Optimal. Leaf size=31 \[ 5-\frac {2}{5 x \left (x-\log \left (\frac {x^2}{(x+x (1+x) \log (x))^2}\right )\right )} \]
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Rubi [F]
time = 2.73, 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 {4+8 x+\left (8 x+4 x^2\right ) \log (x)+(-2+(-2-2 x) \log (x)) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )}{5 x^4+\left (5 x^4+5 x^5\right ) \log (x)+\left (-10 x^3+\left (-10 x^3-10 x^4\right ) \log (x)\right ) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )+\left (5 x^2+\left (5 x^2+5 x^3\right ) \log (x)\right ) \log ^2\left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4+8 x-2 \log \left (\frac {1}{(1+\log (x)+x \log (x))^2}\right )+\log (x) \left (4 x (2+x)-2 (1+x) \log \left (\frac {1}{(1+\log (x)+x \log (x))^2}\right )\right )}{5 x^2 (1+(1+x) \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx\\ &=\frac {1}{5} \int \frac {4+8 x-2 \log \left (\frac {1}{(1+\log (x)+x \log (x))^2}\right )+\log (x) \left (4 x (2+x)-2 (1+x) \log \left (\frac {1}{(1+\log (x)+x \log (x))^2}\right )\right )}{x^2 (1+(1+x) \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx\\ &=\frac {1}{5} \int \left (\frac {4}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2}+\frac {8}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2}+\frac {4 \log (x)}{(1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2}+\frac {8 \log (x)}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2}-\frac {2 \log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2}-\frac {2 \log (x) \log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2}-\frac {2 \log (x) \log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2}\right ) \, dx\\ &=-\left (\frac {2}{5} \int \frac {\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx\right )-\frac {2}{5} \int \frac {\log (x) \log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx-\frac {2}{5} \int \frac {\log (x) \log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx+\frac {4}{5} \int \frac {1}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx+\frac {4}{5} \int \frac {\log (x)}{(1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx+\frac {8}{5} \int \frac {1}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx+\frac {8}{5} \int \frac {\log (x)}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx\\ &=-\left (\frac {2}{5} \int \left (\frac {1}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2}-\frac {1}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )}\right ) \, dx\right )-\frac {2}{5} \int \left (\frac {\log (x)}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2}-\frac {\log (x)}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )}\right ) \, dx-\frac {2}{5} \int \left (\frac {\log (x)}{(1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2}-\frac {\log (x)}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )}\right ) \, dx+\frac {4}{5} \int \frac {1}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx+\frac {4}{5} \int \frac {\log (x)}{(1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx+\frac {8}{5} \int \frac {1}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx+\frac {8}{5} \int \frac {\log (x)}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx\\ &=-\left (\frac {2}{5} \int \frac {1}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx\right )-\frac {2}{5} \int \frac {\log (x)}{(1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx-\frac {2}{5} \int \frac {\log (x)}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx+\frac {2}{5} \int \frac {1}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )} \, dx+\frac {2}{5} \int \frac {\log (x)}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )} \, dx+\frac {2}{5} \int \frac {\log (x)}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )} \, dx+\frac {4}{5} \int \frac {1}{x^2 (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx+\frac {4}{5} \int \frac {\log (x)}{(1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx+\frac {8}{5} \int \frac {1}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx+\frac {8}{5} \int \frac {\log (x)}{x (1+\log (x)+x \log (x)) \left (x-\log \left (\frac {1}{(1+(1+x) \log (x))^2}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.24, size = 24, normalized size = 0.77 \begin {gather*} \frac {2}{5 x \left (-x+\log \left (\frac {1}{(1+\log (x)+x \log (x))^2}\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 2.73, size = 105, normalized size = 3.39
method | result | size |
default | \(-\frac {4 i}{5 \left (\pi \mathrm {csgn}\left (i \left (1+\ln \left (x \right ) \left (x +1\right )\right )\right )^{2} \mathrm {csgn}\left (i \left (1+\ln \left (x \right ) \left (x +1\right )\right )^{2}\right )-2 \pi \,\mathrm {csgn}\left (i \left (1+\ln \left (x \right ) \left (x +1\right )\right )\right ) \mathrm {csgn}\left (i \left (1+\ln \left (x \right ) \left (x +1\right )\right )^{2}\right )^{2}+\pi \mathrm {csgn}\left (i \left (1+\ln \left (x \right ) \left (x +1\right )\right )^{2}\right )^{3}+2 i x +4 i \ln \left (1+\ln \left (x \right ) \left (x +1\right )\right )\right ) x}\) | \(105\) |
risch | \(-\frac {4 i}{5 x \left (\pi \mathrm {csgn}\left (i \left (x \ln \left (x \right )+\ln \left (x \right )+1\right )\right )^{2} \mathrm {csgn}\left (i \left (x \ln \left (x \right )+\ln \left (x \right )+1\right )^{2}\right )-2 \pi \,\mathrm {csgn}\left (i \left (x \ln \left (x \right )+\ln \left (x \right )+1\right )\right ) \mathrm {csgn}\left (i \left (x \ln \left (x \right )+\ln \left (x \right )+1\right )^{2}\right )^{2}+\pi \mathrm {csgn}\left (i \left (x \ln \left (x \right )+\ln \left (x \right )+1\right )^{2}\right )^{3}+2 i x +4 i \ln \left (x \ln \left (x \right )+\ln \left (x \right )+1\right )\right )}\) | \(105\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 20, normalized size = 0.65 \begin {gather*} -\frac {2}{5 \, {\left (x^{2} + 2 \, x \log \left ({\left (x + 1\right )} \log \left (x\right ) + 1\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 36, normalized size = 1.16 \begin {gather*} -\frac {2}{5 \, {\left (x^{2} - x \log \left (\frac {1}{{\left (x^{2} + 2 \, x + 1\right )} \log \left (x\right )^{2} + 2 \, {\left (x + 1\right )} \log \left (x\right ) + 1}\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.17, size = 36, normalized size = 1.16 \begin {gather*} \frac {2}{- 5 x^{2} + 5 x \log {\left (\frac {1}{\left (2 x + 2\right ) \log {\left (x \right )} + \left (x^{2} + 2 x + 1\right ) \log {\left (x \right )}^{2} + 1} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.74, size = 41, normalized size = 1.32 \begin {gather*} -\frac {2}{5 \, {\left (x^{2} + x \log \left (x^{2} \log \left (x\right )^{2} + 2 \, x \log \left (x\right )^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {8\,x-\ln \left (\frac {1}{\left (x^2+2\,x+1\right )\,{\ln \left (x\right )}^2+\left (2\,x+2\right )\,\ln \left (x\right )+1}\right )\,\left (\ln \left (x\right )\,\left (2\,x+2\right )+2\right )+\ln \left (x\right )\,\left (4\,x^2+8\,x\right )+4}{\ln \left (x\right )\,\left (5\,x^5+5\,x^4\right )-\ln \left (\frac {1}{\left (x^2+2\,x+1\right )\,{\ln \left (x\right )}^2+\left (2\,x+2\right )\,\ln \left (x\right )+1}\right )\,\left (\ln \left (x\right )\,\left (10\,x^4+10\,x^3\right )+10\,x^3\right )+{\ln \left (\frac {1}{\left (x^2+2\,x+1\right )\,{\ln \left (x\right )}^2+\left (2\,x+2\right )\,\ln \left (x\right )+1}\right )}^2\,\left (\ln \left (x\right )\,\left (5\,x^3+5\,x^2\right )+5\,x^2\right )+5\,x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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