Optimal. Leaf size=27 \[ 4 \left (1-x+\frac {1}{4} \log \left (\frac {4}{3}\right ) \left (-x+\frac {\log (x)}{x}\right )^2\right ) \]
________________________________________________________________________________________
Rubi [B] time = 0.06, antiderivative size = 80, normalized size of antiderivative = 2.96, number of steps used = 7, number of rules used = 3, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {14, 2304, 2305} \begin {gather*} \frac {\log \left (\frac {4}{3}\right ) \log ^2(x)}{x^2}+\frac {1}{2} x^2 \log \left (\frac {16}{9}\right )-\frac {\log \left (\frac {16}{9}\right ) \log (x)}{2 x^2}+\frac {\log \left (\frac {4}{3}\right ) \log (x)}{x^2}-\frac {\log \left (\frac {16}{9}\right )}{4 x^2}+\frac {\log \left (\frac {4}{3}\right )}{2 x^2}-4 x-2 \log \left (\frac {4}{3}\right ) \log (x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 2304
Rule 2305
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-4 x-2 \log \left (\frac {4}{3}\right )+x^2 \log \left (\frac {16}{9}\right )}{x}+\frac {\log \left (\frac {16}{9}\right ) \log (x)}{x^3}-\frac {2 \log \left (\frac {4}{3}\right ) \log ^2(x)}{x^3}\right ) \, dx\\ &=-\left (\left (2 \log \left (\frac {4}{3}\right )\right ) \int \frac {\log ^2(x)}{x^3} \, dx\right )+\log \left (\frac {16}{9}\right ) \int \frac {\log (x)}{x^3} \, dx+\int \frac {-4 x-2 \log \left (\frac {4}{3}\right )+x^2 \log \left (\frac {16}{9}\right )}{x} \, dx\\ &=-\frac {\log \left (\frac {16}{9}\right )}{4 x^2}-\frac {\log \left (\frac {16}{9}\right ) \log (x)}{2 x^2}+\frac {\log \left (\frac {4}{3}\right ) \log ^2(x)}{x^2}-\left (2 \log \left (\frac {4}{3}\right )\right ) \int \frac {\log (x)}{x^3} \, dx+\int \left (-4-\frac {2 \log \left (\frac {4}{3}\right )}{x}+x \log \left (\frac {16}{9}\right )\right ) \, dx\\ &=-4 x+\frac {\log \left (\frac {4}{3}\right )}{2 x^2}-\frac {\log \left (\frac {16}{9}\right )}{4 x^2}+\frac {1}{2} x^2 \log \left (\frac {16}{9}\right )-2 \log \left (\frac {4}{3}\right ) \log (x)+\frac {\log \left (\frac {4}{3}\right ) \log (x)}{x^2}-\frac {\log \left (\frac {16}{9}\right ) \log (x)}{2 x^2}+\frac {\log \left (\frac {4}{3}\right ) \log ^2(x)}{x^2}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [B] time = 0.02, size = 80, normalized size = 2.96 \begin {gather*} -4 x+\frac {\log \left (\frac {4}{3}\right )}{2 x^2}-\frac {\log \left (\frac {16}{9}\right )}{4 x^2}+\frac {1}{2} x^2 \log \left (\frac {16}{9}\right )-2 \log \left (\frac {4}{3}\right ) \log (x)+\frac {\log \left (\frac {4}{3}\right ) \log (x)}{x^2}-\frac {\log \left (\frac {16}{9}\right ) \log (x)}{2 x^2}+\frac {\log \left (\frac {4}{3}\right ) \log ^2(x)}{x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.73, size = 33, normalized size = 1.22 \begin {gather*} -\frac {x^{4} \log \left (\frac {3}{4}\right ) - 2 \, x^{2} \log \left (\frac {3}{4}\right ) \log \relax (x) + 4 \, x^{3} + \log \left (\frac {3}{4}\right ) \log \relax (x)^{2}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.24, size = 43, normalized size = 1.59 \begin {gather*} -x^{2} {\left (\log \relax (3) - 2 \, \log \relax (2)\right )} + 2 \, {\left (\log \relax (3) - 2 \, \log \relax (2)\right )} \log \relax (x) - 4 \, x - \frac {{\left (\log \relax (3) - 2 \, \log \relax (2)\right )} \log \relax (x)^{2}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 48, normalized size = 1.78
method | result | size |
risch | \(\frac {\left (-\ln \relax (3)+2 \ln \relax (2)\right ) \ln \relax (x )^{2}}{x^{2}}+2 x^{2} \ln \relax (2)-x^{2} \ln \relax (3)-4 \ln \relax (2) \ln \relax (x )+2 \ln \relax (3) \ln \relax (x )-4 x\) | \(48\) |
norman | \(\frac {\left (-\ln \relax (3)+2 \ln \relax (2)\right ) x^{4}+\left (-\ln \relax (3)+2 \ln \relax (2)\right ) \ln \relax (x )^{2}+\left (2 \ln \relax (3)-4 \ln \relax (2)\right ) x^{2} \ln \relax (x )-4 x^{3}}{x^{2}}\) | \(53\) |
default | \(-x^{2} \ln \relax (3)+2 x^{2} \ln \relax (2)+2 \ln \relax (3) \left (-\frac {\ln \relax (x )^{2}}{2 x^{2}}-\frac {\ln \relax (x )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )-4 \ln \relax (2) \left (-\frac {\ln \relax (x )^{2}}{2 x^{2}}-\frac {\ln \relax (x )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )+2 \ln \relax (3) \ln \relax (x )-4 \ln \relax (2) \ln \relax (x )-4 x -2 \ln \relax (3) \left (-\frac {\ln \relax (x )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )+4 \ln \relax (2) \left (-\frac {\ln \relax (x )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )\) | \(117\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.37, size = 51, normalized size = 1.89 \begin {gather*} -x^{2} \log \left (\frac {3}{4}\right ) + \frac {1}{2} \, {\left (\frac {2 \, \log \relax (x)}{x^{2}} + \frac {1}{x^{2}}\right )} \log \left (\frac {3}{4}\right ) + 2 \, \log \left (\frac {3}{4}\right ) \log \relax (x) - 4 \, x - \frac {{\left (2 \, \log \relax (x)^{2} + 2 \, \log \relax (x) + 1\right )} \log \left (\frac {3}{4}\right )}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.15, size = 25, normalized size = 0.93 \begin {gather*} x^2\,\ln \left (\frac {4}{3}\right )-4\,x+\ln \left (\frac {9}{16}\right )\,\ln \relax (x)+\frac {\ln \left (\frac {4}{3}\right )\,{\ln \relax (x)}^2}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.23, size = 44, normalized size = 1.63 \begin {gather*} x^{2} \left (- \log {\relax (3 )} + 2 \log {\relax (2 )}\right ) - 4 x - 2 \left (- \log {\relax (3 )} + 2 \log {\relax (2 )}\right ) \log {\relax (x )} + \frac {\left (- \log {\relax (3 )} + 2 \log {\relax (2 )}\right ) \log {\relax (x )}^{2}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________