Optimal. Leaf size=31 \[ x-\frac {2 \left (1+5 \left (2 x-\left (5+\log ^2\left (\frac {9}{\log (5)}\right )\right )^2\right )\right )}{\log (x)} \]
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Rubi [A]
time = 0.25, antiderivative size = 39, normalized size of antiderivative = 1.26, number of steps
used = 10, number of rules used = 7, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {6873, 6874,
2335, 2395, 2334, 2339, 30} \begin {gather*} x+\frac {2 \left (124+5 \log ^4\left (\frac {9}{\log (5)}\right )+50 \log ^2\left (\frac {9}{\log (5)}\right )\right )}{\log (x)}-\frac {20 x}{\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2334
Rule 2335
Rule 2339
Rule 2395
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {20 x-20 x \log (x)+x \log ^2(x)-248 \left (1+\frac {5}{124} \log ^2\left (\frac {9}{\log (5)}\right ) \left (10+\log ^2\left (\frac {9}{\log (5)}\right )\right )\right )}{x \log ^2(x)} \, dx\\ &=\int \left (1-\frac {20}{\log (x)}+\frac {2 \left (-124+10 x-50 \log ^2\left (\frac {9}{\log (5)}\right )-5 \log ^4\left (\frac {9}{\log (5)}\right )\right )}{x \log ^2(x)}\right ) \, dx\\ &=x+2 \int \frac {-124+10 x-50 \log ^2\left (\frac {9}{\log (5)}\right )-5 \log ^4\left (\frac {9}{\log (5)}\right )}{x \log ^2(x)} \, dx-20 \int \frac {1}{\log (x)} \, dx\\ &=x-20 \text {li}(x)+2 \int \left (\frac {10}{\log ^2(x)}+\frac {-124-50 \log ^2\left (\frac {9}{\log (5)}\right )-5 \log ^4\left (\frac {9}{\log (5)}\right )}{x \log ^2(x)}\right ) \, dx\\ &=x-20 \text {li}(x)+20 \int \frac {1}{\log ^2(x)} \, dx-\left (2 \left (124+50 \log ^2\left (\frac {9}{\log (5)}\right )+5 \log ^4\left (\frac {9}{\log (5)}\right )\right )\right ) \int \frac {1}{x \log ^2(x)} \, dx\\ &=x-\frac {20 x}{\log (x)}-20 \text {li}(x)+20 \int \frac {1}{\log (x)} \, dx-\left (2 \left (124+50 \log ^2\left (\frac {9}{\log (5)}\right )+5 \log ^4\left (\frac {9}{\log (5)}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )\\ &=x-\frac {20 x}{\log (x)}+\frac {2 \left (124+50 \log ^2\left (\frac {9}{\log (5)}\right )+5 \log ^4\left (\frac {9}{\log (5)}\right )\right )}{\log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 35, normalized size = 1.13 \begin {gather*} x-\frac {2 \left (-124+10 x-50 \log ^2\left (\frac {9}{\log (5)}\right )-5 \log ^4\left (\frac {9}{\log (5)}\right )\right )}{\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(109\) vs.
\(2(28)=56\).
