \frac {\log ^3\left (-\frac {25}{4} \left (-1+e^{e^{x^2}} x\right ) \log (x)\right )}{x \left (-1+e^{e^{x^2}} x\right )} \, dx+4 \int \frac {(1+\log (x)) \log ^3\left (-\frac {25}{4} \left (-1+e^{e^{x^2}} x\right ) \log (x)\right )}{x \log (x)} \, dx+8 \int \frac {e^{e^{x^2}+x^2} x^2 \log ^3\left (-\frac {25}{4} \left (-1+e^{e^{x^2}} x\right ) \log (x)\right )}{-1+e^{e^{x^2}} x} \, dx\\ &=4 \int \frac {\log ^3\left (-\frac {25}{4} \left (-1+e^{e^{x^2}} x\right ) \log (x)\right )}{x \left (-1+e^{e^{x^2}} x\right )} \, dx+4 \int \left (\frac {\log ^3\left (-\frac {25}{4} \left (-1+e^{e^{x^2}} x\right ) \log (x)\right )}{x}+\frac {\log ^3\left (-\frac {25}{4} \left (-1+e^{e^{x^2}} x\right ) \log (x)\right )}{x \log (x)}\right ) \, dx+8 \int \frac {e^{e^{x^2}+x^2} x^2 \log ^3\left (-\frac {25}{4} \left (-1+e^{e^{x^2}} x\right ) \log (x)\right )}{-1+e^{e^{x^2}} x} \, dx\\ &=4 \int \frac {\log ^3\left (-\frac {25}{4} \left (-1+e^{e^{x^2}} x\right ) \log (x)\right )}{x} \, dx+4 \int \frac {\log ^3\left (-\frac {25}{4} \left (-1+e^{e^{x^2}} x\right ) \log (x)\right )}{x \left (-1+e^{e^{x^2}} x\right )} \, dx+4 \int \frac {\log ^3\left (-\frac {25}{4} \left (-1+e^{e^{x^2}} x\right ) \log (x)\right )}{x \log (x)} \, dx+8 \int \frac {e^{e^{x^2}+x^2} x^2 \log ^3\left (-\frac {25}{4} \left (-1+e^{e^{x^2}} x\right ) \log (x)\right )}{-1+e^{e^{x^2}} x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.47, size = 20, normalized size = 0.95 \begin {gather*} \log ^4\left (-\frac {25}{4} \left (-1+e^{e^{x^2}} x\right ) \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-4 + E^E^x^2*(4*x + (4*x + 8*E^x^2*x^3)*Log[x]))*Log[(25*Log[x] - 25*E^E^x^2*x*Log[x])/4]^3)/(-(x*
Log[x]) + E^E^x^2*x^2*Log[x]),x]

[Out]

Log[(-25*(-1 + E^E^x^2*x)*Log[x])/4]^4

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fricas [A]  time = 0.52, size = 18, normalized size = 0.86 \begin {gather*} \log \left (-\frac {25}{4} \, x e^{\left (e^{\left (x^{2}\right )}\right )} \log \relax (x) + \frac {25}{4} \, \log \relax (x)\right )^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^3*exp(x^2)+4*x)*log(x)+4*x)*exp(exp(x^2))-4)*log(-25/4*x*log(x)*exp(exp(x^2))+25/4*log(x))^3/
(x^2*log(x)*exp(exp(x^2))-x*log(x)),x, algorithm="fricas")

[Out]

log(-25/4*x*e^(e^(x^2))*log(x) + 25/4*log(x))^4

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 \, {\left ({\left ({\left (2 \, x^{3} e^{\left (x^{2}\right )} + x\right )} \log \relax (x) + x\right )} e^{\left (e^{\left (x^{2}\right )}\right )} - 1\right )} \log \left (-\frac {25}{4} \, x e^{\left (e^{\left (x^{2}\right )}\right )} \log \relax (x) + \frac {25}{4} \, \log \relax (x)\right )^{3}}{x^{2} e^{\left (e^{\left (x^{2}\right )}\right )} \log \relax (x) - x \log \relax (x)}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^3*exp(x^2)+4*x)*log(x)+4*x)*exp(exp(x^2))-4)*log(-25/4*x*log(x)*exp(exp(x^2))+25/4*log(x))^3/
(x^2*log(x)*exp(exp(x^2))-x*log(x)),x, algorithm="giac")

