Optimal. Leaf size=31 \[ \frac {e^{\frac {5}{3}+\frac {x-\log \left (x^2\right )}{x}}+\log (x)}{\log \left (\log \left (\frac {5}{x}\right )\right )} \]
________________________________________________________________________________________
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*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\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*}
________________________________________________________________________________________
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*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.87, size = 78, normalized size = 2.52
method | result | size |
risch | \(\frac {{\mathrm e}^{-\frac {-3 i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+6 i \pi \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )-3 i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}+12 \ln \relax (x )-16 x}{6 x}}+\ln \relax (x )}{\ln \left (\ln \relax (5)-\ln \relax (x )\right )}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________