Optimal. Leaf size=24 \[ \frac {900 \left (3-\log \left (\frac {x^2}{\log (4+x)}\right )\right )^2}{(-5+x)^2} \]
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Rubi [F]
time = 9.36, 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 {-27000 x+5400 x^2+\left (216000-54000 x-27000 x^2\right ) \log (4+x)+\left (9000 x-1800 x^2+\left (-72000+39600 x+14400 x^2\right ) \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )+\left (-7200 x-1800 x^2\right ) \log (4+x) \log ^2\left (\frac {x^2}{\log (4+x)}\right )}{\left (-500 x+175 x^2+15 x^3-11 x^4+x^5\right ) \log (4+x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1800 \left (3-\log \left (\frac {x^2}{\log (4+x)}\right )\right ) \left (-((-5+x) x)-(4+x) \log (4+x) \left (10-5 x+x \log \left (\frac {x^2}{\log (4+x)}\right )\right )\right )}{(5-x)^3 x (4+x) \log (4+x)} \, dx\\ &=1800 \int \frac {\left (3-\log \left (\frac {x^2}{\log (4+x)}\right )\right ) \left (-((-5+x) x)-(4+x) \log (4+x) \left (10-5 x+x \log \left (\frac {x^2}{\log (4+x)}\right )\right )\right )}{(5-x)^3 x (4+x) \log (4+x)} \, dx\\ &=1800 \int \left (-\frac {15}{(-5+x)^3}+\frac {30}{(-5+x)^3 x}+\frac {3}{(-5+x)^2 (4+x) \log (4+x)}+\frac {\left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3 x (4+x) \log (4+x)}-\frac {\log ^2\left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3}\right ) \, dx\\ &=\frac {13500}{(5-x)^2}+1800 \int \frac {\left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3 x (4+x) \log (4+x)} \, dx-1800 \int \frac {\log ^2\left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3} \, dx+5400 \int \frac {1}{(-5+x)^2 (4+x) \log (4+x)} \, dx+54000 \int \frac {1}{(-5+x)^3 x} \, dx\\ &=\frac {13500}{(5-x)^2}-1800 \int \frac {\log ^2\left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3} \, dx+1800 \int \left (\frac {\left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{45 (-5+x)^3 \log (4+x)}-\frac {14 \left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{2025 (-5+x)^2 \log (4+x)}+\frac {151 \left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{91125 (-5+x) \log (4+x)}-\frac {\left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{500 x \log (4+x)}+\frac {\left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{2916 (4+x) \log (4+x)}\right ) \, dx+5400 \int \left (\frac {1}{9 (-5+x)^2 \log (4+x)}-\frac {1}{81 (-5+x) \log (4+x)}+\frac {1}{81 (4+x) \log (4+x)}\right ) \, dx+54000 \int \left (\frac {1}{5 (-5+x)^3}-\frac {1}{25 (-5+x)^2}+\frac {1}{125 (-5+x)}-\frac {1}{125 x}\right ) \, dx\\ &=\frac {8100}{(5-x)^2}-\frac {2160}{5-x}+432 \log (5-x)-432 \log (x)+\frac {50}{81} \int \frac {\left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{(4+x) \log (4+x)} \, dx+\frac {1208}{405} \int \frac {\left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x) \log (4+x)} \, dx-\frac {18}{5} \int \frac {\left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{x \log (4+x)} \, dx-\frac {112}{9} \int \frac {\left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^2 \log (4+x)} \, dx+40 \int \frac {\left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3 \log (4+x)} \, dx-\frac {200}{3} \int \frac {1}{(-5+x) \log (4+x)} \, dx+\frac {200}{3} \int \frac {1}{(4+x) \log (4+x)} \, dx+600 \int \frac {1}{(-5+x)^2 \log (4+x)} \, dx-1800 \int \frac {\log ^2\left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3} \, dx\\ &=\frac {8100}{(5-x)^2}-\frac {2160}{5-x}+432 \log (5-x)-432 \log (x)+\frac {50}{81} \int \frac {\left (-((-5+x) x)+\left (-40+22 x+8 x^2\right ) \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{(4+x) \log (4+x)} \, dx+\frac {1208}{405} \int \left (-\frac {40 \log \left (\frac {x^2}{\log (4+x)}\right )}{-5+x}+\frac {22 x \log \left (\frac {x^2}{\log (4+x)}\right )}{-5+x}+\frac {8 x^2 \log \left (\frac {x^2}{\log (4+x)}\right )}{-5+x}+\frac {5 x \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x) \log (4+x)}-\frac {x^2 \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x) \log (4+x)}\right ) \, dx-\frac {18}{5} \int \frac {\left (-((-5+x) x)+\left (-40+22 x+8 x^2\right ) \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{x \log (4+x)} \, dx-\frac {112}{9} \int \left (-\frac {40 \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^2}+\frac {22 x \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^2}+\frac {8 x^2 \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^2}+\frac {5 x \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^2 \log (4+x)}-\frac {x^2 \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^2 \log (4+x)}\right ) \, dx+40 \int \left (-\frac {40 \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3}+\frac {22 x \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3}+\frac {8 x^2 \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3}+\frac {5 x \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3 \log (4+x)}-\frac {x^2 \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3 \log (4+x)}\right ) \, dx-\frac {200}{3} \int \frac {1}{(-5+x) \log (4+x)} \, dx+\frac {200}{3} \text {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,4+x\right )+600 \int \frac {1}{(-5+x)^2 \log (4+x)} \, dx-1800 \int \frac {\log ^2\left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.11, size = 22, normalized size = 0.92 \begin {gather*} \frac {900 \left (-3+\log \left (\frac {x^2}{\log (4+x)}\right )\right )^2}{(-5+x)^2} \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 = 7.54, size = 1389, normalized size = 57.88
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1389\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 44, normalized size = 1.83 \begin {gather*} \frac {900 \, {\left (4 \, \log \left (x\right )^{2} - 2 \, {\left (2 \, \log \left (x\right ) - 3\right )} \log \left (\log \left (x + 4\right )\right ) + \log \left (\log \left (x + 4\right )\right )^{2} - 12 \, \log \left (x\right ) + 9\right )}}{x^{2} - 10 \, x + 25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 40, normalized size = 1.67 \begin {gather*} \frac {900 \, {\left (\log \left (\frac {x^{2}}{\log \left (x + 4\right )}\right )^{2} - 6 \, \log \left (\frac {x^{2}}{\log \left (x + 4\right )}\right ) + 9\right )}}{x^{2} - 10 \, x + 25} \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. 53 vs.
\(2 (19) = 38\).
time = 0.24, size = 53, normalized size = 2.21 \begin {gather*} \frac {16200}{2 x^{2} - 20 x + 50} + \frac {900 \log {\left (\frac {x^{2}}{\log {\left (x + 4 \right )}} \right )}^{2}}{x^{2} - 10 x + 25} - \frac {5400 \log {\left (\frac {x^{2}}{\log {\left (x + 4 \right )}} \right )}}{x^{2} - 10 x + 25} \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. 101 vs.
\(2 (22) = 44\).
time = 0.59, size = 101, normalized size = 4.21 \begin {gather*} -1800 \, {\left (\frac {\log \left (x^{2}\right )}{x^{2} - 10 \, x + 25} - \frac {3}{x^{2} - 10 \, x + 25}\right )} \log \left (\log \left (x + 4\right )\right ) + \frac {900 \, \log \left (x^{2}\right )^{2}}{x^{2} - 10 \, x + 25} + \frac {900 \, \log \left (\log \left (x + 4\right )\right )^{2}}{x^{2} - 10 \, x + 25} - \frac {5400 \, \log \left (x^{2}\right )}{x^{2} - 10 \, x + 25} + \frac {8100}{x^{2} - 10 \, x + 25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.42, size = 22, normalized size = 0.92 \begin {gather*} \frac {900\,{\left (\ln \left (\frac {x^2}{\ln \left (x+4\right )}\right )-3\right )}^2}{{\left (x-5\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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