Optimal. Leaf size=24 \[ \frac {\log (x)-\log ^2\left (\frac {1}{3} (20+2 x)\right )}{45 x^2} \]
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Rubi [A]
time = 0.56, antiderivative size = 29, normalized size of antiderivative = 1.21, number of steps
used = 18, number of rules used = 11, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.196, Rules used = {1607, 6874,
2458, 12, 2389, 2379, 2438, 2351, 31, 2445, 2340} \begin {gather*} \frac {\log (x)}{45 x^2}-\frac {\log ^2\left (\frac {2 x}{3}+\frac {20}{3}\right )}{45 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 1607
Rule 2340
Rule 2351
Rule 2379
Rule 2389
Rule 2438
Rule 2445
Rule 2458
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {10+x+(-20-2 x) \log (x)-2 x \log \left (\frac {1}{3} (20+2 x)\right )+(20+2 x) \log ^2\left (\frac {1}{3} (20+2 x)\right )}{x^3 (450+45 x)} \, dx\\ &=\int \left (\frac {2 \log \left (\frac {20}{3}+\frac {2 x}{3}\right )}{45 (-10-x) x^2}+\frac {2 \log ^2\left (\frac {20}{3}+\frac {2 x}{3}\right )}{45 x^3}+\frac {1-2 \log (x)}{45 x^3}\right ) \, dx\\ &=\frac {1}{45} \int \frac {1-2 \log (x)}{x^3} \, dx+\frac {2}{45} \int \frac {\log \left (\frac {20}{3}+\frac {2 x}{3}\right )}{(-10-x) x^2} \, dx+\frac {2}{45} \int \frac {\log ^2\left (\frac {20}{3}+\frac {2 x}{3}\right )}{x^3} \, dx\\ &=-\frac {\log ^2\left (\frac {20}{3}+\frac {2 x}{3}\right )}{45 x^2}+\frac {\log (x)}{45 x^2}+\frac {4}{135} \int \frac {\log \left (\frac {20}{3}+\frac {2 x}{3}\right )}{\left (\frac {20}{3}+\frac {2 x}{3}\right ) x^2} \, dx+\frac {1}{15} \text {Subst}\left (\int -\frac {2 \log (x)}{3 x \left (-10+\frac {3 x}{2}\right )^2} \, dx,x,\frac {20}{3}+\frac {2 x}{3}\right )\\ &=-\frac {\log ^2\left (\frac {20}{3}+\frac {2 x}{3}\right )}{45 x^2}+\frac {\log (x)}{45 x^2}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.06, size = 26, normalized size = 1.08 \begin {gather*} \frac {1}{45} \left (\frac {\log (x)}{x^2}-\frac {\log ^2\left (\frac {2 (10+x)}{3}\right )}{x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.73, size = 22, normalized size = 0.92
method | result | size |
risch | \(-\frac {\ln \left (\frac {2 x}{3}+\frac {20}{3}\right )^{2}}{45 x^{2}}+\frac {\ln \left (x \right )}{45 x^{2}}\) | \(22\) |
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. 81 vs.
\(2 (18) = 36\).
time = 0.55, size = 81, normalized size = 3.38 \begin {gather*} -\frac {100 \, \log \left (3\right )^{2} - 200 \, \log \left (3\right ) \log \left (2\right ) + 100 \, \log \left (2\right )^{2} - {\left (x^{2} + 200 \, \log \left (3\right ) - 200 \, \log \left (2\right )\right )} \log \left (x + 10\right ) + 100 \, \log \left (x + 10\right )^{2} + {\left (x^{2} - 100\right )} \log \left (x\right ) + 10 \, x - 50}{4500 \, x^{2}} + \frac {x - 5}{450 \, x^{2}} - \frac {1}{4500} \, \log \left (x + 10\right ) + \frac {1}{4500} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 18, normalized size = 0.75 \begin {gather*} -\frac {\log \left (\frac {2}{3} \, x + \frac {20}{3}\right )^{2} - \log \left (x\right )}{45 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 24, normalized size = 1.00 \begin {gather*} \frac {\log {\left (x \right )}}{45 x^{2}} - \frac {\log {\left (\frac {2 x}{3} + \frac {20}{3} \right )}^{2}}{45 x^{2}} \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. 50 vs.
\(2 (18) = 36\).
time = 0.38, size = 50, normalized size = 2.08 \begin {gather*} \frac {2 \, \log \left (3\right ) \log \left (2 \, x + 20\right )}{45 \, x^{2}} - \frac {\log \left (2 \, x + 20\right )^{2}}{45 \, x^{2}} - \frac {\log \left (3\right )^{2} - \log \left (3\right )}{45 \, x^{2}} + \frac {\log \left (\frac {1}{3} \, x\right )}{45 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.50, size = 18, normalized size = 0.75 \begin {gather*} \frac {\ln \left (x\right )-{\ln \left (\frac {2\,x}{3}+\frac {20}{3}\right )}^2}{45\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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