time = 0.22, size = 110, normalized size = 3.55
method | result | size |
risch | \(x +\frac {160 \ln \left (3\right )^{4}-320 \ln \left (3\right )^{3} \ln \left (\ln \left (5\right )\right )+240 \ln \left (3\right )^{2} \ln \left (\ln \left (5\right )\right )^{2}-80 \ln \left (3\right ) \ln \left (\ln \left (5\right )\right )^{3}+10 \ln \left (\ln \left (5\right )\right )^{4}+400 \ln \left (3\right )^{2}-400 \ln \left (3\right ) \ln \left (\ln \left (5\right )\right )+100 \ln \left (\ln \left (5\right )\right )^{2}-20 x +248}{\ln \left (x \right )}\) | \(76\) |
norman | \(\frac {x \ln \left (x \right )-20 x +248+160 \ln \left (3\right )^{4}-320 \ln \left (3\right )^{3} \ln \left (\ln \left (5\right )\right )+240 \ln \left (3\right )^{2} \ln \left (\ln \left (5\right )\right )^{2}-80 \ln \left (3\right ) \ln \left (\ln \left (5\right )\right )^{3}+10 \ln \left (\ln \left (5\right )\right )^{4}+400 \ln \left (3\right )^{2}-400 \ln \left (3\right ) \ln \left (\ln \left (5\right )\right )+100 \ln \left (\ln \left (5\right )\right )^{2}}{\ln \left (x \right )}\) | \(77\) |
default | \(x +\frac {160 \ln \left (3\right )^{4}}{\ln \left (x \right )}-\frac {320 \ln \left (3\right )^{3} \ln \left (\ln \left (5\right )\right )}{\ln \left (x \right )}+\frac {240 \ln \left (3\right )^{2} \ln \left (\ln \left (5\right )\right )^{2}}{\ln \left (x \right )}-\frac {80 \ln \left (3\right ) \ln \left (\ln \left (5\right )\right )^{3}}{\ln \left (x \right )}+\frac {10 \ln \left (\ln \left (5\right )\right )^{4}}{\ln \left (x \right )}+\frac {400 \ln \left (3\right )^{2}}{\ln \left (x \right )}-\frac {400 \ln \left (3\right ) \ln \left (\ln \left (5\right )\right )}{\ln \left (x \right )}+\frac {100 \ln \left (\ln \left (5\right )\right )^{2}}{\ln \left (x \right )}-\frac {20 x}{\ln \left (x \right )}+\frac {248}{\ln \left (x \right )}\) | \(110\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.38, size = 51, normalized size = 1.65 \begin {gather*} \frac {10 \, \log \left (\frac {9}{\log \left (5\right )}\right )^{4}}{\log \left (x\right )} + x + \frac {100 \, \log \left (\frac {9}{\log \left (5\right )}\right )^{2}}{\log \left (x\right )} + \frac {248}{\log \left (x\right )} - 20 \, {\rm Ei}\left (\log \left (x\right )\right ) + 20 \, \Gamma \left (-1, -\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 36, normalized size = 1.16 \begin {gather*} \frac {10 \, \log \left (\frac {9}{\log \left (5\right )}\right )^{4} + x \log \left (x\right ) + 100 \, \log \left (\frac {9}{\log \left (5\right )}\right )^{2} - 20 \, x + 248}{\log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs.
\(2 (22) = 44\).
time = 0.04, size = 87, normalized size = 2.81 \begin {gather*} x + \frac {- 20 x - 400 \log {\left (3 \right )} \log {\left (\log {\left (5 \right )} \right )} - 320 \log {\left (3 \right )}^{3} \log {\left (\log {\left (5 \right )} \right )} - 80 \log {\left (3 \right )} \log {\left (\log {\left (5 \right )} \right )}^{3} + 10 \log {\left (\log {\left (5 \right )} \right )}^{4} + 100 \log {\left (\log {\left (5 \right )} \right )}^{2} + 240 \log {\left (3 \right )}^{2} \log {\left (\log {\left (5 \right )} \right )}^{2} + 160 \log {\left (3 \right )}^{4} + 248 + 400 \log {\left (3 \right )}^{2}}{\log {\left (x \right )}} \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. 75 vs.
\(2 (28) = 56\).
time = 0.41, size = 75, normalized size = 2.42 \begin {gather*} x + \frac {2 \, {\left (80 \, \log \left (3\right )^{4} - 160 \, \log \left (3\right )^{3} \log \left (\log \left (5\right )\right ) + 120 \, \log \left (3\right )^{2} \log \left (\log \left (5\right )\right )^{2} - 40 \, \log \left (3\right ) \log \left (\log \left (5\right )\right )^{3} + 5 \, \log \left (\log \left (5\right )\right )^{4} + 200 \, \log \left (3\right )^{2} - 200 \, \log \left (3\right ) \log \left (\log \left (5\right )\right ) + 50 \, \log \left (\log \left (5\right )\right )^{2} - 10 \, x + 124\right )}}{\log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.31, size = 34, normalized size = 1.10 \begin {gather*} x+\frac {100\,{\ln \left (\frac {9}{\ln \left (5\right )}\right )}^2-20\,x+10\,{\ln \left (\frac {9}{\ln \left (5\right )}\right )}^4+248}{\ln \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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