Optimal. Leaf size=22 \[ \left (x+\log (x) \left (x+\frac {\log ^2(x)}{4}\right )\right ) (x+\log (3 x)) \]
________________________________________________________________________________________
Rubi [B] time = 0.29, antiderivative size = 70, normalized size of antiderivative = 3.18, number of steps used = 25, number of rules used = 11, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {12, 14, 2313, 2296, 2295, 2346, 2302, 30, 6742, 2361, 2366} \begin {gather*} -\frac {x^2}{2}+\left (x^2+x\right ) \log (x)+\frac {1}{6} (3 x+1)^2-x+\frac {1}{4} x \log ^3(x)+\frac {1}{4} \log (3 x) \log ^3(x)-x \log (x)+x \log (3 x) \log (x)+x \log (3 x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 30
Rule 2295
Rule 2296
Rule 2302
Rule 2313
Rule 2346
Rule 2361
Rule 2366
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {4 x+12 x^2+\left (4 x+8 x^2\right ) \log (x)+3 x \log ^2(x)+(1+x) \log ^3(x)+\left (8 x+4 x \log (x)+3 \log ^2(x)\right ) \log (3 x)}{x} \, dx\\ &=\frac {1}{4} \int \left (\frac {4 x+12 x^2+4 x \log (x)+8 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)+x \log ^3(x)}{x}+\frac {\left (8 x+4 x \log (x)+3 \log ^2(x)\right ) \log (3 x)}{x}\right ) \, dx\\ &=\frac {1}{4} \int \frac {4 x+12 x^2+4 x \log (x)+8 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)+x \log ^3(x)}{x} \, dx+\frac {1}{4} \int \frac {\left (8 x+4 x \log (x)+3 \log ^2(x)\right ) \log (3 x)}{x} \, dx\\ &=\frac {1}{4} \int \left (4 (1+3 x)+4 (1+2 x) \log (x)+3 \log ^2(x)+\frac {(1+x) \log ^3(x)}{x}\right ) \, dx+\frac {1}{4} \int \left (8 \log (3 x)+4 \log (x) \log (3 x)+\frac {3 \log ^2(x) \log (3 x)}{x}\right ) \, dx\\ &=\frac {1}{6} (1+3 x)^2+\frac {1}{4} \int \frac {(1+x) \log ^3(x)}{x} \, dx+\frac {3}{4} \int \log ^2(x) \, dx+\frac {3}{4} \int \frac {\log ^2(x) \log (3 x)}{x} \, dx+2 \int \log (3 x) \, dx+\int (1+2 x) \log (x) \, dx+\int \log (x) \log (3 x) \, dx\\ &=-2 x+\frac {1}{6} (1+3 x)^2+\left (x+x^2\right ) \log (x)+\frac {3}{4} x \log ^2(x)+x \log (3 x)+x \log (x) \log (3 x)+\frac {1}{4} \log ^3(x) \log (3 x)+\frac {1}{4} \int \log ^3(x) \, dx+\frac {1}{4} \int \frac {\log ^3(x)}{x} \, dx-\frac {3}{4} \int \frac {\log ^3(x)}{3 x} \, dx-\frac {3}{2} \int \log (x) \, dx-\int (1+x) \, dx-\int (-1+\log (x)) \, dx\\ &=-\frac {x}{2}-\frac {x^2}{2}+\frac {1}{6} (1+3 x)^2-\frac {3}{2} x \log (x)+\left (x+x^2\right ) \log (x)+\frac {3}{4} x \log ^2(x)+\frac {1}{4} x \log ^3(x)+x \log (3 x)+x \log (x) \log (3 x)+\frac {1}{4} \log ^3(x) \log (3 x)-\frac {1}{4} \int \frac {\log ^3(x)}{x} \, dx+\frac {1}{4} \operatorname {Subst}\left (\int x^3 \, dx,x,\log (x)\right )-\frac {3}{4} \int \log ^2(x) \, dx-\int \log (x) \, dx\\ &=\frac {x}{2}-\frac {x^2}{2}+\frac {1}{6} (1+3 x)^2-\frac {5}{2} x \log (x)+\left (x+x^2\right ) \log (x)+\frac {1}{4} x \log ^3(x)+\frac {\log ^4(x)}{16}+x \log (3 x)+x \log (x) \log (3 x)+\frac {1}{4} \log ^3(x) \log (3 x)-\frac {1}{4} \operatorname {Subst}\left (\int x^3 \, dx,x,\log (x)\right )+\frac {3}{2} \int \log (x) \, dx\\ &=-x-\frac {x^2}{2}+\frac {1}{6} (1+3 x)^2-x \log (x)+\left (x+x^2\right ) \log (x)+\frac {1}{4} x \log ^3(x)+x \log (3 x)+x \log (x) \log (3 x)+\frac {1}{4} \log ^3(x) \log (3 x)\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [B] time = 0.02, size = 70, normalized size = 3.18 \begin {gather*} x^2-\frac {1}{4} x \log (81)+\frac {1}{4} x \log (6561)+x \log (x)+x^2 \log (x)+\frac {1}{4} x \log (81) \log (x)+x \log ^2(x)+\frac {1}{4} x \log ^3(x)+\frac {1}{12} \log (27) \log ^3(x)+\frac {\log ^4(x)}{4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.69, size = 42, normalized size = 1.91 \begin {gather*} \frac {1}{4} \, {\left (x + \log \relax (3)\right )} \log \relax (x)^{3} + \frac {1}{4} \, \log \relax (x)^{4} + x \log \relax (x)^{2} + x^{2} + x \log \relax (3) + {\left (x^{2} + x \log \relax (3) + x\right )} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.30, size = 43, normalized size = 1.95 \begin {gather*} \frac {1}{4} \, {\left (x + \log \relax (3)\right )} \log \relax (x)^{3} + \frac {1}{4} \, \log \relax (x)^{4} + x \log \relax (x)^{2} + x^{2} + x \log \relax (3) + {\left (x^{2} + x {\left (\log \relax (3) + 1\right )}\right )} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.08, size = 49, normalized size = 2.23
method | result | size |
risch | \(\frac {\ln \relax (x )^{4}}{4}+\frac {\left (\ln \relax (3)+x \right ) \ln \relax (x )^{3}}{4}+x \ln \relax (x )^{2}+\frac {\left (4 x \ln \relax (3)+4 x^{2}+4 x \right ) \ln \relax (x )}{4}+x \ln \relax (3)+x^{2}\) | \(49\) |
default | \(\frac {x \ln \relax (x )^{3}}{4}+x \ln \relax (x )^{2}+x \ln \relax (x )+\frac {\ln \relax (x )^{4}}{4}+\frac {\ln \relax (3) \ln \relax (x )^{3}}{4}+\ln \relax (3) \left (x \ln \relax (x )-x \right )+x^{2} \ln \relax (x )+x^{2}+2 x \ln \relax (3)\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.77, size = 108, normalized size = 4.91 \begin {gather*} \frac {1}{16} \, \log \left (3 \, x\right )^{4} - \frac {1}{4} \, \log \left (3 \, x\right )^{3} \log \relax (x) + \frac {3}{8} \, \log \left (3 \, x\right )^{2} \log \relax (x)^{2} + \frac {1}{16} \, \log \relax (x)^{4} + x^{2} \log \relax (x) + \frac {1}{4} \, {\left (\log \relax (x)^{3} - 3 \, \log \relax (x)^{2} + 6 \, \log \relax (x) - 6\right )} x + \frac {3}{4} \, {\left (\log \relax (x)^{2} - 2 \, \log \relax (x) + 2\right )} x + x^{2} - x {\left (\log \relax (3) - 2\right )} + 2 \, x \log \left (3 \, x\right ) + {\left (x \log \left (3 \, x\right ) - x\right )} \log \relax (x) - 2 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.09, size = 21, normalized size = 0.95 \begin {gather*} \frac {\left (x+\ln \relax (3)+\ln \relax (x)\right )\,\left ({\ln \relax (x)}^3+4\,x\,\ln \relax (x)+4\,x\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.30, size = 48, normalized size = 2.18 \begin {gather*} x^{2} + x \log {\relax (x )}^{2} + x \log {\relax (3 )} + \left (\frac {x}{4} + \frac {\log {\relax (3 )}}{4}\right ) \log {\relax (x )}^{3} + \left (x^{2} + x + x \log {\relax (3 )}\right ) \log {\relax (x )} + \frac {\log {\relax (x )}^{4}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________