Optimal. Leaf size=22 \[ \log \left (5-x+\frac {2 x^2}{\left (x+\log \left (\frac {\log (x)}{x}\right )\right )^2}\right ) \]
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Rubi [F]
time = 25.94, 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 x+\left (-4 x+x^3\right ) \log (x)+\left (-4 x+3 x^2\right ) \log (x) \log \left (\frac {\log (x)}{x}\right )+3 x \log (x) \log ^2\left (\frac {\log (x)}{x}\right )+\log (x) \log ^3\left (\frac {\log (x)}{x}\right )}{\left (-7 x^3+x^4\right ) \log (x)+\left (-17 x^2+3 x^3\right ) \log (x) \log \left (\frac {\log (x)}{x}\right )+\left (-15 x+3 x^2\right ) \log (x) \log ^2\left (\frac {\log (x)}{x}\right )+(-5+x) \log (x) \log ^3\left (\frac {\log (x)}{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 \left (\frac {1}{-5+x}-\frac {2 (1-\log (x)+x \log (x))}{x \log (x) \left (x+\log \left (\frac {\log (x)}{x}\right )\right )}+\frac {2 \left (25 x-10 x^2+x^3-25 x \log (x)+45 x^2 \log (x)-12 x^3 \log (x)+x^4 \log (x)+25 \log \left (\frac {\log (x)}{x}\right )-10 x \log \left (\frac {\log (x)}{x}\right )+x^2 \log \left (\frac {\log (x)}{x}\right )-25 \log (x) \log \left (\frac {\log (x)}{x}\right )+35 x \log (x) \log \left (\frac {\log (x)}{x}\right )-11 x^2 \log (x) \log \left (\frac {\log (x)}{x}\right )+x^3 \log (x) \log \left (\frac {\log (x)}{x}\right )\right )}{(-5+x) x \log (x) \left (-7 x^2+x^3-10 x \log \left (\frac {\log (x)}{x}\right )+2 x^2 \log \left (\frac {\log (x)}{x}\right )-5 \log ^2\left (\frac {\log (x)}{x}\right )+x \log ^2\left (\frac {\log (x)}{x}\right )\right )}\right ) \, dx\\ &=\log (5-x)-2 \int \frac {1-\log (x)+x \log (x)}{x \log (x) \left (x+\log \left (\frac {\log (x)}{x}\right )\right )} \, dx+2 \int \frac {25 x-10 x^2+x^3-25 x \log (x)+45 x^2 \log (x)-12 x^3 \log (x)+x^4 \log (x)+25 \log \left (\frac {\log (x)}{x}\right )-10 x \log \left (\frac {\log (x)}{x}\right )+x^2 \log \left (\frac {\log (x)}{x}\right )-25 \log (x) \log \left (\frac {\log (x)}{x}\right )+35 x \log (x) \log \left (\frac {\log (x)}{x}\right )-11 x^2 \log (x) \log \left (\frac {\log (x)}{x}\right )+x^3 \log (x) \log \left (\frac {\log (x)}{x}\right )}{(-5+x) x \log (x) \left (-7 x^2+x^3-10 x \log \left (\frac {\log (x)}{x}\right )+2 x^2 \log \left (\frac {\log (x)}{x}\right )-5 \log ^2\left (\frac {\log (x)}{x}\right )+x \log ^2\left (\frac {\log (x)}{x}\right )\right )} \, dx\\ &=\log (5-x)-2 \log \left (x+\log \left (\frac {\log (x)}{x}\right )\right )+2 \int \frac {-(-5+x)^2 \left (x+\log \left (\frac {\log (x)}{x}\right )\right )-\log (x) \left (x \left (-25+45 x-12 x^2+x^3\right )+(-5+x)^2 (-1+x) \log \left (\frac {\log (x)}{x}\right )\right )}{(5-x) x \log (x) \left ((-7+x) x^2+2 (-5+x) x \log \left (\frac {\log (x)}{x}\right )+(-5+x) \log ^2\left (\frac {\log (x)}{x}\right )\right )} \, dx\\ &=\log (5-x)-2 \log \left (x+\log \left (\frac {\log (x)}{x}\right )\right )+2 \int \left (\frac {-25 x+10 x^2-x^3+25 x \log (x)-45 x^2 \log (x)+12 x^3 \log (x)-x^4 \log (x)-25 \log \left (\frac {\log (x)}{x}\right )+10 x \log \left (\frac {\log (x)}{x}\right )-x^2 \log \left (\frac {\log (x)}{x}\right )+25 \log (x) \log \left (\frac {\log (x)}{x}\right )-35 x \log (x) \log \left (\frac {\log (x)}{x}\right )+11 x^2 \log (x) \log \left (\frac {\log (x)}{x}\right )-x^3 \log (x) \log \left (\frac {\log (x)}{x}\right )}{5 x \log (x) \left (-7 x^2+x^3-10 x \log \left (\frac {\log (x)}{x}\right )+2 x^2 \log \left (\frac {\log (x)}{x}\right )-5 \log ^2\left (\frac {\log (x)}{x}\right )+x \log ^2\left (\frac {\log (x)}{x}\right )\right )}+\frac {25 x-10 x^2+x^3-25 x \log (x)+45 x^2 \log (x)-12 x^3 \log (x)+x^4 \log (x)+25 \log \left (\frac {\log (x)}{x}\right )-10 x \log \left (\frac {\log (x)}{x}\right )+x^2 \log \left (\frac {\log (x)}{x}\right )-25 \log (x) \log \left (\frac {\log (x)}{x}\right )+35 x \log (x) \log \left (\frac {\log (x)}{x}\right )-11 x^2 \log (x) \log \left (\frac {\log (x)}{x}\right )+x^3 \log (x) \log \left (\frac {\log (x)}{x}\right )}{5 (-5+x) \log (x) \left (-7 x^2+x^3-10 x \log \left (\frac {\log (x)}{x}\right )+2 x^2 \log \left (\frac {\log (x)}{x}\right )-5 \log ^2\left (\frac {\log (x)}{x}\right )+x \log ^2\left (\frac {\log (x)}{x}\right )\right )}\right ) \, dx\\ &=\log (5-x)-2 \log \left (x+\log \left (\frac {\log (x)}{x}\right )\right )+\frac {2}{5} \int \frac {-25 x+10 x^2-x^3+25 x \log (x)-45 x^2 \log (x)+12 x^3 \log (x)-x^4 \log (x)-25 \log \left (\frac {\log (x)}{x}\right )+10 x \log \left (\frac {\log (x)}{x}\right )-x^2 \log \left (\frac {\log (x)}{x}\right )+25 \log (x) \log \left (\frac {\log (x)}{x}\right )-35 x \log (x) \log \left (\frac {\log (x)}{x}\right )+11 x^2 \log (x) \log \left (\frac {\log (x)}{x}\right )-x^3 \log (x) \log \left (\frac {\log (x)}{x}\right )}{x \log (x) \left (-7 x^2+x^3-10 x \log \left (\frac {\log (x)}{x}\right )+2 x^2 \log \left (\frac {\log (x)}{x}\right )-5 \log ^2\left (\frac {\log (x)}{x}\right )+x \log ^2\left (\frac {\log (x)}{x}\right )\right )} \, dx+\frac {2}{5} \int \frac {25 x-10 x^2+x^3-25 x \log (x)+45 x^2 \log (x)-12 x^3 \log (x)+x^4 \log (x)+25 \log \left (\frac {\log (x)}{x}\right )-10 x \log \left (\frac {\log (x)}{x}\right )+x^2 \log \left (\frac {\log (x)}{x}\right )-25 \log (x) \log \left (\frac {\log (x)}{x}\right )+35 x \log (x) \log \left (\frac {\log (x)}{x}\right )-11 x^2 \log (x) \log \left (\frac {\log (x)}{x}\right )+x^3 \log (x) \log \left (\frac {\log (x)}{x}\right )}{(-5+x) \log (x) \left (-7 x^2+x^3-10 x \log \left (\frac {\log (x)}{x}\right )+2 x^2 \log \left (\frac {\log (x)}{x}\right )-5 \log ^2\left (\frac {\log (x)}{x}\right )+x \log ^2\left (\frac {\log (x)}{x}\right )\right )} \, dx\\ &=\log (5-x)-2 \log \left (x+\log \left (\frac {\log (x)}{x}\right )\right )+\frac {2}{5} \int \frac {-(-5+x)^2 \left (x+\log \left (\frac {\log (x)}{x}\right )\right )-\log (x) \left (x \left (-25+45 x-12 x^2+x^3\right )+(-5+x)^2 (-1+x) \log \left (\frac {\log (x)}{x}\right )\right )}{(5-x) \log (x) \left ((-7+x) x^2+2 (-5+x) x \log \left (\frac {\log (x)}{x}\right )+(-5+x) \log ^2\left (\frac {\log (x)}{x}\right )\right )} \, dx+\frac {2}{5} \int \frac {-(-5+x)^2 \left (x+\log \left (\frac {\log (x)}{x}\right )\right )-\log (x) \left (x \left (-25+45 x-12 x^2+x^3\right )+(-5+x)^2 (-1+x) \log \left (\frac {\log (x)}{x}\right )\right )}{x \log (x) \left ((-7+x) x^2+2 (-5+x) x \log \left (\frac {\log (x)}{x}\right )+(-5+x) \log ^2\left (\frac {\log (x)}{x}\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(67\) vs. \(2(22)=44\).
