Optimal. Leaf size=21 \[ x^4+\frac {\log \left (\log \left (5 \left (1-\frac {x}{\log (x)}\right )\right )\right )}{x} \]
________________________________________________________________________________________
Rubi [F] time = 2.49, 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 {x-x \log (x)+\left (-4 x^6 \log (x)+4 x^5 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )+\left (x \log (x)-\log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )\right )}{\left (-x^3 \log (x)+x^2 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x+x \log (x)-\left (-4 x^6 \log (x)+4 x^5 \log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )-\left (x \log (x)-\log ^2(x)\right ) \log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {-5 x+5 \log (x)}{\log (x)}\right )\right )}{x^2 (x-\log (x)) \log (x) \log \left (-\frac {5 (x-\log (x))}{\log (x)}\right )} \, dx\\ &=\int \left (\frac {-1+\log (x)+4 x^5 \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )-4 x^4 \log ^2(x) \log \left (5-\frac {5 x}{\log (x)}\right )}{x (x-\log (x)) \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )}-\frac {\log \left (\log \left (5-\frac {5 x}{\log (x)}\right )\right )}{x^2}\right ) \, dx\\ &=\int \frac {-1+\log (x)+4 x^5 \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )-4 x^4 \log ^2(x) \log \left (5-\frac {5 x}{\log (x)}\right )}{x (x-\log (x)) \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )} \, dx-\int \frac {\log \left (\log \left (5-\frac {5 x}{\log (x)}\right )\right )}{x^2} \, dx\\ &=\int \left (4 x^3+\frac {-1+\log (x)}{x (x-\log (x)) \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )}\right ) \, dx-\int \frac {\log \left (\log \left (5-\frac {5 x}{\log (x)}\right )\right )}{x^2} \, dx\\ &=x^4+\int \frac {-1+\log (x)}{x (x-\log (x)) \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )} \, dx-\int \frac {\log \left (\log \left (5-\frac {5 x}{\log (x)}\right )\right )}{x^2} \, dx\\ &=x^4+\int \left (\frac {1}{x (x-\log (x)) \log \left (5-\frac {5 x}{\log (x)}\right )}-\frac {1}{x (x-\log (x)) \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )}\right ) \, dx-\int \frac {\log \left (\log \left (5-\frac {5 x}{\log (x)}\right )\right )}{x^2} \, dx\\ &=x^4+\int \frac {1}{x (x-\log (x)) \log \left (5-\frac {5 x}{\log (x)}\right )} \, dx-\int \frac {1}{x (x-\log (x)) \log (x) \log \left (5-\frac {5 x}{\log (x)}\right )} \, dx-\int \frac {\log \left (\log \left (5-\frac {5 x}{\log (x)}\right )\right )}{x^2} \, dx\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 19, normalized size = 0.90 \begin {gather*} x^4+\frac {\log \left (\log \left (5-\frac {5 x}{\log (x)}\right )\right )}{x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 22, normalized size = 1.05 \begin {gather*} \frac {x^{5} + \log \left (\log \left (-\frac {5 \, {\left (x - \log \relax (x)\right )}}{\log \relax (x)}\right )\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.30, size = 24, normalized size = 1.14 \begin {gather*} x^{4} + \frac {\log \left (\log \left (-5 \, x + 5 \, \log \relax (x)\right ) - \log \left (\log \relax (x)\right )\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.14, size = 135, normalized size = 6.43
method | result | size |
risch | \(\frac {\ln \left (\ln \relax (5)+i \pi -\ln \left (\ln \relax (x )\right )+\ln \left (x -\ln \relax (x )\right )+\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-x \right )}{\ln \relax (x )}\right ) \left (\mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-x \right )}{\ln \relax (x )}\right )+\mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right )\right ) \left (\mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-x \right )}{\ln \relax (x )}\right )-\mathrm {csgn}\left (i \left (\ln \relax (x )-x \right )\right )\right )}{2}+i \pi \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-x \right )}{\ln \relax (x )}\right )^{2} \left (-\mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-x \right )}{\ln \relax (x )}\right )-1\right )\right )}{x}+x^{4}\) | \(135\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.53, size = 24, normalized size = 1.14 \begin {gather*} \frac {x^{5} + \log \left (\log \relax (5) + \log \left (-x + \log \relax (x)\right ) - \log \left (\log \relax (x)\right )\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.61, size = 22, normalized size = 1.05 \begin {gather*} \frac {\ln \left (\ln \left (-\frac {5\,\left (x-\ln \relax (x)\right )}{\ln \relax (x)}\right )\right )}{x}+x^4 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.55, size = 19, normalized size = 0.90 \begin {gather*} x^{4} + \frac {\log {\left (\log {\left (\frac {- 5 x + 5 \log {\relax (x )}}{\log {\relax (x )}} \right )} \right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________