[Out]

integrate(4*(((2*x^3*e^(x^2) + x)*log(x) + x)*e^(e^(x^2)) - 1)*log(-25/4*x*e^(e^(x^2))*log(x) + 25/4*log(x))^3
/(x^2*e^(e^(x^2))*log(x) - x*log(x)), x)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (\left (8 x^{3} {\mathrm e}^{x^{2}}+4 x \right ) \ln \relax (x )+4 x \right ) {\mathrm e}^{{\mathrm e}^{x^{2}}}-4\right ) \ln \left (-\frac {25 x \ln \relax (x ) {\mathrm e}^{{\mathrm e}^{x^{2}}}}{4}+\frac {25 \ln \relax (x )}{4}\right )^{3}}{x^{2} \ln \relax (x ) {\mathrm e}^{{\mathrm e}^{x^{2}}}-x \ln \relax (x )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((8*x^3*exp(x^2)+4*x)*ln(x)+4*x)*exp(exp(x^2))-4)*ln(-25/4*x*ln(x)*exp(exp(x^2))+25/4*ln(x))^3/(x^2*ln(x)
*exp(exp(x^2))-x*ln(x)),x)

[Out]

int((((8*x^3*exp(x^2)+4*x)*ln(x)+4*x)*exp(exp(x^2))-4)*ln(-25/4*x*ln(x)*exp(exp(x^2))+25/4*ln(x))^3/(x^2*ln(x)
*exp(exp(x^2))-x*ln(x)),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 4 \, \int \frac {{\left ({\left ({\left (2 \, x^{3} e^{\left (x^{2}\right )} + x\right )} \log \relax (x) + x\right )} e^{\left (e^{\left (x^{2}\right )}\right )} - 1\right )} \log \left (-\frac {25}{4} \, x e^{\left (e^{\left (x^{2}\right )}\right )} \log \relax (x) + \frac {25}{4} \, \log \relax (x)\right )^{3}}{x^{2} e^{\left (e^{\left (x^{2}\right )}\right )} \log \relax (x) - x \log \relax (x)}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x^3*exp(x^2)+4*x)*log(x)+4*x)*exp(exp(x^2))-4)*log(-25/4*x*log(x)*exp(exp(x^2))+25/4*log(x))^3/
(x^2*log(x)*exp(exp(x^2))-x*log(x)),x, algorithm="maxima")

[Out]

4*integrate((((2*x^3*e^(x^2) + x)*log(x) + x)*e^(e^(x^2)) - 1)*log(-25/4*x*e^(e^(x^2))*log(x) + 25/4*log(x))^3
/(x^2*e^(e^(x^2))*log(x) - x*log(x)), x)

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mupad [B]  time = 2.65, size = 19, normalized size = 0.90 \begin {gather*} {\left (\ln \left (\ln \relax (x)-x\,{\mathrm {e}}^{{\mathrm {e}}^{x^2}}\,\ln \relax (x)\right )+\ln \left (\frac {25}{4}\right )\right )}^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log((25*log(x))/4 - (25*x*exp(exp(x^2))*log(x))/4)^3*(exp(exp(x^2))*(4*x + log(x)*(4*x + 8*x^3*exp(x^2))
) - 4))/(x*log(x) - x^2*exp(exp(x^2))*log(x)),x)

[Out]

(log(log(x) - x*exp(exp(x^2))*log(x)) + log(25/4))^4

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sympy [A]  time = 3.40, size = 24, normalized size = 1.14 \begin {gather*} \log {\left (- \frac {25 x e^{e^{x^{2}}} \log {\relax (x )}}{4} + \frac {25 \log {\relax (x )}}{4} \right )}^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x**3*exp(x**2)+4*x)*ln(x)+4*x)*exp(exp(x**2))-4)*ln(-25/4*x*ln(x)*exp(exp(x**2))+25/4*ln(x))**3
/(x**2*ln(x)*exp(exp(x**2))-x*ln(x)),x)

[Out]

log(-25*x*exp(exp(x**2))*log(x)/4 + 25*log(x)/4)**4

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