time = 0.12, size = 67, normalized size = 3.05 \begin {gather*} -2 \log \left (x+\log \left (\frac {\log (x)}{x}\right )\right )+\log \left (-7 x^2+x^3-10 x \log \left (\frac {\log (x)}{x}\right )+2 x^2 \log \left (\frac {\log (x)}{x}\right )-5 \log ^2\left (\frac {\log (x)}{x}\right )+x \log ^2\left (\frac {\log (x)}{x}\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 = 17.00, size = 1099, normalized size = 49.95
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1099\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 80 vs.
\(2 (22) = 44\).
time = 0.34, size = 80, normalized size = 3.64 \begin {gather*} -2 \, \log \left (x - \log \left (x\right ) + \log \left (\log \left (x\right )\right )\right ) + \log \left (x - 5\right ) + \log \left (\frac {x^{3} + {\left (x - 5\right )} \log \left (x\right )^{2} + {\left (x - 5\right )} \log \left (\log \left (x\right )\right )^{2} - 7 \, x^{2} - 2 \, {\left (x^{2} - 5 \, x\right )} \log \left (x\right ) + 2 \, {\left (x^{2} - {\left (x - 5\right )} \log \left (x\right ) - 5 \, x\right )} \log \left (\log \left (x\right )\right )}{x - 5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 62 vs.
\(2 (22) = 44\).
time = 0.32, size = 62, normalized size = 2.82 \begin {gather*} -2 \, \log \left (x + \log \left (\frac {\log \left (x\right )}{x}\right )\right ) + \log \left (x - 5\right ) + \log \left (\frac {x^{3} + {\left (x - 5\right )} \log \left (\frac {\log \left (x\right )}{x}\right )^{2} - 7 \, x^{2} + 2 \, {\left (x^{2} - 5 \, x\right )} \log \left (\frac {\log \left (x\right )}{x}\right )}{x - 5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs.
\(2 (22) = 44\).
time = 0.98, size = 92, normalized size = 4.18 \begin {gather*} \log \left (x^{3} - 2 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2} + 2 \, x^{2} \log \left (\log \left (x\right )\right ) - 2 \, x \log \left (x\right ) \log \left (\log \left (x\right )\right ) + x \log \left (\log \left (x\right )\right )^{2} - 7 \, x^{2} + 10 \, x \log \left (x\right ) - 5 \, \log \left (x\right )^{2} - 10 \, x \log \left (\log \left (x\right )\right ) + 10 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) - 5 \, \log \left (\log \left (x\right )\right )^{2}\right ) - 2 \, \log \left (-x + \log \left (x\right ) - \log \left (\log \left (x\right )\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 = -0.05 \begin {gather*} \int -\frac {\ln \left (x\right )\,{\ln \left (\frac {\ln \left (x\right )}{x}\right )}^3+3\,x\,\ln \left (x\right )\,{\ln \left (\frac {\ln \left (x\right )}{x}\right )}^2-\ln \left (x\right )\,\left (4\,x-3\,x^2\right )\,\ln \left (\frac {\ln \left (x\right )}{x}\right )+4\,x-\ln \left (x\right )\,\left (4\,x-x^3\right )}{-\ln \left (x\right )\,\left (x-5\right )\,{\ln \left (\frac {\ln \left (x\right )}{x}\right )}^3+\ln \left (x\right )\,\left (15\,x-3\,x^2\right )\,{\ln \left (\frac {\ln \left (x\right )}{x}\right )}^2+\ln \left (x\right )\,\left (17\,x^2-3\,x^3\right )\,\ln \left (\frac {\ln \left (x\right )}{x}\right )+\ln \left (x\right )\,\left (7\,x^3-x^4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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