∫ e5 _3 +x−log(x2) x x+x log(x)+(x log( 5x)+ex−log(x2) _____________x (−2e5∕3 log( 5 x)+e5∕3 log( 5 x) log(x2))) log(log( 5 x)) ______________________________________________________________________________________________________________________________

Optimal. Leaf size=31 \[ \frac {e^{\frac {5}{3}+\frac {x-\log \left (x^2\right )}{x}}+\log (x)}{\log \left (\log \left (\frac {5}{x}\right )\right )} \]

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Rubi [F] time = 2.44, 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 {5}{3}+\frac {x-\log \left (x^2\right )}{x}} x+x \log (x)+\left (x \log \left (\frac {5}{x}\right )+e^{\frac {x-\log \left (x^2\right )}{x}} \left (-2 e^{5/3} \log \left (\frac {5}{x}\right )+e^{5/3} \log \left (\frac {5}{x}\right ) \log \left (x^2\right )\right )\right ) \log \left (\log \left (\frac {5}{x}\right )\right )}{x^2 \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx \end {gather*}

Int[(E^(5/3 + (x - Log[x^2])/x)*x + x*Log[x] + (x*Log[5/x] + E^((x - Log[x^2])/x)*(-2*E^(5/3)*Log[5/x] + E^(5/

-LogIntegral[Log[5/x]] + E^(8/3)*Defer[Int][(x*(x^2)^(-1 - x^(-1)))/(Log[5/x]*Log[Log[5/x]]^2), x] + Defer[Int

][Log[x]/(x*Log[5/x]*Log[Log[5/x]]^2), x] - 2*E^(8/3)*Defer[Int][(x^2)^(-1 - x^(-1))/Log[Log[5/x]], x] + E^(8/

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {\log (x)+\log \left (\frac {5}{x}\right ) \log \left (\log \left (\frac {5}{x}\right )\right )}{x \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )}+\frac {e^{8/3} \left (x^2\right )^{-1-\frac {1}{x}} \left (x-2 \log \left (\frac {5}{x}\right ) \log \left (\log \left (\frac {5}{x}\right )\right )+\log \left (\frac {5}{x}\right ) \log \left (x^2\right ) \log \left (\log \left (\frac {5}{x}\right )\right )\right )}{\log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )}\right ) \, dx\\ &=e^{8/3} \int \frac {\left (x^2\right )^{-1-\frac {1}{x}} \left (x-2 \log \left (\frac {5}{x}\right ) \log \left (\log \left (\frac {5}{x}\right )\right )+\log \left (\frac {5}{x}\right ) \log \left (x^2\right ) \log \left (\log \left (\frac {5}{x}\right )\right )\right )}{\log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx+\int \frac {\log (x)+\log \left (\frac {5}{x}\right ) \log \left (\log \left (\frac {5}{x}\right )\right )}{x \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx\\ &=e^{8/3} \int \frac {\left (x^2\right )^{-1-\frac {1}{x}} \left (x+\log \left (\frac {5}{x}\right ) \left (-2+\log \left (x^2\right )\right ) \log \left (\log \left (\frac {5}{x}\right )\right )\right )}{\log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx+\int \left (\frac {\log (x)}{x \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )}+\frac {1}{x \log \left (\log \left (\frac {5}{x}\right )\right )}\right ) \, dx\\ &=e^{8/3} \int \left (\frac {x \left (x^2\right )^{-1-\frac {1}{x}}}{\log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )}+\frac {\left (x^2\right )^{-1-\frac {1}{x}} \left (-2+\log \left (x^2\right )\right )}{\log \left (\log \left (\frac {5}{x}\right )\right )}\right ) \, dx+\int \frac {\log (x)}{x \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx+\int \frac {1}{x \log \left (\log \left (\frac {5}{x}\right )\right )} \, dx\\ &=e^{8/3} \int \frac {x \left (x^2\right )^{-1-\frac {1}{x}}}{\log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx+e^{8/3} \int \frac {\left (x^2\right )^{-1-\frac {1}{x}} \left (-2+\log \left (x^2\right )\right )}{\log \left (\log \left (\frac {5}{x}\right )\right )} \, dx+\int \frac {\log (x)}{x \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx-\operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,\log \left (\frac {5}{x}\right )\right )\\ &=-\text {li}\left (\log \left (\frac {5}{x}\right )\right )+e^{8/3} \int \left (-\frac {2 \left (x^2\right )^{-1-\frac {1}{x}}}{\log \left (\log \left (\frac {5}{x}\right )\right )}+\frac {\left (x^2\right )^{-1-\frac {1}{x}} \log \left (x^2\right )}{\log \left (\log \left (\frac {5}{x}\right )\right )}\right ) \, dx+e^{8/3} \int \frac {x \left (x^2\right )^{-1-\frac {1}{x}}}{\log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx+\int \frac {\log (x)}{x \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx\\ &=-\text {li}\left (\log \left (\frac {5}{x}\right )\right )+e^{8/3} \int \frac {x \left (x^2\right )^{-1-\frac {1}{x}}}{\log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx+e^{8/3} \int \frac {\left (x^2\right )^{-1-\frac {1}{x}} \log \left (x^2\right )}{\log \left (\log \left (\frac {5}{x}\right )\right )} \, dx-\left (2 e^{8/3}\right ) \int \frac {\left (x^2\right )^{-1-\frac {1}{x}}}{\log \left (\log \left (\frac {5}{x}\right )\right )} \, dx+\int \frac {\log (x)}{x \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.12, size = 28, normalized size = 0.90 \begin {gather*} \frac {e^{8/3} \left (x^2\right )^{-1/x}+\log (x)}{\log \left (\log \left (\frac {5}{x}\right )\right )} \end {gather*}

