Optimal. Leaf size=22 \[ \frac {3 (x-\log (\log (4)))}{1+\frac {3 x}{\log \left (\frac {1}{x}\right )}} \]
________________________________________________________________________________________
Rubi [F] time = 0.40, 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 {-9 x+3 \log ^2\left (\frac {1}{x}\right )+\left (9+9 \log \left (\frac {1}{x}\right )\right ) \log (\log (4))}{9 x^2+6 x \log \left (\frac {1}{x}\right )+\log ^2\left (\frac {1}{x}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-9 x+3 \log ^2\left (\frac {1}{x}\right )+\left (9+9 \log \left (\frac {1}{x}\right )\right ) \log (\log (4))}{\left (3 x+\log \left (\frac {1}{x}\right )\right )^2} \, dx\\ &=\int \left (3+\frac {9 (-1+3 x) (x-\log (\log (4)))}{\left (3 x+\log \left (\frac {1}{x}\right )\right )^2}-\frac {9 (2 x-\log (\log (4)))}{3 x+\log \left (\frac {1}{x}\right )}\right ) \, dx\\ &=3 x+9 \int \frac {(-1+3 x) (x-\log (\log (4)))}{\left (3 x+\log \left (\frac {1}{x}\right )\right )^2} \, dx-9 \int \frac {2 x-\log (\log (4))}{3 x+\log \left (\frac {1}{x}\right )} \, dx\\ &=3 x-9 \int \left (\frac {2 x}{3 x+\log \left (\frac {1}{x}\right )}-\frac {\log (\log (4))}{3 x+\log \left (\frac {1}{x}\right )}\right ) \, dx+9 \int \left (\frac {3 x^2}{\left (3 x+\log \left (\frac {1}{x}\right )\right )^2}+\frac {\log (\log (4))}{\left (3 x+\log \left (\frac {1}{x}\right )\right )^2}-\frac {x (1+3 \log (\log (4)))}{\left (3 x+\log \left (\frac {1}{x}\right )\right )^2}\right ) \, dx\\ &=3 x-18 \int \frac {x}{3 x+\log \left (\frac {1}{x}\right )} \, dx+27 \int \frac {x^2}{\left (3 x+\log \left (\frac {1}{x}\right )\right )^2} \, dx+(9 \log (\log (4))) \int \frac {1}{\left (3 x+\log \left (\frac {1}{x}\right )\right )^2} \, dx+(9 \log (\log (4))) \int \frac {1}{3 x+\log \left (\frac {1}{x}\right )} \, dx-(9 (1+3 \log (\log (4)))) \int \frac {x}{\left (3 x+\log \left (\frac {1}{x}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 24, normalized size = 1.09 \begin {gather*} 3 x-\frac {9 x (x-\log (\log (4)))}{3 x+\log \left (\frac {1}{x}\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.52, size = 27, normalized size = 1.23 \begin {gather*} \frac {3 \, {\left (3 \, x \log \left (2 \, \log \relax (2)\right ) + x \log \left (\frac {1}{x}\right )\right )}}{3 \, x + \log \left (\frac {1}{x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.14, size = 31, normalized size = 1.41 \begin {gather*} 3 \, x - \frac {9 \, {\left (x^{2} - x \log \relax (2) - x \log \left (\log \relax (2)\right )\right )}}{3 \, x - \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.12, size = 27, normalized size = 1.23
method | result | size |
risch | \(3 x +\frac {9 \left (-x +\ln \relax (2)+\ln \left (\ln \relax (2)\right )\right ) x}{\ln \left (\frac {1}{x}\right )+3 x}\) | \(27\) |
norman | \(\frac {\left (-3 \ln \relax (2)-3 \ln \left (\ln \relax (2)\right )\right ) \ln \left (\frac {1}{x}\right )+3 x \ln \left (\frac {1}{x}\right )}{\ln \left (\frac {1}{x}\right )+3 x}\) | \(35\) |
derivativedivides | \(\frac {3 \ln \left (\frac {1}{x}\right )}{\frac {\ln \left (\frac {1}{x}\right )}{x}+3}+\frac {9 \ln \relax (2)}{\frac {\ln \left (\frac {1}{x}\right )}{x}+3}+\frac {9 \ln \left (\ln \relax (2)\right )}{\frac {\ln \left (\frac {1}{x}\right )}{x}+3}\) | \(53\) |
default | \(\frac {3 \ln \left (\frac {1}{x}\right )}{\frac {\ln \left (\frac {1}{x}\right )}{x}+3}+\frac {9 \ln \relax (2)}{\frac {\ln \left (\frac {1}{x}\right )}{x}+3}+\frac {9 \ln \left (\ln \relax (2)\right )}{\frac {\ln \left (\frac {1}{x}\right )}{x}+3}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.49, size = 27, normalized size = 1.23 \begin {gather*} \frac {3 \, {\left (3 \, x {\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} - x \log \relax (x)\right )}}{3 \, x - \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.13, size = 28, normalized size = 1.27 \begin {gather*} 3\,x+\frac {x\,\ln \left ({\ln \relax (4)}^9\right )-9\,x^2}{3\,x+\ln \left (\frac {1}{x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.12, size = 31, normalized size = 1.41 \begin {gather*} 3 x + \frac {- 9 x^{2} + 9 x \log {\left (\log {\relax (2 )} \right )} + 9 x \log {\relax (2 )}}{3 x + \log {\left (\frac {1}{x} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________