Optimal. Leaf size=27 \[ x+x^2 \log ^2(x)+\frac {25 \log ^4\left (\log \left (\frac {x}{2}\right )\right )}{(-1+\log (4))^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 29, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 7, integrand size = 120, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.058, Rules used = {6, 12, 6688, 2304, 2305, 2302, 30} \begin {gather*} x^2 \log ^2(x)+x+\frac {25 \log ^4\left (\log \left (\frac {x}{2}\right )\right )}{(1-\log (4))^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 6
Rule 12
Rule 30
Rule 2302
Rule 2304
Rule 2305
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (x-2 x \log (4)+x \log ^2(4)\right ) \log \left (\frac {x}{2}\right )+\left (2 x^2-4 x^2 \log (4)+2 x^2 \log ^2(4)\right ) \log \left (\frac {x}{2}\right ) \log (x)+\left (2 x^2-4 x^2 \log (4)+2 x^2 \log ^2(4)\right ) \log \left (\frac {x}{2}\right ) \log ^2(x)+100 \log ^3\left (\log \left (\frac {x}{2}\right )\right )}{\left (x (1-2 \log (4))+x \log ^2(4)\right ) \log \left (\frac {x}{2}\right )} \, dx\\ &=\int \frac {\left (x-2 x \log (4)+x \log ^2(4)\right ) \log \left (\frac {x}{2}\right )+\left (2 x^2-4 x^2 \log (4)+2 x^2 \log ^2(4)\right ) \log \left (\frac {x}{2}\right ) \log (x)+\left (2 x^2-4 x^2 \log (4)+2 x^2 \log ^2(4)\right ) \log \left (\frac {x}{2}\right ) \log ^2(x)+100 \log ^3\left (\log \left (\frac {x}{2}\right )\right )}{x \left (1-2 \log (4)+\log ^2(4)\right ) \log \left (\frac {x}{2}\right )} \, dx\\ &=\frac {\int \frac {\left (x-2 x \log (4)+x \log ^2(4)\right ) \log \left (\frac {x}{2}\right )+\left (2 x^2-4 x^2 \log (4)+2 x^2 \log ^2(4)\right ) \log \left (\frac {x}{2}\right ) \log (x)+\left (2 x^2-4 x^2 \log (4)+2 x^2 \log ^2(4)\right ) \log \left (\frac {x}{2}\right ) \log ^2(x)+100 \log ^3\left (\log \left (\frac {x}{2}\right )\right )}{x \log \left (\frac {x}{2}\right )} \, dx}{1-2 \log (4)+\log ^2(4)}\\ &=\frac {\int \left ((-1+\log (4))^2 \left (1+2 x \log (x)+2 x \log ^2(x)\right )+\frac {100 \log ^3\left (\log \left (\frac {x}{2}\right )\right )}{x \log \left (\frac {x}{2}\right )}\right ) \, dx}{1-2 \log (4)+\log ^2(4)}\\ &=\frac {100 \int \frac {\log ^3\left (\log \left (\frac {x}{2}\right )\right )}{x \log \left (\frac {x}{2}\right )} \, dx}{(1-\log (4))^2}+\int \left (1+2 x \log (x)+2 x \log ^2(x)\right ) \, dx\\ &=x+2 \int x \log (x) \, dx+2 \int x \log ^2(x) \, dx+\frac {100 \operatorname {Subst}\left (\int \frac {\log ^3(x)}{x} \, dx,x,\log \left (\frac {x}{2}\right )\right )}{(1-\log (4))^2}\\ &=x-\frac {x^2}{2}+x^2 \log (x)+x^2 \log ^2(x)-2 \int x \log (x) \, dx+\frac {100 \operatorname {Subst}\left (\int x^3 \, dx,x,\log \left (\log \left (\frac {x}{2}\right )\right )\right )}{(1-\log (4))^2}\\ &=x+x^2 \log ^2(x)+\frac {25 \log ^4\left (\log \left (\frac {x}{2}\right )\right )}{(1-\log (4))^2}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 27, normalized size = 1.00 \begin {gather*} x+x^2 \log ^2(x)+\frac {25 \log ^4\left (\log \left (\frac {x}{2}\right )\right )}{(-1+\log (4))^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.58, size = 119, normalized size = 4.41 \begin {gather*} \frac {4 \, x^{2} \log \relax (2)^{4} - 4 \, x^{2} \log \relax (2)^{3} + 25 \, \log \left (\log \left (\frac {1}{2} \, x\right )\right )^{4} + {\left (x^{2} + 4 \, x\right )} \log \relax (2)^{2} + {\left (4 \, x^{2} \log \relax (2)^{2} - 4 \, x^{2} \log \relax (2) + x^{2}\right )} \log \left (\frac {1}{2} \, x\right )^{2} - 4 \, x \log \relax (2) + 2 \, {\left (4 \, x^{2} \log \relax (2)^{3} - 4 \, x^{2} \log \relax (2)^{2} + x^{2} \log \relax (2)\right )} \log \left (\frac {1}{2} \, x\right ) + x}{4 \, \log \relax (2)^{2} - 4 \, \log \relax (2) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 36, normalized size = 1.33 \begin {gather*} x^{2} \log \relax (x)^{2} + \frac {25 \, \log \left (-\log \relax (2) + \log \relax (x)\right )^{4}}{4 \, \log \relax (2)^{2} - 4 \, \log \relax (2) + 1} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.11, size = 37, normalized size = 1.37
method | result | size |
default | \(x +x^{2} \ln \relax (x )^{2}+\frac {25 \ln \left (\ln \relax (x )-\ln \relax (2)\right )^{4}}{4 \ln \relax (2)^{2}-4 \ln \relax (2)+1}\) | \(37\) |
risch | \(x +x^{2} \ln \relax (x )^{2}+\frac {25 \ln \left (\ln \relax (x )-\ln \relax (2)\right )^{4}}{4 \ln \relax (2)^{2}-4 \ln \relax (2)+1}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.59, size = 354, normalized size = 13.11 \begin {gather*} \frac {75 \, \log \left (-\log \relax (2) + \log \relax (x)\right )^{4}}{4 \, \log \relax (2)^{2} - 4 \, \log \relax (2) + 1} - \frac {150 \, \log \left (-\log \relax (2) + \log \relax (x)\right )^{2} \log \left (\log \left (\frac {1}{2} \, x\right )\right )^{2}}{4 \, \log \relax (2)^{2} - 4 \, \log \relax (2) + 1} + \frac {100 \, \log \left (-\log \relax (2) + \log \relax (x)\right ) \log \left (\log \left (\frac {1}{2} \, x\right )\right )^{3}}{4 \, \log \relax (2)^{2} - 4 \, \log \relax (2) + 1} + \frac {2 \, {\left (2 \, x^{2} \log \relax (x)^{2} - 2 \, x^{2} \log \relax (x) + x^{2}\right )} \log \relax (2)^{2}}{4 \, \log \relax (2)^{2} - 4 \, \log \relax (2) + 1} + \frac {2 \, {\left (2 \, x^{2} \log \relax (x) - x^{2}\right )} \log \relax (2)^{2}}{4 \, \log \relax (2)^{2} - 4 \, \log \relax (2) + 1} + \frac {4 \, x \log \relax (2)^{2}}{4 \, \log \relax (2)^{2} - 4 \, \log \relax (2) + 1} - \frac {2 \, {\left (2 \, x^{2} \log \relax (x)^{2} - 2 \, x^{2} \log \relax (x) + x^{2}\right )} \log \relax (2)}{4 \, \log \relax (2)^{2} - 4 \, \log \relax (2) + 1} - \frac {2 \, {\left (2 \, x^{2} \log \relax (x) - x^{2}\right )} \log \relax (2)}{4 \, \log \relax (2)^{2} - 4 \, \log \relax (2) + 1} - \frac {4 \, x \log \relax (2)}{4 \, \log \relax (2)^{2} - 4 \, \log \relax (2) + 1} + \frac {2 \, x^{2} \log \relax (x)^{2} - 2 \, x^{2} \log \relax (x) + x^{2}}{2 \, {\left (4 \, \log \relax (2)^{2} - 4 \, \log \relax (2) + 1\right )}} + \frac {2 \, x^{2} \log \relax (x) - x^{2}}{2 \, {\left (4 \, \log \relax (2)^{2} - 4 \, \log \relax (2) + 1\right )}} + \frac {x}{4 \, \log \relax (2)^{2} - 4 \, \log \relax (2) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 6.66, size = 58, normalized size = 2.15 \begin {gather*} \frac {\left (8\,{\ln \relax (2)}^2-\ln \left (256\right )+2\right )\,x^2\,{\ln \relax (x)}^2}{2\,\left (4\,{\ln \relax (2)}^2-\ln \left (16\right )+1\right )}+x+\frac {{\ln \left (\ln \left (\frac {x}{2}\right )\right )}^4}{4\,\left (\frac {{\ln \relax (2)}^2}{25}-\frac {\ln \left (16\right )}{100}+\frac {1}{100}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.51, size = 34, normalized size = 1.26 \begin {gather*} x^{2} \log {\relax (x )}^{2} + x + \frac {25 \log {\left (\log {\relax (x )} - \log {\relax (2 )} \right )}^{4}}{- 4 \log {\relax (2 )} + 1 + 4 \log {\relax (2 )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________