Integrate[(E^(5/3 + (x - Log[x^2])/x)*x + x*Log[x] + (x*Log[5/x] + E^((x - Log[x^2])/x)*(-2*E^(5/3)*Log[5/x] +

E^(5/3)*Log[5/x]*Log[x^2]))*Log[Log[5/x]])/(x^2*Log[5/x]*Log[Log[5/x]]^2),x]

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fricas [A] time = 0.73, size = 43, normalized size = 1.39 \begin {gather*} \frac {e^{\left (\frac {2 \, {\left (4 \, x - 3 \, \log \relax (5) + 3 \, \log \left (\frac {5}{x}\right )\right )}}{3 \, x}\right )} + \log \relax (5) - \log \left (\frac {5}{x}\right )}{\log \left (\log \left (\frac {5}{x}\right )\right )} \end {gather*}

integrate((((exp(5/3)*log(5/x)*log(x^2)-2*exp(5/3)*log(5/x))*exp((-log(x^2)+x)/x)+x*log(5/x))*log(log(5/x))+x*

exp(5/3)*exp((-log(x^2)+x)/x)+x*log(x))/x^2/log(5/x)/log(log(5/x))^2,x, algorithm="fricas")

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giac [A] time = 0.71, size = 36, normalized size = 1.16 \begin {gather*} \frac {\log \relax (x)}{\log \left (\log \relax (5) - \log \relax (x)\right )} + \frac {e^{\frac {8}{3}}}{x^{\frac {2}{x}} \log \left (\log \relax (5) - \log \relax (x)\right )} \end {gather*}

integrate((((exp(5/3)*log(5/x)*log(x^2)-2*exp(5/3)*log(5/x))*exp((-log(x^2)+x)/x)+x*log(5/x))*log(log(5/x))+x*

exp(5/3)*exp((-log(x^2)+x)/x)+x*log(x))/x^2/log(5/x)/log(log(5/x))^2,x, algorithm="giac")

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method	result	size

risch	\(\frac {{\mathrm e}^{-\frac {-3 i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+6 i \pi \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )-3 i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}+12 \ln \relax (x )-16 x}{6 x}}+\ln \relax (x )}{\ln \left (\ln \relax (5)-\ln \relax (x )\right )}\)	\(78\)

int((((exp(5/3)*ln(5/x)*ln(x^2)-2*exp(5/3)*ln(5/x))*exp((-ln(x^2)+x)/x)+x*ln(5/x))*ln(ln(5/x))+x*exp(5/3)*exp(

(exp(-1/6*(-3*I*Pi*csgn(I*x^2)^3+6*I*Pi*csgn(I*x^2)^2*csgn(I*x)-3*I*Pi*csgn(I*x^2)*csgn(I*x)^2+12*ln(x)-16*x)/

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maxima [A] time = 0.66, size = 33, normalized size = 1.06 \begin {gather*} \frac {x^{\frac {2}{x}} \log \relax (x) + e^{\frac {8}{3}}}{x^{\frac {2}{x}} \log \left (\log \relax (5) - \log \relax (x)\right )} \end {gather*}

integrate((((exp(5/3)*log(5/x)*log(x^2)-2*exp(5/3)*log(5/x))*exp((-log(x^2)+x)/x)+x*log(5/x))*log(log(5/x))+x*

exp(5/3)*exp((-log(x^2)+x)/x)+x*log(x))/x^2/log(5/x)/log(log(5/x))^2,x, algorithm="maxima")

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mupad [B] time = 0.86, size = 42, normalized size = 1.35 \begin {gather*} \ln \left (\frac {1}{x}\right )+\ln \relax (x)+\frac {\ln \relax (x)}{\ln \left (\ln \left (\frac {1}{x}\right )+\ln \relax (5)\right )}+\frac {{\mathrm {e}}^{8/3}}{\ln \left (\ln \left (\frac {1}{x}\right )+\ln \relax (5)\right )\,{\left (x^2\right )}^{1/x}} \end {gather*}

int((log(log(5/x))*(x*log(5/x) - exp((x - log(x^2))/x)*(2*exp(5/3)*log(5/x) - log(x^2)*exp(5/3)*log(5/x))) + x

*log(x) + x*exp((x - log(x^2))/x)*exp(5/3))/(x^2*log(log(5/x))^2*log(5/x)),x)

log(1/x) + log(x) + log(x)/log(log(1/x) + log(5)) + exp(8/3)/(log(log(1/x) + log(5))*(x^2)^(1/x))

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sympy [A] time = 0.51, size = 34, normalized size = 1.10 \begin {gather*} \frac {e^{\frac {5}{3}} e^{\frac {x - 2 \log {\relax (x )}}{x}}}{\log {\left (- \log {\relax (x )} + \log {\relax (5 )} \right )}} + \frac {\log {\relax (x )}}{\log {\left (- \log {\relax (x )} + \log {\relax (5 )} \right )}} \end {gather*}

integrate((((exp(5/3)*ln(5/x)*ln(x**2)-2*exp(5/3)*ln(5/x))*exp((-ln(x**2)+x)/x)+x*ln(5/x))*ln(ln(5/x))+x*exp(5

exp(5/3)*exp((x - 2*log(x))/x)/log(-log(x) + log(5)) + log(x)/log(-log(x) + log(5))

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3.7.4 \(\int \frac {e^{\frac {5}{3}+\frac {x-\log (x^2)}{x}} x+x \log (x)+(x \log (\frac {5}{x})+e^{\frac {x-\log (x^2)}{x}} (-2 e^{5/3} \log (\frac {5}{x})+e^{5/3} \log (\frac {5}{x}) \log (x^2))) \log (\log (\frac {5}{x}))}{x^2 \log (\frac {5}{x}) \log ^2(\log (\frac {5}{x}))} \, dx\)