Optimal. Leaf size=20 \[ \frac {1+\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x} \]
________________________________________________________________________________________
Rubi [F] time = 1.46, 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 {-6-6 \log (x)+\left (2+3 x^2\right ) \log ^2(x)+\left (-6 \log (x)+\left (2-3 x^2\right ) \log ^2(x)\right ) \log \left (\frac {6+\left (-2+3 x^2\right ) \log (x)}{3 \log (x)}\right )}{6 x^2 \log (x)+\left (-2 x^2+3 x^4\right ) \log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-6-6 \log (x)+\left (2+3 x^2\right ) \log ^2(x)+\left (-6 \log (x)+\left (2-3 x^2\right ) \log ^2(x)\right ) \log \left (\frac {6+\left (-2+3 x^2\right ) \log (x)}{3 \log (x)}\right )}{x^2 \log (x) \left (6-2 \log (x)+3 x^2 \log (x)\right )} \, dx\\ &=\int \left (\frac {-6-6 \log (x)+2 \log ^2(x)+3 x^2 \log ^2(x)}{x^2 \log (x) \left (6-2 \log (x)+3 x^2 \log (x)\right )}-\frac {\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x^2}\right ) \, dx\\ &=\int \frac {-6-6 \log (x)+2 \log ^2(x)+3 x^2 \log ^2(x)}{x^2 \log (x) \left (6-2 \log (x)+3 x^2 \log (x)\right )} \, dx-\int \frac {\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x^2} \, dx\\ &=\int \left (\frac {2+3 x^2}{x^2 \left (-2+3 x^2\right )}-\frac {1}{x^2 \log (x)}-\frac {36}{\left (-2+3 x^2\right ) \left (6-2 \log (x)+3 x^2 \log (x)\right )}+\frac {-2+3 x^2}{x^2 \left (6-2 \log (x)+3 x^2 \log (x)\right )}\right ) \, dx-\int \frac {\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x^2} \, dx\\ &=-\left (36 \int \frac {1}{\left (-2+3 x^2\right ) \left (6-2 \log (x)+3 x^2 \log (x)\right )} \, dx\right )+\int \frac {2+3 x^2}{x^2 \left (-2+3 x^2\right )} \, dx-\int \frac {1}{x^2 \log (x)} \, dx+\int \frac {-2+3 x^2}{x^2 \left (6-2 \log (x)+3 x^2 \log (x)\right )} \, dx-\int \frac {\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x^2} \, dx\\ &=\frac {1}{x}+6 \int \frac {1}{-2+3 x^2} \, dx-36 \int \left (-\frac {1}{2 \sqrt {2} \left (\sqrt {2}-\sqrt {3} x\right ) \left (6-2 \log (x)+3 x^2 \log (x)\right )}-\frac {1}{2 \sqrt {2} \left (\sqrt {2}+\sqrt {3} x\right ) \left (6-2 \log (x)+3 x^2 \log (x)\right )}\right ) \, dx+\int \left (\frac {3}{6-2 \log (x)+3 x^2 \log (x)}-\frac {2}{x^2 \left (6-2 \log (x)+3 x^2 \log (x)\right )}\right ) \, dx-\int \frac {\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x^2} \, dx-\operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )\\ &=\frac {1}{x}-\sqrt {6} \tanh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\text {Ei}(-\log (x))-2 \int \frac {1}{x^2 \left (6-2 \log (x)+3 x^2 \log (x)\right )} \, dx+3 \int \frac {1}{6-2 \log (x)+3 x^2 \log (x)} \, dx+\left (9 \sqrt {2}\right ) \int \frac {1}{\left (\sqrt {2}-\sqrt {3} x\right ) \left (6-2 \log (x)+3 x^2 \log (x)\right )} \, dx+\left (9 \sqrt {2}\right ) \int \frac {1}{\left (\sqrt {2}+\sqrt {3} x\right ) \left (6-2 \log (x)+3 x^2 \log (x)\right )} \, dx-\int \frac {\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x^2} \, dx\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.29, size = 22, normalized size = 1.10 \begin {gather*} \frac {1}{x}+\frac {\log \left (-\frac {2}{3}+x^2+\frac {2}{\log (x)}\right )}{x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.54, size = 25, normalized size = 1.25 \begin {gather*} \frac {\log \left (\frac {{\left (3 \, x^{2} - 2\right )} \log \relax (x) + 6}{3 \, \log \relax (x)}\right ) + 1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.22, size = 32, normalized size = 1.60 \begin {gather*} \frac {\log \left (3 \, x^{2} \log \relax (x) - 2 \, \log \relax (x) + 6\right )}{x} - \frac {\log \left (3 \, \log \relax (x)\right )}{x} + \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.11, size = 181, normalized size = 9.05
method | result | size |
risch | \(\frac {\ln \left (x^{2} \ln \relax (x )-\frac {2 \ln \relax (x )}{3}+2\right )}{x}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (i \left (x^{2} \ln \relax (x )-\frac {2 \ln \relax (x )}{3}+2\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )-\frac {2 \ln \relax (x )}{3}+2\right )}{\ln \relax (x )}\right )-i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )-\frac {2 \ln \relax (x )}{3}+2\right )}{\ln \relax (x )}\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left (x^{2} \ln \relax (x )-\frac {2 \ln \relax (x )}{3}+2\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )-\frac {2 \ln \relax (x )}{3}+2\right )}{\ln \relax (x )}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i \left (x^{2} \ln \relax (x )-\frac {2 \ln \relax (x )}{3}+2\right )}{\ln \relax (x )}\right )^{3}+2 \ln \left (\ln \relax (x )\right )-2}{2 x}\) | \(181\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.49, size = 27, normalized size = 1.35 \begin {gather*} -\frac {\log \relax (3) - \log \left ({\left (3 \, x^{2} - 2\right )} \log \relax (x) + 6\right ) + \log \left (\log \relax (x)\right ) - 1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 10.79, size = 25, normalized size = 1.25 \begin {gather*} \frac {\ln \left (\frac {\frac {\ln \relax (x)\,\left (3\,x^2-2\right )}{3}+2}{\ln \relax (x)}\right )+1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.47, size = 22, normalized size = 1.10 \begin {gather*} \frac {\log {\left (\frac {\frac {\left (3 x^{2} - 2\right ) \log {\relax (x )}}{3} + 2}{\log {\relax (x )}} \right )}}{x} + \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________